FINAL REPORT Of MAJOR RESEARCH PROJECT [F. No. 42-10/2013 (SR) dt. 12/3/2013] to

UNIVERSITY GRANTS COMMISSION, NEW DELHI by

!

Rudra Kanta Deka Department of Mathematics Gauhati University Guwahati-781014 Email: [email protected]

Annexure – VIII UNIVERSITY GRANTS COMMISSION BAHADUR SHAH ZAFAR MARG NEW DELHI – 110 002 PROFORMA FOR SUBMISSION OF INFORMATION AT THE TIME OF SENDING THE FINAL REPORT OF THE WORK DONE ON THE PROJECT 1. NAME AND ADDRESS OF THE PRINCIPAL INVESTIGATOR: Rudra Kanta Deka 2. NAME AND ADDRESS OF THE INSTITUTION: Mathematics Department, Gauhati University, Guwahati-781014 3. UGC APPROVAL NO. AND DATE: F. No. 42-10/2013 (SR) dt. 12/3/2013 4. DATE OF IMPLEMENTATION: 1/4/2013 5. TENURE OF THE PROJECT: Three years 6. TOTAL GRANT ALLOCATED: Rs.10,09,910/ 7. TOTAL GRANT RECEIVED: Rs. 9,43,399/ 8. FINAL EXPENDITURE : Rs. 9,43,196/ 9. TITLE OF THE PROJECT : Analysis of heat and mass transfer with variable viscosity including multihase flows and MHD Bio-fluid 10. OBJECTIVES OF THE PROJECT: The fluid viscosity is dependent on temperature of fluids. The objective of the project is to study the effect of variable viscosity on flows with heat and mass transfer. Multiphase flows and MHD bio-fluid are to be considered, taking variable viscosity into account. 11. WHETHER OBJECTIVES WERE ACHIEVED: (GIVE DETAILS) The objectives are achieved. Few papers on the objectives are completed. Few of these are published in national and international journals. Some are communicated. 12. ACHIEVEMENTS FROM THE PROJECT: Two students have been trained for numerical computation. They were trained to run MATLAB. They are registered for Ph.D. degree under my supervision. As consideration of variable viscosity leads to highly non-linear coupled equations. Therefore, accurate numerical method is necessary to compete with the international workers. Further, in finding solutions, separation of flow is inevitable. Therefore, in finding solutions, utmost care has been taken, so that accuracy is more. Different numerical as well as semi-numerical methods are implemented. For example, Runge-KuttaFehlberg methods with Newton’s method of iteration, Adaptive multi-step differential transform method, Perturbation methods are few of them.

PUBLICATIONS OUT OF THE PROJECT

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(i)

Effect of variable viscosity on flow past a porous wedge with suction or injection: new results. Afr. Mat. 2014, DOI 10.1007/s13370-014-0284-5

(ii)

Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature. IJMA, 5(9), 2014, 129-137

(iii)

Effects of variable viscosity on mixed convection heat and mass transfer past a wedge with variable temperature. IJMA, 5(5), 2014, 150-157

(iv)

Magnetohydrodynamic mixed convection flow past a wedge under variable temperature and chemical reaction. American Journal of Computational and Appled Mathematics 2013, 3(2), 74-80

(v)

Effect of variable viscosity on a nanofluid over a porous wedge. Intl. J. Fluid Mechanics Research, Begell House Publication.

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Author's personal copy Afr. Mat. DOI 10.1007/s13370-014-0284-5

Effect of variable viscosity on flow past a porous wedge with suction or injection: new results Rudra Kanta Deka · Mwblib Basumatary

Received: 4 December 2013 / Accepted: 18 August 2014 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014

Abstract An analysis is carried out to study the problem of the steady flow and heat transfer of an incompressible fluid over a static wedge in the presence of suction/injection with variable viscosity. The viscosity is assumed to vary as inverse linear function of temperature. The governing partial differential equations are first transformed into a system of non-linear ordinary differential equations using similarity transformations, and later solved numerically by using Runge–Kutta–Fehlberg method with shooting technique. The velocity and temperature profiles, the skin friction and the rate of heat transfer are computed and discussed for various values of suction/injection parameter, viscosity parameter and Hartree pressure gradient parameter for gases and liquids. Keywords Boundary layer · Variable viscosity · Porous wedge · Similarity solution · Flow separation Mathematics Subject Classification

34B15 · 34B40 · 76D10 · 80A20

List of symbols a Cf f fw k m Nu

Constant used in equation (5) Skin friction coefficient Dimensionless stream function Suction/injection parameter Thermal conductivity Falkner–Skan power law parameter Local Nusselt number

R. K. Deka (B) Department of Mathematics, Gauhati University, Guwahati 781014, Assam, India e-mail: [email protected] M. Basumatary Department of Mathematics, Lumding College, Lumding 782447, Assam, India e-mail: [email protected]

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Author's personal copy R. K. Deka, M. Basumatary

Pr Re T, Tw , T∞ Tr u, v U∞ Vw x, y

Prandtl number Reynolds number Temperature of fluid, wall and free stream Reference temperature Velocity components Free stream velocity Suction/injection velocity at the wall Cartesian coordinates

Greek symbols α β γ τ η θ θr µ µ∞ ν ρ∞ ψ

Thermal diffusivity Wedge angle parameter A constant Shear stress Similarity variable Dimensionless temperature Variable viscosity parameter Dynamic viscosity Viscosity of the free stream Kinematic viscosity Density Stream function

Subscripts w ∞

Condition at the wall Free stream condition

Superscripts ′

Differentiation with respect to η

1 Introduction Historically, the steady laminar flow over a wedge was first analyzed by Falkner and Skan [1] to illustrate the application of Prandtl’s boundary layer theory. Later on, Hartree [2] investigated the same problem with similarity transformation and gave numerical results for wall shear stress for different values of the wedge angle. Stewartson [3] made an attempt to establish further solutions of the Falkner–Skan equation. Cebeci and Keller [4] applied shooting and parallel shooting methods requiring asymptotic boundary condition to be imposed at a changing unknown boundary in the computation process. Thereafter, many solutions have been obtained for different aspects of this class of boundary layer problems ([5–8]). A large amount of literatures on this problem has been cited in books by Schlichting and Gersten [9] as well as in Leal [10] and in the recent papers by Ishak et al. [11], Yacob et al. [12], Bararnia et al. [13], Parand et al. [14], Postelnicu et al. [15], Afzal et al. [16] and Ashwini et al. [17]. Koh and Hartnett [5] have solved for skin friction and heat transfer for incompressible laminar flow over porous wedges with suction and variable wall temperature. Yih

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[18] presented an analysis of the forced convection boundary layer flow over a wedge with uniform suction/blowing, whereas Watanabe [8] investigated the behavior of the boundary layer over a wedge with suction or injection in forced flow. In all of the above mentioned studies, fluid viscosity was assumed to be constant. However, it is known that the physical properties of the fluid may change significantly, when temperature changes. To predict accurately the flow behavior and heat transfer rate, it is necessary to take this variation of viscosity with temperature into account. In view of this importance, many researchers investigated the effects of variable properties for fluid viscosity on flow and heat transfer over a surface; for example, Soundalgekar et al. [19] studied the flow of an incompressible viscous fluid past a continuously moving semi-infinite plate along with variable viscosity and variable temperature. Pantokratoras [20] extended the works of Soundalgekar et al. [19] and Hady et al. [21] assuming both viscosity and Prandlt number variable inside the boundary layer. Mukhopadhyay [22] investigated the effects of radiation and variable fluid viscosity on flow and heat transfer along a symmetric wedge, while Salem [23] studied the effects of temperature-dependent viscosity on non-Darcy hydrodynamic free convection heat transfer from a vertical wedge in porous media. Recently, Muhaimin et al. [24] have analyzed the effect of thermophoresis particle deposition and chemical reaction on unsteady MHD mixed convective flow over a porous wedge in the presence of temperaturedependent viscosity. Thereafter, many solutions have been obtained for different aspects of this class of boundary layer problems. The aim of this work is to study the effect of variable viscosity on flow past a porous wedge with suction or injection. However, the analyses are focused on the case of the occurrence of flow separation. It is well known that the occurrence of flow separation has several undesirable effects in so far as it leads to an increase in the drag on a body immersed in the flow. The effects of injection on the boundary layer flow are of interest in reducing the drag force (see [25]). A similar problem in a viscous fluid with prescribed wall temperature was considered by Riley and Weidman [26] and Ishak et al. [11]. The mathematical formulation of the problem is presented in Sect. 2, followed by numerical solution procedure in Sect. 3. Results and discussion are presented in Sect. 4, and finally conclusions are given in Sect. 5. 2 Mathematical formulations Let us consider a steady two dimensional laminar flow of a viscous incompressible fluid past a porous wedge at a constant temperature Tw . Let the velocity and temperature of uniform mainstream be U∞ and T∞ (with Tw > T∞ ) respectively. We consider a Cartesian coordinate system (x, y) as shown in Fig. 1, where x and y are the coordinates measured along the surface of the wedge and normal to it respectively. The porous wedge is considered permeable with a lateral mass flux Vw and the outer flow velocity, U (x) = U∞ x m . Under these assumptions, the boundary layer equations governing the flow and heat transfer can be written as, Continuity equation ∂u ∂v + =0 ∂x ∂y

(1)

Momentum equation ∂u dU 1 ∂ ∂u +v =U + u ∂x ∂y dx ρ∞ ∂ y

! " ∂u µ ∂y

(2)

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Author's personal copy R. K. Deka, M. Basumatary Fig. 1 Physical model and coordinate system

Energy equation u

∂T ∂T ∂2T +v =α 2 ∂x ∂y ∂y

(3)

along with the boundary conditions u = 0, v = Vw , T = Tw at y = 0

as y → ∞

u = U (x), T = T∞

#

(4)

where u and v are velocity components along x and y directions respectively, ρ∞ is the fluid density, α is the thermal diffusivity, µ is the coefficient of viscosity, considered to vary as a function of temperature given by (see [19]) the following: µ−1 = µ−1 ∞ [1 + γ (T − T∞ )] = a(T − Tr ) where a = γ /µ∞ and Tr = T∞ − γ

−1

(5) (6)

In the relation (6) above, both a and Tr are constants and their values depend on the reference state andγ , a thermal property of the fluid. In general, a > 0 for liquids, and a < 0 for gases. To solve Eqs. (1), (2) and (3) subjected to the boundary conditions (4), we introduce the following similarity variables: ⎫ ψ = [2ν∞ U x/(m + 1)]1/2 f (η) = [2ν∞ U∞ /(m + 1)]1/2 x (m+1)/2 f (η) ⎪ ⎬ 1/2 1/2 (m−1)/2 (7) η = y[U (m + 1)/(2ν∞ x)] = y[U∞ (m + 1)/(2ν∞ )] x ⎪ ⎭ θ (η) = (T − T∞ )/(Tw − T∞ ), T − Tr = (θ − θr )(Tw − T∞ )

where ψ(x, y) is the stream function, defined by u = ∂ψ/∂ y, v = −∂ψ/∂ x and θr = (Tr − T∞ )/(Tw − T∞ ) = −1/[γ (Tw − T∞ )], is the parameter characterizing the influence of viscosity. It is noticed that the continuity Eq. (1) is satisfied automatically, then (2) and (3) yield f ′′′ + (1 − θ/θr ){ f f ′′ + β(1 − f ′2 )} + θ ′ f ′′ /[θr (1 − θ/θr )] = 0 ′′

′

θ + Pr f θ = 0

(8) (9)

The corresponding boundary conditions (4) become, f = f w , f ′ = 0, θ = 1 at η = 0 f ′ = 1, θ = 0

123

at η → ∞

#

(10)

Author's personal copy Effect of variable viscosity

where primes denote the differentiation with respect to η. β = 2m/(m + 1), where β is the Hartree pressure gradient parameter that corresponds to β = Ω/π for a total angle Ω of the wedge. It is worth mentioning that β is a measure of pressure gradient, dp/dx. If β is positive, the pressure gradient is negative or favorable, and negative β denotes an unfavorable positive pressure gradient, while β = 0 denotes the flat plate (see [27]). The parameter f w is defined by, f w = −Vw [2x/{(m + 1)ν∞ U }]1/2

(11)

Notice that Vw > 0 (i.e. f w < 0) is for mass injection and Vw < 0 (i.e. f w > 0) is for mass suction, while Vw = 0 (i.e. f w = 0) is for impermeable surface and Pr = v/α is the Prandtl number. From Eq. (5), µ can be readily put into the form µ = µ∞ /[1 − θ (η)/θr ], which clearly shows that, as θr → ∞ (or as γ = 0), this leads to µ → µ∞ (constant), i.e., the viscosity variation in the boundary layer is negligible. Also, according to the definition of variable viscosity parameter, θr = (Tr − T∞ )/(Tw − T∞ ), for a given reference temperature Tr , variation of θr means the variation of the temperature difference -T = Tw − T∞ . The effect of variable viscosity is important, if θr is small. Since the viscosity of liquids decreases with increasing temperature, while it increases for gases, θr is negative for liquids and positive for gases. The concept of this parameter θr was first introduced by Ling and Dybbs [28] in their study of forced convection flow in porous media. The physical quantities of most interest for the present problem are the local skin-friction coefficient (dimensionless wall shear stress) and the local Nusselt number (rate of heat transfer) which are defined respectively by the following relations: C f = 2τw /(ρ∞ U 2 ) and N u = xqw /[k(Tw − T∞ )]

(12)

Now the wall shear stress on the surface τw and the rate of heat transfer qw are given by, τw = µ(∂u/∂ y) y=0

and

qw = −k(∂ T /∂ y) y=0

(13)

Using (5), (7) and (13), the quantities in (12) can be written as, (C f Re1/2 )/2 = [(m + 1)/2]1/2 /(1 − 1/θr ) f ′′ (0) N u Re−1/2 = −[(m + 1)/2]1/2 θ ′ (0)

(14) (15)

where Re = U x/ν∞ is the local Reynolds number. 3 Method of solution The system of nonlinear ordinary differential equations (8) and (9) under the boundary conditions (10) have been solved numerically by using shooting technique along with Runge– Kutta Fehlberg method. The computations were done by a program which uses a symbolic and computational computer language Matlab. A step size of -η = 0.01 was selected to be satisfactory for a convergence criterion of 10−6 in nearly all cases. The value of η∞ is found in each iteration loop by assigning η∞ = η∞ + -η. The maximum value of η∞ , to each group of parameters f w , m, β, θr and Pr, was chosen when the values of unknown boundary conditions at η = 0 do not change to successful loop with error less than 10−6 . The velocity profiles f ′ (η), temperature profiles θ (η), skin friction f ′′ (0) and the rate of heat transfer coefficient {−θ ′ (0)} are calculated for various values of parameters f w , m, β, θr and Pr. The numerical results thus obtained are represented in tables and figures.

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4 Results and discussions In order to analyze the results, numerical computations have been carried out for various values of the variable viscosity parameter (θr ), pressure gradient parameter (β), suction/injection parameter ( f w ) and the Prandtl number Pr. We consider the values of f w within the range −1 ≤ f w ≤ 1. It is worthy to mention that when θr → ∞, the present study reduces to constant viscosity. For this particular case, we have compared the values of the reduced skin friction coefficient f ′′ (0) and rate of heat transfer coefficient {−θ ′ (0)} with the established results of the research carried out by Watanabe [8] and Kumari et al. [29] for different values of the Falkner–Skan exponent m with Pr = 0.73 and the suction/injection parameter f w = 0 (impermeable wedge) as shown in Table 1. The table shows excellent agreement. Based on our computation, the boundary layer flow is about to separate when θr → ∞ with f w = 0 at β = −0.198837 which is excellent agreement as compared to Hartree [2], and Cebeci and Keller [4]. Figures 2 and 3 show the velocity and temperature profiles for first and second solutions for various values of the pressure gradient parameter (β) when f w = 0.5 with θr = 2 for air (Pr = 0.71) and θr = −2 for water (Pr = 7) respectively. From the first solution, it is clear that an increase in β leads to an increase in the normal flow velocity profiles near the surface of the wedge, while the temperature profiles decrease. It is due to the fact that as β increases, the flow acceleration increases, thereby thinning the boundary layer. Also, the shear stress at the surface increases for the fact that the temperature of the fluid along the surface of the wedge reduces. For accelerated flows (β > 0), we obtain velocity profiles without a point of inflection. On the other hand, for decelerated flows (β < 0), we obtain velocity profile with a point of inflection. In case of air, the lowest value of β = −0.382938 corresponds to the separation of velocity profile with skin-friction coefficient f ′′ (0) = 0.000001 and in case of water, the lowest value of β = −0.427781 corresponds to the separation velocity profile with skin-friction coefficient f ′′ (0) = 0.000011 for which solutions exist within the boundary layer. The second solution profiles prove the existence of dual solutions. Also the results show that as the adverse pressure gradient parameter (−β) is decreased, the magnitude of the reverse flow velocity decreases and the boundary layer thickness increases. The term normal flow indicates that the flow velocity has a unique direction, and the reverse flow means that the velocity is both positive and negative in the interval of integration.

Table 1 Comparison of f ′′ (0) and −θ ′ (0) in our study with those of Watanbe [8] and Kumari et al. [29] for various values of m with Pr = 0.73 when θr → ∞ m

f ′′ (0)

−θ ′ (0)

Present work

Watanabe [8]

Kumari et al. [29]

Present work

Watanabe [8]

Kumari et al. [29]

0

0.469601

0.46960

0.46975

0.420160

0.42015

0.42079

0.0141

0.504615

0.50461

0.50472

0.425785

0.42578

0.42635

0.0435

0.568978

0.56898

0.56904

0.435492

0.43548

0.43597

0.0909

0.654979

0.65498

0.65501

0.447312

0.44730

0.44770

0.1429

0.731999

0.73200

0.73202

0.456951

0.45693

0.45728

0.2

0.802126

0.80213

0.80214

0.465051

0.46503

0.46534

0.3333

0.927654

0.92765

0.92766

0.478158

0.47814

0.47840

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Author's personal copy Effect of variable viscosity 1.2

β = 0.6, 0.2, 0.0, −0.25, −0.3, −0.382938

1

f ’(η), θ(η) →

0.8 0.6

β = −0.3, −0.25 (second solution)

0.4

Pr = 0.71, f = 0.5, w θ =2

0.2

r

0 −0.2

0

1

2

3

4

5

η→

6

7

8

Fig. 2 Velocity and temperature profiles of air (Pr = 0.71) that show the existence of two solutions for β = −0.25 and −0.3, when f w = 0.5 and θr = 2 1.2 1

β = 0.6, 0.2, 0.0, −0.25, −0.3, −0.427781

f ’(η), θ(η) →

0.8 0.6

β = −0.3, −0.25 (second solution)

0.4 0.2

Pr = 7, f = 0.5 w θ = −2 r

0 −0.2

0

1

2

3

η→

4

5

6

7

Fig. 3 Velocity and temperature profiles of water (Pr = 7) that show the existence of two solutions for β = −0.25 and −0.3, when f w = 0.5 and θr = −2

The viscosity distributions are plotted in Figs. 4 and 5 for different values of θr when f w = 0.5, β = 0.2 for air and water respectively. From these figures it can be seen that when |θr | is large (|θr | → ∞), the viscosity becomes constant inside the boundary layer and it approaches unity at the outer edge of the boundary layer. It is also seen that a decrease in θr leads to an increase in the viscosity distribution of air. The viscosity of air increases by 5.26, 11.11 and 100 %, when variable viscosity parameter θr decreases from ∞ (i.e. ambient viscosity of gases) to 20, 10 and 2 respectively at the surface of the wedge (η = 0). On the other hand an increase in θr leads to a decrease in the viscosity distribution of water. The viscosity of water decreases by 4.76, 9.09, 33.33, 55.56 and 90.91 %, when variable viscosity parameter θr increases from −∞ (i.e. ambient viscosity of liquids) to −20, −10, −2, −0.8

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µ / µinfinity →

1.8

θr = 2.0 = 10 = 20 → infinity

1.6

1.4

Pr = 0.71, fw = 0.5

1.2

β = 0.2 1

0.8

0

1

2

η→

3

4

5

Fig. 4 Viscosity profiles of air (Pr = 0.71) for various values of θr when f w = 0.5, β = 0.2 1.4

Pr = 7, fw = 0.5,

1.2

β = 0.2

µ/ µ

infinity

→

1 0.8 0.6

θ = −0.1 r

= −0.8 = −2 = −10 = −20 → infinity

0.4 0.2 0

0

0.5

1

1.5

η→

2

2.5

3

Fig. 5 Viscosity profiles of water (Pr = 7) for various values of θr when f w = 0.5, β = 0.2

and −0.1 respectively at the surface of the wedge (η = 0). So, in the presence of variable viscosity parameter θr , the viscosity of air inside the boundary layer is much larger than the ambient one, whereas the viscosity of water inside the boundary layer is less than the ambient one. The skin friction coefficient f ′′ (0) of gases as a function of β for various values of f w and Pr with θr = 2 is shown in Fig. 6 and for various values ofθr with f w = 0.5, −0.5 in Fig. 7. These figures show that there is only one solution when β ≥ 0, two solutions when βc ≤ β < 0 and no solution when β < βc . Here βc is the critical value of β such that the skin friction coefficient f ′′ (0) almost vanishes. The values of βc are different for various values of parameters f w , Pr and θr . In the range βc ≤ β < 0, it is seen two values of f ′′ (0) one is f ′′ (0) ≥ 0 and the other is f ′′ (0) < 0. Physically, f ′′ (0) > 0 means that the fluid

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Author's personal copy Effect of variable viscosity 1

θ =2 r f = 1, 0.5, 0.0, −0.5, −1

0.8

w

f "(0) →

0.6 0.4 0.2 0

Pr = 0.2 Pr = 0.71 β

−0.2

c

−0.4 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

β→ Fig. 6 Skin friction coefficient as a function of β for various values of f w , Pr with θr = 2 1.4 1.2 1

fw = 0.5 = −0.5

Pr = 0.2 θr → infinity = 20 = 10 =2

β

c

f "(0) →

0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

β→ Fig. 7 Skin friction coefficient as a function of β for various values of θr with Pr = 0.2, f w = 0.5 and −0.5

exerts a drag force on the wedge while f ′′ (0) < 0 means the opposite and f ′′ (0) ≈ 0 at βc means there is no wall shear stress. It is seen from Fig. 6 that, for fixed suction, the skin friction coefficient f ′′ (0) decreases as Pr increases till certain β = β ∗ . The values of β ∗ are shown in Table 2. For β < β ∗ , it increases positively (i.e. f ′′ (0) > 0) with increasing Pr. On the other hand for the values of f w = 0.0 and f w = −0.5, increasing Pr is to decrease the skin friction coefficient f ′′ (0). The skin friction coefficient f ′′ (0) almost vanishes and the flow will separate completely for the Prandtl numbers Pr = 0.2 and 0.71 respectively at βc = −0.612486 and −0.655800 when f w = 1; again at βc = −0.368841 and −0.382938 when f w = 0.5, then at βc = −0.183926 and −0.181893 when f w = 0.0, and the same result will come out at βc = −0.059125 and −0.055829 when f w = −0.5. These show that the critical values of adverse pressure gradient parameter, {−βc } increase as the value

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Author's personal copy R. K. Deka, M. Basumatary Table 2 Values of cross over point β ∗

θr

fw

Pr

β∗

2

1

0.2 and 0.71

2

0.5

0.2 and 0.71

−0.292540

2 and 10

−0.5

0.2

20 and ∞

−0.5

0.2

−2

10 and 20 −0.8

−0.5

0.2

0.5

7 and 100

0.5

7 and 100

−0.264600 −0.040916 −0.017790 −0.012220 −0.150910 −0.133431

of f w increases, and in the presence of suction, increasing Prandtl number is to increase those critical values, but in the presence of injection, the critical values are slightly decreased as Prandtl number increases. Thus, suction delays the separation. It also reveals that the separation is delayed with the increasing Prandtl number but it is just opposite for the values of f w = 0.0 and f w = −0.5. It is also observed that the effect of Pr on the skin friction coefficient f ′′ (0) is more efficient for increasing suction than the injection. On the other hand, when f w = −1 (large injection), the effect of Pr on the skin friction coefficient f ′′ (0) is not significant. It is also mentionable here that the separation is found to occur for very small non-negative values of β, i.e. for β = 0.0 (flat plate) and β = 0.001 with the Prandtl numbers Pr = 0.2 and Pr = 0.71 respectively when f w = −1. From Fig. 7 it can be seen that when f w = −0.5, the skin friction coefficient f ′′ (0) tends to zero and the flow will separate for the values of variable viscosity parameter θr = 2, 10, 20 and ∞ at βc = −0.059125, −0.052358, −0.051280 and −0.050177 respectively, and it increases as θr increases till certain β = β ∗ . For β < β ∗ (see Table 2 for β ∗ ), it slightly decreases with θr . It is also seen that when f w = 0.5, increasing θr leads to an increase in the skin friction coefficient f ′′ (0) and it approaches to zero for the values of variable viscosity parameter θr = 2, 10, 20 and ∞ at βc = −0.368841, −0.412228, −0.417166 and −0.422019 respectively. It reveals that the separation is delayed with the larger value of θr for suction ( f w = 0.5), but it is just opposite for injection ( f w = −0.5). From both these figures, we also observe that the skin-friction coefficient ( f ′′ (0) > 0) is greatly increased as β increases. Figures 8 and 9 respectively show the variation of the skin friction coefficient f ′′ (0) and rate of heat transfer coefficient {−θ ′ (0)} for gases (Pr = 0.2 and 0.71) as a function of f w for various values of the pressure gradient parameter β in the presence of variable viscosity parameter θr = 2. The results show that the skin friction coefficient and rate of heat transfer coefficient are enhanced by suction ( f w > 0) for fixed β and Pr but effect of injection ( f w < 0) is just opposite. This result admits of a physical interpretation. Since suction ( f w > 0) reduces both the momentum and thermal boundary layer thickness and the effect of injection ( f w < 0) is just reverse. Hence f ′′ (0) and {−θ ′ (0)} increases with suction ( f w > 0). It is also observed that for β = 0.0, there exists no solution when the effect of injection is strong ( f w ≤ −1). This result is consistent with the above mentioned result that for flat plate, separation occurs when f w = −1. For θr = 2, increasing β is to increase the critical value of {− f w } for which solutions exist. Thus larger values of β delays the flow separation. Further it is seen that for the fixed values ofβ, the rate of heat transfer coefficient {−θ ′ (0)} increases as Pr increases till certain injection f w = f w∗ . The values of f w∗ are shown in Table 3. For f w < f w∗ , the rate of heat transfer decreases as Pr increases. The skin friction coefficient f ′′ (0)of liquids (Pr = 7 and 100) as a function of β and f w are shown in Figs. 10 and 11 respectively. From Fig. 10 it can be seen that when f w = 0.5,

123

Author's personal copy Effect of variable viscosity 1 0.9 0.8

Pr = 0.2, θ = 2 r

Pr = 0.71, θ = 2 r

β = 0.6, 0.2, 0.0

f "(0) →

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −1

−0.5

0

f →

0.5

1

w

Fig. 8 Skin friction coefficient as a function of f w for various values of β when θr = 2 for Pr = 0.2 and 0.71 1 0.9 0.8

Pr = 0.2,θr = 2 Pr = 0.71, θ = 2 r

−θ ’(0) →

0.7 0.6 0.5

β = 0.6, 0.2, 0.0

0.4 0.3 0.2 0.1 0 −1

−0.5

0

f →

0.5

1

w

Fig. 9 Rate of heat transfer as a function of f w for various values of β when θr = 2 for Pr = 0.2 and 0.71 Table 3 Values of cross over point f w∗

θr

β

Pr

f w∗

2

0.0

0.2 and 0.71

2

0.2

0.2 and 0.71

−0.448056

2

0.6

0.2 and 0.71

−0.568660 −0.681063

the skin friction coefficient f ′′ (0) increases with the increasing values of β and θr . It is also seen that for the small values of |θr | with f w = 0.5, increasing Prandtl number is to increase the skin friction coefficient f ′′ (0) till certain β = β ∗ . For β < β ∗ , it decreases with increasing Prandtl number. But for the large values of |θr |, the effect of Prandtl num-

123

Author's personal copy R. K. Deka, M. Basumatary 3 2.5

" f (0) →

2

θ = −0.8, −2, −10, −20, −infinity r

1.5 1 0.5 0

−0.5 −1 −0.8

−0.6

−0.4

−0.2

0

Pr = 7

f = 0.5

= 100 βc

= 0.5

0.2

w

0.4

0.6

β→ Fig. 10 Skin friction coefficient for Pr = 7 and 100 as a function of β for various values of θr with f w = 0.5 3 Pr = 7 2.5

= 100

r

= −2

β = 0.6, 0.2, 0.0

1.5

"

f (0) →

2

θ = −2

1 0.5 0 −1

−0.5

0

f →

0.5

1

w

Fig. 11 Skin friction coefficient for Pr = 7 and 100 as a function of f w for various values of β with θr = −2

ber on the skin friction coefficient f ′′ (0) is not significant. Based on our computation, when Pr = 7, the values of βc are −0.436133, −0.427781, −0.423185, −0.422603 and −0.422019 for θr = −0.8, −2, −10, −20 and θr → −∞ respectively, and when Pr = 100 these values are −0.422141, −0.422069, −0.422029, −0.422024 and −0.422019 for θr = −0.8, −2, −10, −20 and θr → −∞ respectively. Thus, the values of {−βc } increases as the value of Pr decreases with the small values of|θr |. Hence, the flow separation is seen earlier in oil (Pr = 100) than water (Pr = 7). From Fig. 11, it is observed that the skin friction coefficient f ′′ (0) increases with increasing values of f w . The rate of heat transfer coefficients, {−θ ′ (0)} for Pr = 7 and 100 as a function of f w with different values of β are shown in Figs. 12 and 13 respectively. From both these figures, we observe that the rate of heat transfer coefficient is greatly increased as f w increases. It is

123

Author's personal copy Effect of variable viscosity 8 β = 0.0 θ = −2 Pr = 7 r

7

= 0.2 = 0.6

=7 =7

5 4

’

−θ (0) →

6

= −2 = −2

3 2 1 0 −1

−0.5

0

0.5

f →

1

w

Fig. 12 Rate of heat transfer coefficient as a function of f w for various values of β with θr = −2 and Pr = 7 120 β = 0.0 θr = −2 Pr = 100 = 0.2 = 0.6

100

= −2 = −2

= 100 = 100

’

−θ (0) →

80 60 40 20 0 −0.2

0

0.2

0.4

f →

0.6

0.8

1

1.2

w

Fig. 13 Rate of heat transfer coefficient as a function of f w for various values of β with θr = −2 and Pr = 100

also observed that, for Pr = 7, {−θ ′ (0)} is slightly increased in the range −0.5 < f w < 0.5 with the increasing values of β and almost vanishes, when f w = −0.5. But the effect of β on the rate of heat transfer coefficient is insignificant for Pr = 100 and almost vanishes, when f w = −0.1. Figures 14 and 15 illustrate the influence of ambient Prandtl number and variable viscosity parameter θr on the velocity and temperature profiles in the boundary layer for gases and liquids respectively. From both these figures (for gases and liquids), it is observed that the velocity profiles increase, whereas the temperature profiles decrease with the increasing values of variable viscosity parameter θr . This can be explained physically. As the value of |θr | increases, the temperature difference of the porous wedge and the ambient fluid within

123

Author's personal copy R. K. Deka, M. Basumatary 1.2 1 Pr = 0.2, fw = 0.5, β = 0.2

f ’(η), θ(η) →

0.8

= 0.71,

= 0.5,

= 0.2

0.6 θr = 1.01, 2, 10, 20 0.4 0.2 0 −0.2

0

2

4

η→

6

8

10

Fig. 14 Velocity and temperature profiles for Pr = 0.2 and 0.71 for various values of θr with f w = 0.5 and β = 0.2 1.2 1 Pr = 7, f = 0.5, β = 0.2 w

= 100, = 0.5, = 0.2

0.6 0.4 θ = −20, −10, −2, −0.8, −0.1

’

f (η), θ(η) →

0.8

a

0.2 0 −0.2

0

0.5

1

1.5

η→

2

2.5

3

Fig. 15 Velocity and temperature profiles for Pr = 7 and 100 for various values of θr with f w = 0.5 and β = 0.2

the boundary layer decreases. Thus the viscosity of gases decreases, whereas liquids viscosity increases, which results in the reduction of the boundary layer thickness. But the variable viscosity parameter does not have significant effect on the temperature profiles of the oil. It should be noticed in both the cases that for large values of |θr |, the velocity and temperature profiles are close to those of constant viscosity (|θr | → ∞). Large values of |θr | implies the variable viscosity effect can be neglected and so it can be concluded that, for oil, the variable viscosity is not effective. It is noted that, for gases (Pr < 1), increasing Prandtl number slightly decrease the fluid velocity up to the certain point of η say η∗ after this point the velocity is found to increase with the increasing Prandtl number. But it is just opposite in the case of liquids (Pr >> 1).

123

Author's personal copy Effect of variable viscosity 1.2 fw = 1, 0.5, 0.0, −0.5, −1

1

f ’(η), θ(η) →

0.8 θ = 2, Pr = 0.2, β = 0.2 r

0.6

θr → infinity, Pr = 0.2, β = 0.2

0.4 0.2 0 −0.2

0

2

4

6

η→

8

10

Fig. 16 Velocity and temperature profiles of air (Pr = 0.2) for various values of f w with β = 0.2 when θr = 2 and θr → ∞ 1.2 fw = 1, 0.5, 0.0, −0.05, −0.1

1

’ f (η), θ(η) →

0.8 0.6

θ = −2, Pr = 7, β = 0.2 r

θr → −infinity, Pr = 7, β = 0.2

0.4 0.2 0 −0.2

0

0.5

1

1.5

η→

2

2.5

3

3.5

Fig. 17 Velocity and temperature profiles of water (Pr = 7) for various values of f w with β = 0.2 when θr = −2 and θr → −∞

It is also clearly seen that for both gases and liquids, the temperature profiles decrease as the Prandtl number increases. The reason for such a behavior is that the higher Prandtl number fluid has a relatively low thermal conductivity which opposes conduction. This results in the reduction of the thermal boundary layer thickness and hence a decrease in the temperature profiles. The effect of suction/injection parameter f w on the velocity and temperature profiles with fixed β = 0.2 in the absence as well as in the presence of variable viscosity for Pr = 0.2 and Pr = 7 are shown in the Figs. 16 and 17 respectively. An increase in f w , when θr is fixed, causes a significant increase in velocity and decrease in temperature profiles throughout the boundary layer for both Pr = 0.2 and 7, which corresponds to either increasing suction

123

Author's personal copy R. K. Deka, M. Basumatary

or decreasing injection. So, increasing suction accelerates the fluid motion and reduces the temperature of the fluid along the wall. It is also seen that the velocity of gas (Pr = 0.2) for the constant viscosity case is greater than that of the variable viscosity case, when there is variation in the values of f w , but it is just opposite in case of water. We have also observed that the temperature of gas (Pr = 0.2) for the constant viscosity case is less than that of the variable viscosity case, when there is variation in the values of f w , but it is just opposite in case of water. 5 Conclusions The present study provides similarity solutions for the occurrence of flow separation at the surface of a wedge with the effect of variable viscosity in the presence of suction or injection. The governing partial differential equations were transformed by using suitable similarity variables into a system of non-linear ordinary differential equations, and then solved numerically using Runge–Kutta–Fehlberg method with Newton Raphson shooting technique. It has been observed that the suction and the larger values of pressure gradient parameter delay the flow separation. For suction, the flow separation occurs earlier in gases with the smaller value of |θr |, but a slight delay in liquids with the smaller value of |θr |. Both the skin friction coefficient and rate of heat transfer coefficient increase as suction/injection parameter increases. An increase in pressure gradient parameter leads to an increase in the velocity, while temperature decreases. The effect of Prandtl number on the skin friction coefficient is important when |θr | is small. Acknowledgments

One of the authors (R. K. Deka) acknowledges the support of UGC, New Delhi.

References 1. Falkner, V.M., Skan, S.W.: Some approximate solutions of the boundary layer equations. Philos. Mag. 12, 865–896 (1931) 2. Hartree, D.R.: On an equation occurring in Falkner and Skan’s approximate treatment of the equations of the boundary layer. Proc. Camb. Philos. Soc. (1937). doi:10.1017/S0305004100019575 3. Stewartson, K.: Further solutions the Falkner–Skan equation. Proc. Camb. Philos. Soc. (1954). doi:10. 1017/S030500410002956X 4. Cebeci, T., Keller, H.B.: Shooting and parallel shooting methods for solving the Falkner–Skan boundary layer equation. J. Comput. Phys. (1971). doi:10.1016/0021-9991(71)90090-8 5. Koh, J.C.Y., Harnett, J.P.: Skin friction and heat transfer for incompressible laminar flow over a porous wedge with suction and variable wall temperature. Int. J. Heat Mass Transf. (1961). doi:10.1016/ 0017-9310(61)90088-6 6. Chen, K.K., Libby, P.A.: Boundary layers with small departure from the Falkner–Skan profile. J. Fluid Mech. (1968). doi:10.1017/S0022112068001291 7. Lin, H.T., Lin, L.K.: Similarity solutions for laminar forced convection heat transfer from wedges to fluids of any Prandtl number. Int. J. Heat Mass Transf. (1987). doi:10.1016/0017-9310(87)90041-X 8. Watanabe, T.: Thermal boundary layer over a wedge with uniform suction and injection in forced flow. Acta Mech. (1990). doi:10.1007/BF01172973 9. Schlichting, H., Gersten, K.: Boundary Layer Theory, 8th revised edn. Springer-Verlag, Berlin (2000) 10. Leal, L.G.: Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press, New York (2007) 11. Ishak, A., Nazar, R., Pop, I.: Falkner–Skan equation for flow past a moving wedge with suction or injection. J. Appl. Math. Comput. (2007). doi:10.1007/BF02832339 12. Yacob, N.A., Ishak, A., Pop, I.: Falkner–Skan problem for a static or moving wedge in nanofluids. Int. J. Therm. Sci. (2011). doi:10.1016/j.ijthermalsci.2010.10.008

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Author's personal copy Effect of variable viscosity 13. Bararnia, H., Haghparast, N., Miansari, M., Barari, A.: Flow analysis for the Falkner–Skan wedge flow. Curr. Sci. 103(2), 169–177 (2012) 14. Parand, K., Rezaei, A., Ghaderi, S.M.: An approximate solution of the MHD Falkner–Skan flow by Hermite functions pseudospectral method. Commun. Nonlinear Sci. Numer. Simul. (2011). doi:10.1016/ j.cnsns.2010.03.022 15. Postelnicu, A., Pop, I.: Falkner–Skan boundary layer flow of a power-law fluid past a stretching wedge. Appl. Math. Comput. (2011). doi:10.1016/j.amc.2010.09.037 16. Afzal, N.: Falkner–Skan equation for flow past a stretching surface with suction or blowing: analytical solutions. Appl. Math. Comput. (2010). doi:10.1016/j.amc.2010.07.080 17. Ashwini, G., Eswara, A.T.: MHD Falkner–Skan boundary layer flow with internal heat generation or absorption. World Acad. Sci. Eng. Technol. 65, 662–665 (2012) 18. Yih, K.A.: Uniform suction/blowing effect on force convection about a wedge: uniform heat flux. Acta Mech. (1998). doi:10.1007/BF01251888 19. Soundalgekar, V.M., Takhar, H.S., Das, U.N., Deka, R.K., Sarmah, A.: Effect of variable viscosity on boundary layer flow along a continuously moving plate with variable surface temperature. Int. J. Heat Mass Transf. 40, 421–424 (2004) 20. Pantokratoras, A.: Forced and mixed convection boundary layer flow along a flat plate with variable viscosity and Prandtl number: new results. Heat Mass Transf. 41, 1085–1094 (2005) 21. Hady, F.M., Bakier, A.Y., Gorla, R.S.R.: Mixed convection boundary layer flow on a continuous flat plate with variable viscosity. Heat Mass Transf. 31, 169–172 (1996) 22. Mukhopadhyay, S.: Effects of radiation and variable fluid viscosity on flow and heat transfer along a symmetric wedge. J. Appl. Fluid Mech. 2(2), 29–34 (2009) 23. Salem, A.M.: Temperature-dependent viscosity effects on non-Darcy hydrodynamic free convection heat transfer from a vertical wedge in porous media. Chin. Phys. Lett. (2010). doi:10.1088/0256-307X/27/6/ 064401 24. Muhaimin, I., Kandasamy, R., Azme, B.K., Roslan, R.: Effect of thermophoresis particle deposition and chemical reaction on unsteady MHD mixed convective flow over a porous wedge in the presence of temperature-dependent viscosity. Nucl. Eng. Des. (2013). doi:10.1016/j.nucengdes.2013.03.015 25. Schlichting, H.: Boundary Layer Theory (translated by J. Kestin ). Mc Graw Hill Inc, New York (1979) 26. Riley, N., Weidman, P.D.: Multiple solutions of the Falkner–Skan equation for flow past a stretching boundary. SIAM J. Appl. Math. (1989). doi:10.1137/0149081 27. White, F.M.: Viscous Fluid Flow, 3rd edn. Mc Graw-Hill, New York (2006) 28. Ling, J.X., Dybbs, A.: Forced convection over a flat plate submersed in a porous medium: variable viscosity case. In: ASME winter meeting conference, Boston, 13 December 1987 (1987) 29. Kumari, M., Takhar, H.S., Nath, G.: Mixed convection flow over a vertical wedge embedded in a highly porous medium. Heat Mass Transf (2001). doi:10.1007/s002310000154

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International Journal of Mathematical Archive-5(9), 2014, 129-137

Available online through www.ijma.info ISSN 2229 – 5046 EFFECT OF VARIABLE VISCOSITY ON FLOW OF A MICROPOLAR FLUID OVER A LINEARLY STRETCHING SHEET WITH VARIABLE SURFACE TEMPERATURE R. K. Deka* and R. Dutta Department of Mathematics, Gauhati University, Guwahati-14, India. (Received On: 28-08-14; Revised & Accepted On: 20-09-14)

ABSTRACT

Flow of viscous incompressible micropolar fluid over a linear stretching sheet is analyzed for the case of variable viscosity and variable sheet temperature. It is assumed that viscosity varies as a function of temperature. The basic partial differential equations are reduced to a system of nonlinear ordinary differential equations, which are solved numerically using shooting technique. Effect on velocity, angular velocity and temperature profile for different parameters are shown graphically. Keywords: Variable viscosity, Linear stretching sheet, Micropolar fluid, Variable surface temperature.

1. INTRODUCTION The problem of boundary layer flow over a constantly moving plate was first studied by Sakiadis [1]. Crane [2] extended his result to a linear stretching sheet. Cortell [3-4] and Vajravelu [5] studied heat transfer characteristics of Newtonian fluid over a non-linear stretching sheet considering various flow conditions. The effect of surface temperature variation on the heat transfer characteristics of a continuous stretching surface was investigated by Carragher and Crane[6], Gurbka and Bobba[7], Chen and Char [8], Ishak et al. [9] etc. The theory of micropolar fluid was first introduced and formulated by Eringen [10-11]. It describes the behaviour of exotic lubricants, polymeric fluids, liquid crystal, animal bloods colloidal and suspension solution etc., for which the classical Navier-Stokes theory is not sufficient to explain these properties. The boundary layer concept in such fluid over stretching surface was observed by Abo-Eldahab and El Aziz[12], Hassanien et al.[13], Ishak et al. [14-15], Soundalgekar and Takhar [16], Hayat et al. [17] etc. In all these studies viscosity was taken as constant. Hassanien [18] studied the effects of variable viscosity on the laminar heat transfer flow of Newtonian fluids along a linear stretching sheet. Variable viscosity in micropolar fluid was studied by Rahman et.al[19] on different types of flow. Recently, effect of variable viscosity and internal heat generation in micropolar fluids over a continuous moving surface was observed by Abdou and El-Zahar [20]. The combined effect of variable viscosity and variable temperature over a non linear vertical stretching sheet of a micropolar was studied by Rahman et al. [19]. In this study, we consider the linear case of horizontal stretching sheet under the weak concentration of micro-elements, in presence of variable viscosity and variable temperature. 2. MATHEMATICAL ANALYSIS Let us consider a steady, laminar, two dimensional flow of a viscous incompressible micropolar fluid of temperature due to a horizontal continuous stretching sheet with velocity , and temperature respectively. The flow is assumed to be in the x-direction, which is taken along the sheet and y is taken normal to the sheet. Two equal and opposite force introduced along x-axis, so that the sheet is stretched keeping the origin fixed. The fluid properties are assumed to be isotropic and constant, except for the Newtonian fluid viscosity , which is assumed to vary as a reciprocal of a linear function of temperature (See Hassanien[18] and Soundalgekar et al. [21]), which is,

Here, is constant, from the sheet.

is the reference viscosity,

are respectively the temperature of the fluid near and far away

Corresponding author: R. K. Deka* Department of Mathematics, Gauhati University, Guwahati-14, India. International Journal of Mathematical Archive- 5(9), Sept. – 2014

129

R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

The above relation can be written as,

, where

and

are constant and their values

depend on the thermal property of the fluid, i.e . In general g > 0 for liquids and g<0 for gases. Under all the above assumptions, the boundary layer equations can be written as, (1) (2) (3) (4) subject to the boundary conditions,

u

bx, v

u

0, N

0, N 0, T

m0

u ,T y

T

T

ax n at y as

0

(5)

y

where u and v are velocity components along x and y directions, N is the microrotation or angular velocity, whose and are micropolar fluid viscosity, thermal conductivity, specific direction of rotation is in the xy-plane. heat at constant temperature, micro-inertia density and density of the ambient fluid respectively. Also, a, b and

are

constant and is the reference length (see Hayat [17]). Here leads to , indicating that the concentrated particle flows in which the microelement close to the wall surface are unable to rotate. This case is also , the weak concentration known as strong concentration of microelement. In the present study we have taken of microelement (see Guram and Smith [22]). This value of tensor.

indicates the vanishing of antisymmetric part of stress

Now we introduce the following non dimensional variables,

Then equations (2)-(4) reduce to, (6) =0

(7) (8)

and the boundary conditions become, (9) Here the primes denote differentiation w.r.t. , , is the Prandtl number, parameter, is the variable viscosity parameter, which is defined by,

is the vortex viscosity

Now we observe that is negative for liquids (Pr>1) and positive for gases (Pr<1) (see Sounndalgekar et al. [21]). Skin friction coefficient and nusselt number coefficient are defined by,

© 2014, IJMA. All Rights Reserved

130

R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

and where, and are shear stress and surface heat flux at the surface. Using non dimensional quantities we have, and 3. RESULTS AND DISCUSSION Equations (6)-(8) subject to boundary conditions (9) are solved numerically using Ruge-Kutta Fehlberg scheme along with shooting technique. When K=0, n=0, equations (6)-(8) reduce to the equations in Hassanien[18]. Table1 and Table 2 show comparison of the values of Hassanien [18] with the present study for different values of . Also in Table3, we with Ishak et al. [9] for various values of n with and .In both cases we found compare the values of good agreement. (0)}, { (0)} in Table 4 and Table5.Figures 1- 8 show the variation of velocity, We have tabulated the values of { microrotation distribution, and temperature distribution for different values of n, K when is positive or negative. with Hassanien [18], for different values of Table-1: Comparison of present values of Hassanien [18] Present result r Pr = 0.7 Pr = 10 Pr= 0.7 Pr= 10 -10 -1.05607 -1.07439 -1.05597 -1.07454 -8 -1.06955 -1.09245 -1.0694 -1.0926 -6 -1.09158 -1.2212 -1.09144 -1.12226 -4 -1.14012 -1.7985 -1.13395 -1.17998 -2 -1.25928 -1.34142 -1.25129 -1.34153 2 -0.64661 -0.56908 -0.64602 -0.569271 4 -0.84422 -0.79817 -0.84187 -0.798342 6 -0.89793 -0.86821 -0.89793 -0.868377 8 -0.92463 -0.90216 -0.92461 -0.902331 10 -0.94025 -0.92221 -0.94583 -0.929607

, when K=0, n=0.

with Hassanien [18] for different values of Table-2: Comparison of present values of Hassanien [18] Present result r Pr = 0.7 Pr = 10 Pr= 0.7 Pr= 10 -10 -0.4486 -2.29730 -0.446534 -2.29725 -8 -0.44471 -2.29469 -0.444714 -2.29464 -6 -0.44423 -2.29038 -0.441732 -2.29033 -4 -0.44078 -2.28199 -0.43595 -2.2819 -2 -0.44168 -2.25799 -0.41988 -2.2579 2 -0.52468 -2.36770 -0.499509 -2.36763 4 -0.50592 -2.33643 -0.474874 -2.33637 6 -0.46912 -2.32666 -0.467578 -2.3266 8 -0.46572 -2.32188 -0.46407 -2.32183 10 -0.46373 -2.31905 -0.462009 -2.319

, when K=0,n= 0

with Ishak et al. [9] for different values of n, when Table-3: Comparison of present values of n Ishak et al. [9] Present result Pr= 1 Pr=3 Pr= 10 Pr=1 Pr=3 Pr=10 -2 -1.0000 -3.0000 -10.0000 -1.00003 -3.000279 - 9.999641 -1 0.0 0.0 0.0 0.0 0.0 0.0 0 0.5820 1.1652 2.3080 0.58207 1.165191 2.307935 1 1.000 1.9237 3.7207 1.00009 1.923598 3.720559 2 1.3333 2.5097 4.7967 1.33333 2.509621 4.796728 © 2014, IJMA. All Rights Reserved

, K= 0

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R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

Table-4: Values of {-f’’(0)} with K=1 0 .5 .589837 .584026 .744598 .742656 .782859 .781979 .968604 .972768 .886679 .888510 .853541 .854515

r /n

Pr .71

2 5 10 -2 -5 -10

7

Table-5: Values of {- ’ (0)} with K=1. 0 . 5 1 .526802 .720243 .887817 .506341 .69674 .862617 .500873 .690401 .855777 1.91035 2.54872 3.08856 1.92418 2.56411 3.10481 1.92953 2.57005 3.11107

r /n

Pr .71

2 5 10 -2 -5 -10

7

1 .579544 .741077 .781237 .975971 .889898 .855244

Table 4 indicates that {-f’’(0)} decreases for gas and increases for liquid with increasing values of N. Also, it is found that increasing values of , skin friction increases for gas and the converse is true for liquid. From Table 5, when N increases heat flux increases for all values of when

, but an increase in

From Fig.1 (a), it is clear that an increase in 1(b), an increase in

leads to an increase in velocity profiles for liquid, whereas in Fig.

leads to a decrease in velocity for gas. Fig. 2(a) and Fig. 2(b) show the variation of microrotation

distribution of liquid and gas for various values of decreases for increase in profile with

leads to a decrease in the heat flux for gases and

increases, heat flux increases for liquid.

with fixed values of n,K. It depicts that microrotation velocity h

in case of liquid, but when

increases h increases in gas. Variation of temperature

for fixed values of n and K, in both gas and liquid s are shown in figures 3(a) and 3(b). As in Fig.3 (a)

temperature profile decreases as

increases and in Fig.3 (b) temperature profile increases as

increases.

Figures 4(a) and 4(b) show the effect of K on velocity. For fixed values of n and , as K increases velocity profile increases for both liquid and gas. The variation of microrotation distribution for different values of K are shown in figures 5(a) and 5(b). It is observed that microrotation velocity continuously decreases with K and becomes zero far away from the sheet, for both liquid and gas. It can also be seen from these figures that the microrotation velocity decreases as K increases in the vicinity of the plate, but the reverse is true as one moves away from it. Figures 6(a) and 6(b) illustrate the temperature profile for various values of K.As K increases temperature decreases for both water and gas for fixed values of n and . Effects of temperature index n on velocity profiles are presented in figures 7(a) and 7(b). As n increases, velocity increases for liquid and decreases for gas for fixed values of K and .But the effect of n on velocity profile is not pronounced. Figures 8(a) and 8(b) show the effect of n on temperature profile. We noticed that when n increases temperature decreases for both liquid and gas. 3. CONCLUSION (i) In case of liquid when increases velocity increases, but both angular velocity and temperature decreases. Reverse is true for gas. (ii) As K increases velocity increases and temperature decreases for both gas and liquid. The microrotation velocity decreases as K increases near the sheet and this behaviour changes far away from the sheet. (iii) Effect of n on the velocity is too less. When n increases temperature decreases for both gas and liquid. (iv) When plate temperature increases skin friction increases for gas and decreases for liquid. But the heat flux increases for both gas and liquid. increases skin friction increases but heat flux decreases. (v) For gas as (vi) In case of liquid as

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increases skin friction decreases but heat flux decreases.

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R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

ACKNOWLEDGEMENT The authors greatly acknowledge the support of UGC, New Delhi to carry out this work. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Sakiadis, B.C. (1961) Boundary-layer behavior on continuous solid surfaces: I. Boundary layer equations for two-dimensional and axisymmetric flow, AIChE J., 7, pp. 26-28. Crane, L.J. (1970) Flow past a stretching plate, Zeitschriftfürangewandte Mathematik und Physik (ZAMP), 21, pp. 645–647. Cortell, R. (2007) Viscous flow and heat transfer over a nonlinearly stretching sheet, Appl. Math. Comput., 184, pp. 864-873. Cortell, R. (2008) Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet, Physics Letter A, 372, pp. 631-636. Vajravelu, K. (2001) Viscous flow over a nonlinearly stretching sheet,Appl. Math. Comput., 124, pp. 281-288. Carragher, P. and Crane, L.J. (1982) Heat transfer on a continuous stretching sheet, Z. Angew. Math. Mech., 62, pp.564-565. Grubka, L. J. and Bobba, K. M. (1985) Heat transfer characteristics of a continuous stretching surface with variable temperature, J. Heat Transfer (Trans ASME), 107, pp. 248–250. Chen, C. K. and Char, M. I. (1988) Heat transfer of a continuous, stretching surface with suction or blowing, J. Math. Anal. Appl., 135, pp. 568-580. Ishak A., Nazar, R. and Pop, I. (2008) Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet, Heat Mass Transfer, 44, pp. 921-927 Eringen, A. C. (1966) Theory of micropolar fluids, J. Math. Mech., 16, pp. 1–18. Eringen, A. C. (1972) Theory of thermomicrofluids, J. Math. Anal. Appl., 38, pp. 480–496. Abo-Eldahab, E. M. and El Aziz, M. A. (2005) Flow and heat transfer in a micropolar fluid past a stretching surface embedded in a non Darcian porous medium with uniform free stream, Appl. Math. Comput., 162, pp. 881-899. Hassanien I.A. and Shamardan A., Moursy N.M. and Gorla R.S.R. (1999) Flow and heat transfer in the boundary layer of a micropolar fluid on a continuous moving surface,Int. J. Numer. Methods Heat Fluid Flow, 9, pp. 643–659. Ishak A., Nazar R. and Pop I. (2006) Flow of a micropolar fluid on a continuous moving surface, Arch. Mech., 58, pp. 529–541. Ishak A., Nazar R. and Pop I. (2008) Heat transfer over a stretching surface with variable heat flux in micropolar fluids, Phys. Lett A, 372, pp. 559–561. Soundalgekar, V. M. and Takhar, H. S. (1983) Flow of a micropolar fluid on a continuous moving plate, Int. J. Eng. Sci., 21, pp.961-965. Hayat, T.,Abbas, Z. and Javed, T. (2008) Mixed convection flow of a micropolar fluid over a non-linearly stretching sheet,Physics Letters A, 372, pp. 637-647. Hassanien, I. A. (1997) The effect of variable viscosity on flow and heat transfer on a continuous stretching surface, Z. Angew. Math. Mech., 77, pp.876-880. Rahman, M. M, Rahman, M. A., Samad, M. A. and Alam, M. S. (2009)Heat transfer in a micropolar fluid along a non-linear stretching sheet with a temperature-dependent viscosity and variable surface temperature, Int. J. Thermophys, 30, pp. 1649–1670. Abdou, M. M. M. and El-Zahar E. R. (2012) Variable viscosity effect on heat transfer over a continuous moving surface with variable internal heat generation in micropolar fluids, Appl. Math. Sci., 6, pp.6365 – 6379. Soundalgekar, V.M., Takhar, H.S., Das, U.N., Deka, R. K. and Sarmah, A. (2004) Effect of variable viscosity on boundary layer flow along a continuously moving plate with variable surface temperature, Int. J. Heat and Mass Transfer, 40, pp. 421–424. Guram, G.S. and Smith, A.C. (1980) Stagnation flow of micropolar fluids with strong and weak interactions,Comp. Math. with Appl., 6, pp. 213–233.

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R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

(b) 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

f'

f'

(a) 1

0.4

0.4

0.3

0.3

0.2

0.2

0.1 0

=2,20,

0.1

=-2,-20, r 0

r

Figure-1: Velocity profiles for different

0

8

6

4

2

rand

K = 0.5, n = 0.3, (a)

, when Pr=7; (b)

(a)

8

6

4

2

0

,when Pr= 0.71

(b)

0.6

0.45 0.4

0.5 0.35 0.4

0.3

h

h

0.25 0.3

0.2 r

0.2

=-2,-20,

0.15 0.1

0.1

0

0.05 0 0

2

4

6

8

Figure-2: Angular velocity profiles for different when Pr=7; (b)

© 2014, IJMA. All Rights Reserved

r

0

=2,20, 2

r

4

and K = 0.5, n = 0.3, (a)

6

8 ,

, when Pr= 0.71

134

R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

(a)

(b)

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3 r

0.2

0.3

=-2,-20,

0.2

0.1

0.1

0

0

1

0.5

0

1.5

3

2.5

2

r

=2,20,

Figure-3: Temperature profiles for different rand K = 0.5, n = 0.3, (a)

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

f'

f'

1

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 K=0,.5,1.5 0

2

4

6

8

0

Figure-4: Velocity profiles for different K and n = 0.3, (a)

© 2014, IJMA. All Rights Reserved

, when Pr=7; (b)

8

10

,when Pr= 0.71

(b)

(a) 1

0

6

4

2

0

K=0,.5,1.5 0

2

4

, when Pr=7; (b)

6

8

10

, when Pr= 0.71

135

R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

(a)

(b) 0.35

0.7 K=0 K=.5 K=1.5

0.6

K=0 K=.5 K=1.5

0.3

0.4

0.2 h

0.25

h

0.5

0.3

0.15

0.2

0.1

0.1

0.05

0

0

8

6

4

2

0

Figure-5: Angular velocity profiles for different K and n =0.3, (a)

, when Pr=7; (b)

(a) 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3 K=0,.5,1.5

, when Pr= 0.71

K=0,.5,1.5

0.2 0.1

0.1 0

8

(b)

1

0.2

6

4

2

0

0

0.5

1

1.5

2

2.5

0

0

Figure-6: Temperature profiles for different K and n = 0.3, (a)

© 2014, IJMA. All Rights Reserved

2

4

, when Pr=7, (b)

6

8

,when Pr= 0.71

136

R. K. Deka* and R. Dutta / Effect of variable viscosity on flow of a micropolar fluid over a linearly stretching sheet with variable surface temperature/ IJMA- 5(9), Sept.-2014.

(b) 1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

f'

f'

(a) 1

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

n=.3,1,2 0

2

4

6

0

8

n=.3,1,2

0

Figure-7: Velocity profiles for differentn and K = 0.5, (a)

2

4

, when Pr=7; (b)

(a)

8

, when Pr= 0.71 (b)

1

1 n=.3 n=1 n=2

0.9 0.8

0.8 0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0 0

0.5

1

1.5

2

2.5

n=.3 n=1 n=2

0.9

0.7

0

6

3

0

Figure-8: Temperature profiles for different n and K = 0.5, (a)

2

4

, when Pr=7; (b)

8

6

10

, when Pr= 0.71

Source of support: UGC, New Delhi, India. Conflict of interest: None Declared [Copy right © 2014. This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.] © 2014, IJMA. All Rights Reserved

137

International Journal of Mathematical Archive-5(5), 2014, 150-157

Available online through www.ijma.info ISSN 2229 – 5046 EFFECTS OF VARIABLE VISCOSITY ON MIXED CONVECTION HEAT AND MASS TRANSFER PAST A WEDGE WITH VARIABLE TEMPERATURE Rudra Kanta Deka*1 and Sarat Sharma2 1Department

of Mathematics, Gauhati University, Guwahati-781014, Assam, India. of Mathematics, Chaiduar College, Gohpur-784168, Assam, India.

2Department

(Received on: 19-04-14; Revised & Accepted on: 12-05-14)

ABSTRACT

Steady

mixed convection flow past a wedge with variable viscosity and variable surface temperature is studied. Variations in velocity, temperature and concentration profiles due to various values of the temperature exponent, viscosity parameter, Prandtl number and Schimdt number are presented. Proportionate numerical results are tabulated for the local skin-friction, local Nusselt number and local Sherwood number. It is observed that the introduction of variable viscosity effects and variable temperature can affect the separation of flow past a wedge. Key words: mixed convection, wedge flow, variable viscosity, variable temperature.

1. INTRODUCTION The analysis of laminar flow and convective heat transfer, under the influence of thermo-physical properties, has gained importance in many scientific and engineering applications. Flow past a wedge belongs to a special class of boundary layer flow problems where the shape of the wedge affects the boundary layer mainly by determining the pressure gradient along the surface of the wedge. The momentum boundary-layer equation for Falkner-Skan flow past a wedge with potential flow velocity was first studied by Falkner and Skan [1] in the early 1930’s, to illustrate the application of Prandtl’s boundary layer theory. Later Hartree [2] investigated the same problem and found similarity solutions. Ishak et al. [3] considered the case of a moving wedge using the Falkner-Skan transformations. Williams and Rhyne [4] have used a group of transformations to encompass both the situations of small and large time in the unsteady flow past a wedge by simple modification of the Falkner and Skan transformations in solving the wedge equations employing the Thomas algorithm. Massoudi [5] employed non-similarity method to solve wedge flow problems. Ibrahim and Hassanien [6], Hassanien [7], Ece [8] and Loganathan et al. [9] presented flow past a wedge with different thermal conditions. While most of the above cited works deal with the constant physical properties of the fluid under consideration. Sometimes this assumption may give erroneous results in practical situations of heat transfer problems since it is known that these physical properties may change with temperature, especially fluid viscosity. It is to be noted that like other thermo-physical properties, temperature-dependent viscosity plays a vital role in surface friction and heat transfer rate near the wall. Hence, from application point of view, to accurately predict the flow behavior, it is necessary to take into account this variation of viscosity. For this, there are different types of viscosity-temperature models used by various authors. Several authors have considered variable viscosity effects in their studies, few of them are, for example, Kassoy and Zebib [10], Lai and Kulacki [11], Hady et al. [12], Kafoussias et al. [13], Hossain and Munir [14], Hossain et al. [15], Pantokratoras [16, 17, 18], Chin et al. [19], Makinde [20], Hossain et al. [21], Hossain et al. [22] and Hassanien, et al. [23]. In all these studies, the plate temperature is assumed constant. However, in some situations the temperature of the wall becomes variable one. Few researchers have studied taking account of this situation for flow past wedge without variable viscosity, for example, Koh and Hartnett [24], Yih [25], Hossain et al. [26], while Soundalgekar et al. [27] included variable wall temperature conditions and variable viscosity for flow past a continuously moving plate. Recently, Salem [28] presented an analysis of the flow past a wedge in a porous media with variable viscosity taking inertia effect and variable surface temperature into account. In this study, we propose to analyze the heat and mass transfer on flow past a wedge with variable plate temperature and variable viscosity.

1Department

Corresponding author: Rudra Kanta Deka*1 of Mathematics, Gauhati University, Guwahati-781014, Assam, India. 1Email: [email protected]

International Journal of Mathematical Archive- 5(5), May – 2014

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Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014.

2. MATHEMATICAL ANALYSIS We consider laminar steady flow of a viscous incompressible fluid past a symmetrical wedge. All fluid properties are assumed to be isotropic and constant, except for the fluid viscosity ( ), which is assumed to be an inverse linear function of the temperature (T) and is a good approximation for most of the fluids (see Lai and Kulacki [11]), viz.,

1

1

[1

(T T )]

(1)

is a reference viscosity and T, T are the temperatures of the fluid near and far away from Here, is a constant and the surface of the wedge. This relation can be written as,

1

a (T Tr )

(2)

where a = / and Tr = T -1/ . Here a and Tr are assumed to be constant, as their values are depending on the reference states and the thermal property of the fluid, . In general, a > 0 for liquids and a < 0 for gases. The governing equations are,

v y

u x

0 1

u

u x

v

u y

u

T x

v

T y

C u x

(3)

u y

y

du dx

u

[ g (T T ) g

(C C )] sin

(4)

2

2 k T Cp y2

(5)

2

C v y

C y2

D

(6)

The boundary conditions are,

y

0, u

0, v

, u

y

0, T

u ( x), T

Tw ( x), C T , C

0

(7)

C

We also assume that the temperature of the surface of the wedge varies as xn and we take it as,

Ax n

Tw ( x) T

(8)

where A is a constant and x is measured from the leading edge. Now we introduce the stream function

m 1 u , 2 x

y G

g

2 m 1

C )x2

(C w 2

, G

and following similarity variables,

xu f ( ), ( )

g (Tw T ) x 2 2

,

1

T T , Tw T

( )

G , Rex2

G , Pr Rex2

2

C C , Rex Cw C , Sc

u x

, (9)

D

All the physical variables and quantities are defined in Nomenclature. Here, the potential flow velocity is taken as, u ( x ) Bx m, and 1=2m/(1+m) , where B is a constant and 1 is the Hartree pressure gradient parameter that corresponds to 1 = / for a total angle of the wedge. If 1 is positive, the pressure gradient is negative or favorable, and negative 1 denotes an unfavorable positive pressure gradient, while 1 = 0 denotes the flat plate (cf. White [30]). Then in view of (9), the equations (4)-(6) reduce to,

f

(1

/

r)

ff

2m (1 m 1

© 2014, IJMA. All Rights Reserved

f 2)

2 m 1

(

1

2

) sin

1 2

f

0

(10)

r

151

1

2

Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014.

Pr

1

f

Sc

1

f

2n f m 1 0

0

(11) (12)

with following boundary conditions:

f (0) Here

f (0) 0, (0) 1, (0) 1, f ( ) 1, ( ) 0, ( ) 0 r

(13)

(T r T ) /(Tw T ) is the viscosity parameter and (T Tr ) /(Tw T )

r

.

The viscosity parameter r is determined by the viscosity of the fluid under consideration and the operating temperature ), the effects of variable viscosity can be neglected. On the other hand, for a smaller difference. If r is large (i.e. r value of r, either the fluid viscosity changes markedly with temperature or operating temperature difference is high. In either case, the variable viscosity effect is expected to become very important. It may be noted hear that the liquid viscosity varies differently with temperature than that of gas and therefore r < 0 for liquids (Pr > 1) and r > 0 for gases (Pr < 1) Knowing the velocity, temperature and concentration field, we derive Skin friction coefficient C f surface shear stress =

=0

=

=0

=

where the surface mass flux

xqw , where the surface heat flux k (Tw T )

; the local Nusselt number, Nu x

and the Local Sherwood number Shx =0

2 w , where the u2

xM w , D(C w C )

.

Then the skin-friction coefficient, local Nusselt number and local Sherwood numbers are respectively derived as,

1 C f ( Rex )1 / 2 2

m 1 2

1/ 2

f (0), Nu x ( Rex )

m 1 2

1/ 2

(0), Shx ( Rex )

m 1 2

1/ 2

(0)

3. SOLUTIONS AND DISCUSSION The transformed coupled non-linear eqs. (10)-(12) together with boundary conditions (13) are solved applying parallel shooting method using MATHEMATICA. To validate our results, we have set m = 0, n = 0 in our work and compare our results with those obtained by Hady et al. [12] for the case of a mixed convection flow past a vertical flat plate (0) corresponding to Pr = 7.0 are presented under variable viscosity effects. The computed results for f (0) and in Table 1. The comparisons are in excellent agrrement. In order to get the physical insight of the flow phenomena, we have considered air (Pr = 0.71) and water (Pr = 7) as working fluid. For the case of fluid heating, Elbashbeshy and Bazid [29] showed that r cannot take the values between 0 and 1 and suggested that r > 1 for gases and r < 0 for liquids. Accordingly, we have chosen values of r. Also, we = 900 and 1 = 2 = 1.0. The velocity, temperature and take Sc = 617, 0.62 (for Cl2 and water vapour), concentration profiles for the flow corresponding to air (Pr = 0.71), water vapour diffusing through air (Sc = 0.62) with n =1 are presented in figure 1 for different values of viscosity parameter r (= 2, 5, 15). It is seen from the figure that viscosity variation induces considerable influence on the flow field. Velocity of the flow increases with increase in the viscosity parameter. For smaller value of the viscosity parameter the increase in velocity is significant but for larger value of the viscosity parameter, the increase in velocity is very low. The difference in velocity for higher viscosity parameter becomes less in comparison to lower values of the parameters. The temperature is seen to decrease with increase in viscosity parameter. For smaller values of the viscosity parameter the decrease in temperature is significant, but for larger value of the viscosity parameter, the decrease in temperature is very low. The species concentration level is also seen to decrease with increase in viscosity parameter. For smaller value of the viscosity parameter the decrease in concentration level is significant, but for larger value of the viscosity parameter, the decrease is insignificant. Similarly, the velocity, temperature and concentration profiles for the flow corresponding to water (Pr = 7.0) and Cl2 (Sc = 617) diffusing through water with n = 1 are presented in figure 2 for different values of r (= -2, -5, -15). We find that the velocity of the flow increases with increase in values of the viscosity parameter. The increase in velocity is low for smaller values but increases significantly for higher values of the parameter. The temperature is seen to decrease with viscosity parameter though the decrease is very small. Viscosity variation has no effect on the concentration level © 2014, IJMA. All Rights Reserved

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Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014.

of the flow for water. The velocity, temperature and concentration profiles for air (Pr = 0.71) and water vapour (Sc=0.62) diffusing through air for different values of n (= 0, 0.3, 2) showing plate temperature variation with r = 2 are presented in figure 3. It is seen that the velocity decreases uniformly with the increase in n. The temperature of the fluid is found to decrease with increase in n. However, the concentration shows no significant change with the increase in the presence of plate temperature variation. Now, we discuss the effect of variable temperature parameter on the local skin friction coefficient, local Nusselt (0) and (0) are presented in Table 2, which number and local Sherwood number. The computed values f (0) , are proportional to the local Skin friction, local Nusselt number and local Sherwood number, respectively under variation of viscosity parameter ( ) and the plate temperature exponent ( ) at m = 1/3. It is observed from these tables that the drag force or force due to skin friction decreases as n increases. It is also observed that the drag force is larger (0) and for gases (Pr <1) compared to fluids (Pr > 1). Similar behavior is seen on the values of (0), with the difference is that these values are smaller for gases compared to fluids. One of the frequently encountered problems in ‘wedge flow’ is the flow separation. It has several undesirable effects in so far as it leads to an increase in the drag on a body immersed in the flow and adversely affects the heat transfer from the surface of the body. We recall that 1 = m/(m+1), as defined earlier, is known as the Hartree pressure gradient of the wedge. Two important cases are the flat plate flow parameter that corresponds to 1 = / for a total angle (m=0) and the stagnation-point flow (m = 1). It is well known that for accelerated flows ( > 0, 1 > 0), velocity profiles exhibit no point of inflexion but for retarded flows ( < 0, 1 < 0) there exists a point of inflexion. The boundary layer thickness is greater at the smaller values of 1 , where 1 < 0 resembles the situation on the trailing side of a blunt object, where the surface is inclined away from the main flow. For fluids with constant viscosity, the case = 0.091, 1 = 0.199 is a limiting case, in which case no solution to the Falkner-Skan equation exists. Thus < 0.199, corresponds to the velocity profile with vanishing wall shear stress (separation). In our present study, we have calculated the different values of 1 for different values of , , Sc and , for which separation of flow = -5, n = 0.5 and Sc = 617 the separation of flow occurs at occurs. For example, corresponding to water, = -0.1044; while in case of air, = 5, n = 1, Sc = 0.62, the separation of flow occurs at 1 = -0.0477. This is clearly 1 displayed in figures 3 and 4 and from this we can conclude that flow separation can effectively be controlled by inclusion of variation of viscosity and plate temperature variation effects into the flow field. 4. CONCLUSIONS The effects of variable viscosity and variable surface temperature on flow, mass, and heat transfer of a steady incompressible Newtonian fluid past a wedge has been studied. The novel result of the study is that varying the surface temperature and the shape of the wedge, the flow separation can be controlled. From this investigation, we can draw the following conclusions: (i) Increasing viscosity parameter enhances the fluid velocity. (ii) For air, the temperature and concentration both tend to decrease with increasing viscosity. (iii) For water, increasing viscosity increases the temperature slightly but the concentration shows no significant change. (iv) Increasing plate temperature retards the fluid velocity. (v) The concentration remains unchanged with an increase in the variation of plate temperature. (vi) Both local skin-friction and Sherwood number decreases with plate temperature but local Nusselt number increases. ACKNOWLEDGEMENT One of the authors (R. K. Deka), acknowledges the financial support of UGC, New Delhi. REFERENCES [1]. Falkner, V. M. and Skan, S. W. (1930) Some approximate solutions of the boundary layer equations, Philosophical Magazine ARCRM, 131(12), pp. 865-896. [2]. Hartree, D. .R. (1937) On equations occurring in Falkner and Skan’s approximate treatment of the equations of boundary layer, Proc. Camb. Phil. Soc., 33, pp. 223-239. [3]. Ishak, A., Nazar, R. and Pop, I. (2007) Falkner-Skan equation for flow past a moving wedge with suction or injection, J. Appl. Math. Computing, 25(1-2), pp. 67-83. [4]. Williams, III J. C. and Rhyne, T. B. (1980) Boundary layer development on a wedge impulsively set into motion, SIAM J. Appl. Math., 38(2), pp. 215-224. [5]. Massoudi, M. (2001) Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge, Int. J. non-Linear Mechanics, 36(6), pp. 961-976. © 2014, IJMA. All Rights Reserved

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Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014.

[6]. Ibrahim, F. S. and Hassanien, I. A. (200) Influence of variable permeability on combined convection along a non-isothermal wedge in a saturated porous medium, Trans. Porous Media, 39(1), pp. 57-71.

[7]. Hassanien, I. A. (2003) Variable permeability effects on mixed convection along a vertical wedge embedded in a porous medium with variable surface heat flux, Appl. Math. Computing, 138(1), pp. 41-59.

[8]. Ece, M. C. (2005) Free convection flow about a wedge under mixed thermal boundary conditions and a magnetic field, Heat and Mass Transfer, 41, pp. 291-297.

[9]. Loganathan, P, Puvi-arasu, P. and Kandasamy, R. (2010) Local non-similarity solution to impact of

chemical reaction on MHD mixed convection heat and mass transfer flow over porous wedge in the presence of suction/injection, Appl. Math. Mech. (Eng. Ed.), 31(12), pp. 1517-1526. [10].Kassoy, D. R. and Zebib, A. (1975) Variable viscosity effects on the onset of convection in porous medium, Phys. Fluids, 18(12), pp. 1649-1651. [11]. Lai, F. C. and Kulacki, F. A. (1990) The effect of variable viscosity on convective heat transfer along a vertical surface in a saturated porous medium, Int. J. Heat Mass Transfer, 33(5), pp. 1028-1031. [12].Hady, F. M., Bakier, A. Y. and Gorla, R. S. R. (1996) Mixed convection boundary layer flow on a continuous flat plate with variable viscosity, Heat and Mass Transfer, 31(3), pp. 169-172. [13].Kafoussias, N. G., Rees, D. A. S. and Daskalakis, J. E. (1998) Numerical study of the combined free-forced convective laminar boundary layer flow past a vertical isothermal flat plate with temperature-dependent viscosity, Acta Mechanica, 127(1-4), pp. 39-50. [14].Hossain, M. A. and Munir, M. S. (2000) Mixed convection flow from a vertical flat plate with temperature dependent viscosity, Int. J. Thermal Science, 39(2), pp. 173-183. [15].Hossain, M. A., Khanafer, K. and Vafai, K. (2001) The effect of radiation on free convection flow of fluid with variable viscosity from a porous vertical plate, Int. J. Therm. Sci., 40, pp. 115-124. [16].Pantokratoras, A. (2002) Laminar free-convection over a vertical isothermal plate with uniform blowing or suction in water with variable physical properties, Int. J. Heat Mass Transfer, 45(5), pp. 963-977. [17].Pantokratoras, A. (2004) Further results on the variable viscosity on flow and heat transfer to a continuous moving flat plate, Int. J. Eng. Sci., 42(17-18), pp. 1891-1896. [18].Pantokratoras, A. (2005) Forced and mixed convection boundary layer flow along a flat plate with variable viscosity and variable Prandtl number: new results, Heat and Mass Transfer, 41, 1085-1094. [19].Chin, K. E., Nazar, R., Arifin, N. M. and Pop, I. (2007) Effect of variable viscosity on mixed convection boundary layer flow over a vertical surface embedded in a porous medium, Int. Communication in Heat and Mass Transfer, 34(4), pp. 464-473. [20].Makinde, O. D. (2012) Effect of variable viscosity on thermal boundary layer over a permeable flat plate with radiation and a convective surface boundary condition, J. Mechanical Science and Technology, 26(5), pp. 1615-1622. [21].Hossain, M. A., Munir, M. S., Hafiz, M. Z. and Takhar, H. S. (2000) Flow of a viscous incompressible fluid of temperature dependent viscosity past a permeable wedge with uniform surface heat flux, Heat and Mass Transfer, 36(4), pp. 333-341. [22].Hossain, M. A., Munir, M. S. and Rees, D. A. S. (2000) Flow of viscous incompressible fluid with temperature dependent viscosity and thermal conductivity past a permeable wedge with uniform surface heat flux, Int. J. Thermal Sciences, 39(6), pp. 635–644. [23].Hassanien,, I. A., Essawy, A. H. and Moursy, N. M. (2003) Variable viscosity and thermal conductivity effects on combined heat and mass transfer in mixed convection over a UHF/UMF wedge in porous media: the entire regime, Appl. Math. and Computing, 145(2-3), pp. 667-682. [24].Koh, J. C. Y. and Hartnett, J. P. (1961) Skin-friction and heat transfer for incompressible laminar flow over porous wedges with suction and variable wall temperature, Int. J. Heat Mass Transfer, 2(3), pp. 185-198. [25].Yih, K. A. (1998) Coupled heat and mass transfer in mixed convection over a wedge with variable wall temperature and concentration in porous media: The entire regime, Int. Communications in Heat Mass Transfer, 26(8), pp. 1145-1158. [26].Hossain, Md. Anwar, Bhowmick, S. and Gorla, R. S. R. (2006) Unsteady mixed-convection boundary layer flow along a symmetric wedge with variable surface temperature, Int. J. Eng. Sci., 44(10), pp. 607-620. [27].Soundalgekar, V.M., Takhar, H. S., Das, U. N., Deka, R. K. and Sarmah, A. (2004) Effect of variable viscosity on boundary layer flow along a continuously moving plate with variable surface temperature, Int. J. Heat and Mass Transfer, 40, pp. 421-424. [28].Salem, A. M. (2010) Temperature-dependent viscosity effects on non-Darcy hydrodynamic free convection heat transfer from a vertical wedge in porous media, Chinese Phys. Lett., 27, 064401. [29].Elbashbeshy, E. M. A. and Bazid, M. A. A. (2004) The effect of temperature-dependent viscosity on heat transfer over a continuous moving surface with variable internal heat generation, Appl. Math.Comput., 153(3), pp. 721-731. [30].White, F. M., Viscous Fluid Flow, 3rd ed., Mc Graw-Hill, New York, 2006. © 2014, IJMA. All Rights Reserved

154

1

2

Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014.

NOMENCLATURE ambient concentration surface concentration specific heat at constant pressure D mass diffusion coefficient g acceleration due to gravity G thermal Grashof number mass Grashof number m Falkner-Skan power-law parameter n temperature exponent Nux Local Nusselt number Prandtl number Local Reynolds number Rex c Schmidt number Local Sherwood number Shx ambient temperature surface temperature fluid temperature ( , ) dimensionless velocity components free stream velocity thermal diffusivity Hartree parameter 1 coefficient of thermal expansion coefficient of thermal expansion with concentration similarity variable 1 , 2 mixed convection parameters viscosity kinematic viscosity total wedge angle dimensionless species concentration viscosity parameter dimensionless temperature Table - 1 Comparison between the results of the present study (A) and those of Hady et al. [10] (B) for m = 0, n = 0 and Pr = 7.0 r

-10 -6 -2 -1 -10 -6 -2 -1

Values of

f (0),

A 1.4110 1.4330 1.5239 1.6213 4.2251 4.2705 4.4457 4.6085

1 1 1 1 5 5 5 5

(0) and -5.0 -5.0 -5.0 -5.0 -5 5 5 5 5 5

© 2014, IJMA. All Rights Reserved

f (0)

1

n 0.0 0.3 1.0 1.5 2.0 0 0.3 1.0 1.5 2.0

Pr 7.0 7.0 7.0 7.0 7.0 0.71 0.71 0.71 0.71 0.71

(0)

B 1.4343 1.4563 1.5474 1.6450 4.2397 4.2854 4.4621 4.6270

A 0.4089 0.4101 0.4146 0.4192 0.5473 0.5479 0.5498 0.5503

B 0.4039 0.4051 0.4098 0.4145 0.5468 0.5474 0.5493 0.5498

Table - 2 (0) for different values of Sc 617 617 617 617 617 0.62 0.62 0.62 0.62 0.62

f (0) 1.6386 1.6053 1.5580 1.5366 1.4893 1.8783 1.8382 1.7794 1.7518 1.7305

(0) 1.2291 1.4719 1.8631 2.0687 2.3485 0.5616 0.6838 0.8829 0.9878 1.0754

r,

n, Pr and Sc at m = 1/3

(0) 5.9964 5.9474 5.8764 5.8434 5.7924 0.5310 0.5271 0.5216 0.5191 0.5173 155

1

2

Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014. 1.4

r=2.0 r=5.0 r=15.0

1.2

f'( ), ( ), ( )

1 0.8

0.6

velocity --------- temperature concentration Pr=0.71, Sc=0.62,n=1.0

r=2.0 r=5.0 r=15.0

0.4 r=2.0 r=5.0 0.2

0

r=15.0

1

0.5

0

Figure - 1. Effect of

r

1.5

2.5

2

3

3.5

4

4.5

5

on velocity, temperature and concentration at m = 0.333 (

1

= 0.5).

1 0.9

r=-2.0 r=-5.0 r=-15.0

0.8

f'( ), ( ), ( )

0.7

velocity ----------temperature concentration Pr=7.0,Sc=617,n=1.0

r=-2.0 r=-5.0 r=-15.0

0.6 0.5 0.4

r=-2.0 r=-5.0 r=-15.0

0.3 0.2 0.1 0

0

Figure - 2. Effect of

0.5

r

1

1.5

2

2.5

3

3.5

4

4.5

5

on velocity, temperature and concentration m = 0.333 (

1

= 0.5).

1

= 0.5).

1.4 n=0 n=0.3 n=2.0

1.2

f'( ), ( ), ( )

1 velocity ---------temperature concentration Pr=0.71,Sc=0.62, r=2.0

0.8 n=0 n=0.3 n=2.0

0.6

0.4 n=0 n=0.3

0.2

n=2.0 0

0

Figure - 3. Effect of © 2014, IJMA. All Rights Reserved

0.5

r

1

1.5

2

2.5

3

3.5

4

4.5

5

on velocity, temperature and concentration m = 0.333 (

156

1

2

Rudra Kanta Deka* and Sarat Sharma */ Effects Of Variable Viscosity on Mixed Convection Heat and Mass Transfer Past A Wedge with Variable Temperature/ IJMA- 5(5), May-2014. 1 0.9 0.8 0.7

f'( )

0.6 1= -0.0850(m= -0.04076)

0.5

1= -0.0900(m= -0.04306) crit= -0.1044(m= -0.04961)

0.4 0.3

velocity Pr=7.0,Sc=617.0, r= -5.0, n=0.5

0.2 0.1 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure - 4. Effect of m on velocity profiles

1.2

1

0.8 1= -0.0400(m= -0.0196) 1= -0.0450(m= -0.0220)

0.6 f'( )

crit= -0.0477(m= -0.0232) 0.4 velocity Pr=0.71,Sc=0.62, r=5, n=1.0

0.2

0

-0.2

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure - 5. Effect of m on velocity profiles Source of support: UGC, New Delhi, India, Conflict of interest: None Declared [Copy right © 2014 This is an Open Access article distributed under the terms of the International Journal of Mathematical Archive (IJMA), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.]

© 2014, IJMA. All Rights Reserved

157

American Journal of Computational and Applied M athematics 2013, 3(2): 74-80 DOI: 10.5923/j.ajcam.20130302.03

Magnetohydrodynamic Mixed Convection Flow Past a Wedge Under Variable Temperature and Chemical Reaction Rudra Kanta Deka1, Sarat Sharma2,* 1 M athematics Department, Gauhati University, Guwahati, 7840014, India Department of M athematical Sciences, Chaiduar College, Gohpur, 784168, India

2

Abstract Steady laminar boundary layer free-forced flow of a Newtonian flu id past a wedge shaped surface is studied. n The temperature of the wedge surface is assumed to vary as x and a first order chemical reaction of the fluid species is considered. The governing boundary layer equations are transformed by the Falknar-Skan transformations and proportionate numerical results are tabulated for the Skin friction, Nusselt number and Sherwood number. Graphical results for the variation in the velocity, temperature and concentration profiles for various values of the temperature exponent, chemical reaction parameter and Sch midt number are p resented.

Keywords MHD, Wedge, Mixed Convection, Chemical Reaction

1. Introduction Bo und ary lay er flo w ov er a h eat ed vert ical su rface emerges in a variety of engineering applicat ions and it has been the subject of extensive research. When the fluid is electrically conducting and also exposed to a magnetic field, flo w and temperatu re fields are gov erned both by t he bu oyan cy and th e Lo ren z fo rces . M ost o f the stu d ies concerning such flows were taken up as steady flows and are considered either for purely free convection or purely forced convection regime only. In dealing with forced convection flow, it is customary to ignore the effects of free convection. Similarly we assume the forced convection as neglig ib le while dealing with free convection flo w. The error involved in ign o ring n at u ral con vect io n wh ile stu dy in g fo rced convect ion is neg lig ib le at h igh velocit ies but may be considerab le at low velocit ies. Situat ions may arise for which free and forced convection effects are co mparable, in wh ich case it is inapprop riate to neg lect either p rocess. Therefore it is desirab le to have a criterion to assess the relative magnitude of natural convection in the presence of forced convection flo ws. Such flo ws situations where both free and forced convection effects are of co mparable order belong to the mixed convection regime. In several practical applicat ions o f heat t ransfer theory to the vert ical plate prob lems th ere exists sign ificant temp erature d ifferen ce between the surface of the hot plate and the free stream. This * Corresponding author: sarat_sharma@si fy.com (Sarat Sharma) Published online at http://journal.sapub.org/ajcam Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved

temperature difference cause density gradients in the fluid med iu m and in the presence of a gravitational body force, free convection effects become important. It has generally been recognized that,

Gr where Gr is the Grashof Re 2

number and Re the Reynolds number is the governing parameter for the laminar boundary layer forced-free mixed convection flow which represents the ratio of the buoyancy forces to the inertial forces inside the boundary layer. Forced convection exists when

0

which occurs at the leading

edge and the free convection limit can be reached if becomes large. We know fro m authoritative work in heat transfer[1] that free convection is negligib le if

1 and

1. Hence co mbined forced convection is negligible if free and forced (or mixed) convection regime is generally 1 . Although the flows with only one o f one for which the two effects involved will have a self similar character, such situations will mathemat ically lead to ordinary differential equations and are easy to solve. But as soon as the two effects occur together, boundary value problem involving partial d ifferential equations will arise and are not easy to solve by conventional procedures. The physical explanation of the co mplexit ies is that the two effects act differently with respect to the characteristic length l and any combination means that the characteristic length is introduced into the problem (i.e., the length fro m the leading edge to the separation position).If both the effects are to be considered, pure forced convection always dominates for l l Most of the works concerning external flows past a vertical

American Journal of Computational and Applied M athematics 2013, 3(2): 74-80

surface are taken up as steady flows and they are either considered for free or forced convection regimes only. However free-fo rced or popularly known as mixed convection flows received considerable attention in the late 1970s to 1980s and numerous literature rev iews are availab le. Soundalgekar et al[2] studied the flow past a semi-infin ite vertical p late under variab le temperature. Wilks[3] and Raju et al[4] have shown several formulations under which a flow can be classified as belonging to the mixed convection regime and values of the mixed convection parameters were used by them. In nature it is rather impossible to find pure flu id unless special efforts are made to obtain it.The most common flu ids like water, air etc. is contaminated with impurities like CO2 , C6 H 6 , H 2 SO4 etc. and generally we have to consider presence of such foreign masses while studying flows past different bodies. In such a case the density difference in the fluid is caused by material constitution in addition to temperature differences. The common examp le of such a flo w is the at mospheric flow wh ich is driven appreciably by both temperature and concentration differences. When such contaminant is present in the flu id under consideration there does occur some chemical reaction e.g. air and benzene react chemically, so also water and sulfuric acid. During such chemical reactions, there is always generation of heat. But when the foreign mass present in the flu id at very low level, we can assume a first order chemical reaction and the heat generated due to chemical reaction can be very negligible. Several authors have done significant works by taking into account a first order chemical reaction on flow past vertical surfaces. Das et al[5] studied the effect of mass transfer and chemical react ion on the flow past a vertical p late. Bala[6] considered a porous, moving vertical plate under variable suction and chemical reaction. Mahmoud[7] studied mixed convection flow under variable viscosity and chemical reaction effects. Flo w past a wedge or ‘wedge flow’ belongs to a special class of boundary layer flo w prob lems where the shape of the wedge affects the boundary layer mainly by determining the pressure gradient along the surface of the wedge. Flow past a wedge of angle 1 yields a different pressure profile for each value of

1,

thereby offering insight into boundary

layer behavior in a number of situations. Two of the special cases are planar stagnation flow ( 1 1) and parallel flow past a flat p late

(

1

0) .Martin and Boyd[8] studied the

flow past a wedge with slip boundary conditions using Falkner-Skan transformations together with a fin ite difference method. Mukhopadhyay[9] studied the effect of radiation and variab le viscosity on the flow past a wedge using a special form o f scaled Lie group of t ransformat ions. Nazar and Pop[10] considered the case of a moving wedge using the Falkner-Skan transformations. Ece[11] used the Thomas algorith m to solve the similar boundary layer equations of flow past a wedge under mixed thermal boundary conditions. Hsu et. al.[12] utilized the series

75

expansion method to study the flow of a second grade fluid past a wedge. Massoudi[13], Mureithi and Mason[14], Loganathan et. al.[15] employed the u Non-similarity method to solve wedge flow problems. Sattar[16] considered the unsteady flo w past a wedge by the Local similarity method. Pop and Watanabe[17], Kandasamy et al.[18], Kandasami et al.[19], Kandasami et al.[20] studied the flow situations past a wedge by introducing a pair of variables ( x, ) instead of or a dimensionless distance along the plate of the wedge. Williams and Rhyne[21] have used a group of transformations to encompass both the situations of small and large t ime in the unsteady flow past a wedge by simp le modification of the Falkner and Skan transformations in solving the wedge equations employing the Thomas algorith m. Ku mari and Nath[22] have successfully implemented these transformations in the study of unsteady flow past a vertical p late and shown their efficacy. Despite numerous solution techniques and transformation methods available for the governing boundary layer equations arising in the flow past a wedge, the Falkner-Skan transformations remains one of the most chosen and used for their simplicity and computational ease. All the above cited works concerning flo w past a wedge were carried out by assuming a constant temperature of the wedge surface. Here we propose to undertake this study by adopting the popular Falkner-Skan transformations to solve our problem by considering a variable temperature along the surface of the wedge under a first order chemical react ion. We will also study the variation in velocity, temperature and concentration profiles for various values of Sch midt numbers.

2. Formulation of the Problem Let us consider the steady two-dimensional hydro magnetic mixed convection boundary layer flo w past the surface of a wedge. We formu late the problem in a body fixed rectangular coordinate system in which x is measured along the surface of the wedge, from the apex, and y is measured normal to the wedge surface. The flow configuration is shown in fig1. The fluid is assumed to be Newtonian and its physical property variations due to temperature is limited to density only. A uniform t ransverse magnetic field o f strength

B0 is applied parallel to the

y-axis and induced magnetic field is neglected. The concentration of diffusing species is very small in comparison to other chemical species and far fro m the surface of the wedge the concentration is assumed infinitesimally s mall and hence the Dufour and Soret effects are neglected. Also there is first order chemical reaction taking place in the flo w. Assuming Boussinesq approximation, the boundary layer governing equations of mo mentu m, energy and diffusion for the flow p roblem are:

u

u

x

y

0

(1)

76

Rudra Kanta Deka et al.: M agnetohydrodynamic M ixed Convection Flow Past a Wedge Under Variable Temperature and Chemical Reaction

u x

u [g

2

u y

v

T T

u y2

B0 T x

u u

C x

v

*

g

m

du dx

C C ]sin

exist for

2

u u

(2) 2

T y

0.091,

0.199 is a limit ing case, no solution

1

to the Falkner-Skan equation for accelerated forward flow

2

v

C y

u

T y2

(3)

k1 C C

(4)

0.199 which corresponds to the velocity

1

profile with vanishing wall shear stress (separation). Nu merical results fro m Hart ree are p lotted for various wedge angles in[1]. The velocity co mponents are given in terms of the stream by the relations: function

u ( x, y )

2

C y2

D

and v ( x, y )

y

(8)

x

The boundary conditions are:

u ( x, y ) v( x, y ) 0, T ( x, y ) T , C ( x, y ) C 0 For t 0, u x, 0 v( x, 0) 0, u ( x, ) u

for t

T ( x, ) T , T ( x, 0) Tw ( x) bx n , C ( x, ) C ,

C ( x, 0) Cw , b

0, n 0

(5)

Here x and y are respectively, the distances along and perpendicular to the surface of the wedge, u and v are the velocity co mponents along x and y d irections respectively, t is the kinematic viscosity, g is the is the time, acceleration due to gravity, is the volumetric coefficient * is the volumetric coefficient of of thermal expansion, is the electrical conductivity, concentration, is the is the thermal diffusivity, D is the effective density, diffusion coefficient and the subscripts w conditions at the surface of the wedge and in the free stream respectively. We now introduce the follo wing Falkner-Skan transformations for wedge flo w as given in[1] and[23] with

y

Figure 1. Flow Configuration

The continuity equation (1) is readily satisfied by (8) and using (6) and (7), equations (2)-(4) and the boundary conditions (5) are reduced to:

f '''

ff ''

m 1u 2 , ( x, y ) xu f ( ), 2 x m 1 T ( x, y ) T (Tw ( x) T ) ( ), (6) C ( x, y ) C (Cw ( x) C ) ( )

(Pr) ( Sc)

where the potential flow velocity can be written as

u ( x)

ax m ,

Where a is a constant and

2m 1 m

1

1

(7)

1

for a total

Here

G

number,

G*

exhibit no point of inflexion but for retarded flows

number,

Re

1

0) there exists a point of inflexion. The case

1

]sin

2

2 M ( f ' 1) 0 m 1 2n ' '' f ' f 0 m 1 2 '' f ' 0 m 1

1

1

2 (9) (10) (11)

f (0) f ' (0) 0, (0) 1, (0) 1, f ' ( ) 1, ( ) 0, ( ) 0

of the wedge. Two important cases are the flat angle plate flow (m=0) and the stagnation-point flow (m=1). For accelerated flo ws ( m 0, 1 0) velocity profiles

(m 0,

2 [ m 1

[1 ( f ' ) 2 ]

The boundary conditions for equations (9)-(11) are:

is the Hartree pressure

gradient parameter that corresponds to

1

g (Tw T ) x 2

g

*

u x

is the thermal Grashof

(Tw T ) x 3 2

(12)

3

is the species Grashof

is the Reynolds number,

1

G is Re 2

American Journal of Computational and Applied M athematics 2013, 3(2): 74-80

the buoyancy parameter due to temperature,

Pr

the buoyancy parameter due to mass,

Sc

Prandtl nu mber,

M

B0 2 x u

k1 x u

the

is the

is the Sch midt number,

D

is

2

G* is Re 2

magnetic

parameter

concentration

and

primes

interest to our study are the Skin friction

Cf

which

shear

given

u y

w

Nu

qw

by

wall

stress

, the Local Nusselt number

T y

given

by

the

heat

flu x

and the Local Sherwood number

xM w D Cw C

Mw

D

C y

given

by

the

mass

flu x

. y 0

In view of (12), the Skin Friction coefficient

C f reduces

1 1 m 1 C f Re 2 2 2 the Nusselt number Nu reduces to

Nu Re

m 1 2

1 2

the Sherwood number Sh reduces to

(Schlichting)

(Present work)

f '' 0

f '' 0

0.000 0.587 0.775 0.928

0.0000 0.5870 0.7748 0.9277

-0.090 0.053 0.176 0.333

We propose to solve equations (9) – (11) under the boundary conditions (12) using the continuation method with the help of Matlab7 which imp lements a fin ite difference code. Continuation method is an imp roved shooting method which reduces the chance of carrying on the computational errors due to poor guesses of the unknown values and ensures greater accuracy. In continuation method firstly we find the unknown boundary conditions for =1 for a guess and put those values for =2 and carry on the process until the values converge to fixed values. Finally the converged initial values are put as the initial values in solving the boundary value problem for the entire range [0, 6] .We have tabulated the values in Table2-Table5

f '' 0

'

,

0

'

and

0

which

f 0

0

(13)

(14)

are

proportional to the Skin friction, Nusselt number and Sherwood number respectively under magnetic, plate temperature variation, chemical reaction and Sch midt number variat ion. We have kept other parameters as constant while analy zing the effects of one parameter wh ich will highlight the changes (if any) more clearly. The velocity, temperature and concentration profiles are presented in Figure2-Figure5. Table 2. Flow separation for various values of magnetic, chemical reaction and temperature exponent

to 1 2

(15)

3. Numerical Solution

of

y 0

Sh

-0.199 0.1 0.3 0.5

is the denote

y 0

xqw k Tw T

k

the

0

Table 1. comparison with standard work

m

2 w u 2

1 2

We have compared our results with a trivial solution given in[1] for the skin-friction to the Falkner-Skan equation for various wedge angles by setting M 0, C 0, 0, n 0 in our code which are in excellent agreement (Table1).

and

Knowing the velocity field, the physical quantities of

is

Sh Re

is the chemical reaction parameter.

dimensionless

m 1 2

77

M 0.5 0.7 0 1.0 0.4

0.2 0.1 0.5 0.4 0.1

n

Sc

Pr

1

2

1.0 0.5 1.5 1.5 0

0.24 0.62 0.78 1.002 1.002

0.71 0.71 0.71 0.71 0.71

0.5 0.4 0.5 1.0 1.0

0.5 0.4 0.5 1.0 1.0

1

-0.3364 -0.5756 0.0686 0.1696 -0.1580

78

Rudra Kanta Deka et al.: M agnetohydrodynamic M ixed Convection Flow Past a Wedge Under Variable Temperature and Chemical Reaction

Table 4. values of skin-friction, heat flux and mass flux for various Schmidt numbers Sc

n

0.24 0.62 0.78 1.002

1 1 1 1

0.5 0.5 0.5 0.5

M

f '' 0

'

3 3 3 3

2.6868 2.6805 2.6780 2.6751

0.9331 0.9308 0.9300 0.9290

0

'

0

0.1141 0.0768 0.0644 0.0443

Figure 2. Magnetic effect on velocity, temperature and concentration profile

Figure 4. Effect of chemical reaction on velocity, temperature and concentration profile

Figure 3. Effect of variable temperature on velocity, temperature and concentration profile Table 3. values of skin-friction, heat flux and mass flux for magnetic, variation in temperature and chemical reaction M

n

0 2 5 10 1 1 1 1 3 3 3 3

1 1 1 1 0 1 2 3 1 1 1 1

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0 0.3 1 1.5

f '' 0

'

1.6283 2.3820 3.1944 4.2116 2.0747 2.0401 2.0223 2.0106 2.6524 2.6667 2.7685 3.2817

0.8588 0.9136 0.9561 0.9944 0.5566 0.8911 1.0954 1.2488 0.9252 0.9280 0.9491 1.0438

0

'

0

0.0261 0.0653 0.0930 0.1160 0.0593 0.0496 0.0452 0.0426 0.5340 0.2922 -1.1218 -7.8680

Figure 5. Effect of Schmidt number on velocity, temperature and concentration profile

One of the frequently encountered problems in ‘wedge flow’ is the flow separation. It has several undesirable effects in so far as it leads to an increase in the drag on a body

American Journal of Computational and Applied M athematics 2013, 3(2): 74-80

immersed in the flow and adversely affects the heat transfer fro m the surface of the body. We have calcu lated the value of 1 for different values of 1 , 2 , M , , Sc, n (Table2), for which separation of flow occurs. The result shows that flow separation can effectively be controlled by introduction of magnetic, chemical reaction and plate temperature variation effects into the flow field.

4. Results and Discussion In our effort to analy ze the effects of t ransverse magnetic field on the velocity, temperature and concentration profiles, we have fixed the values of the plate temperature exponent at 1 and the chemical reaction parameter at 0.5. The values of

f '' 0 ,

'

0 and

'

0 are tabulated in Table3.

The velocity, temperature and concentration profiles for the flow are presented Figure2. It is seen from Figure2 that magnetic field induces considerable influence on the flow field. Increasing magnetic field strength has a enhancing effect on the velocity of the fluid under consideration. Magnetic field tends to decrease the temperature and concentration at equal proportion for the flo w. In Tab le3, values of skin friction, rate of heat flu x and rate of mass flu x show a increasing trend with the increase of magnetic field strength. Effects of variation in the plate temperature upon the velocity, temperature and concentration of the fluid is computed by keeping the magnetic field strength as 1 and chemical reaction parameter as 0.5. The values of f '

0

and

'

0

''

0 ,

are tabulated in Table3. The

velocity and temperature profiles for the flow are presented for co mparison in Figure3. Fo r constant plate temperature the velocity of the flo w is at a high level. But as we increase the plate temperature exponent the velocity becomes lesser gradually. Fo r temperature, there is a uniform lowering effect for the flow with the increase of plate temperature but for concentration the change though minimally lo wering, is not significant. Fro m Table3, it is seen that increase in plate temperature reduces the skin friction and mass flu x of the flow but heat flu x increases. To study the chemical reaction effects we have taken the magnetic field strength as 3 and the plate temperature exponent as 1. The values of f '

''

0 ,

'

0 and

Effects of variation in the species of diffusing species in terms of their Sch midt numbers is studied by taking the magnetic field strength as 1, the p late temperature exponent as 1 and for a low level o f chemical react ion 0.5 . The values of f

''

0 ,

'

0 and

'

0 are tabulated in

Table4.The velocity, temperature and concentration profiles for flow are presented in Figure5. Lighter diffusing species has a higher velocity and concentration compared to the heavier species. Temperature is not changed with a change in the Schmidt number.Table4 shows that in the presence of nominal chemical react ion, the skin frict ion, heat flu x and mass flu x gradually decreases with increase in Sch midt numbers.

5. Conclusions (i) Magnetic field accelerates the velocity and lowers the temperature and concentration. (ii) Skin-friction, heat flu x and mass flu x increase due to magnetic field. (iii) Increasing plate temperature lowers the velocity and temperature considerably but a minor lowering effect for concentration is found. (iv) Skin-friction and mass flu x decrease but heat flux increases with plate temperature. (v) At low level of chemical reaction velocity, temper ature and concentration increase gradually but for higher level the reverse is true. (vi) For lo w level of chemical reaction skin-friction, heat flu x increase but mass flu x decreases. For higher level the reverse is true. (vii) For low Sch midt number the velocity and concentration is higher but temperature remains same. (viii) In the p resence of nominal chemical reaction the skin-frict ion, heat flu x and mass flu x increases with increase in Sch midt number.

6. Nomenclature Gr Grashof number, Re Reynolds number, coefficient of thermal expansion , *

0 are tabulated in Tab le3.The velocity, temperature

and concentration profiles for flow are presented in Figure4. Presence of a first order chemical reaction at low level increases the velocity and concentration of the fluid but temperature co mes down. Table3 shows that in the absence of chemical react ion, the rate of skin friction is comparatively more than the heat and mass flu x. For small change in the chemical reaction strength, skin friction and heat flu x increases but mass flu x decreases. For higher chemical reaction level concentration level increases considerably.

79

1

G

coefficient of thermal expansion with concentration , stream function, Hartree parameter, thermal Grashof number,

*

G B0

mass Grashof number, magnetic field strength, similarity variable,

u , v dimension less velocity components,

u free stream flu id velocity,

80

Rudra Kanta Deka et al.: M agnetohydrodynamic M ixed Convection Flow Past a Wedge Under Variable Temperature and Chemical Reaction

T

free stream temperature,

Tw temperature at the plate, M

T

C

[10] Nazar A. I. R. and Pop, I., 2007, Falkner-Skan equation for flow past a moving wedge with suction or injection, J of Applied M ath. and Computing,25(1-2), 67-83

dimensionless temperature, dimensionless species concentration,

[11] Ece M . C., 2005, Free convection flow about a wedge under mixed thermal boundary conditions and a magnetic field, Heat and M ass Transfer, 41, 291-297

species concentration in the free stream,

g acceleration due to gravity, Pr Prandtl nu mber, chemical reaction parameter,

Sc Sch midt nu mber, kinematic viscosity,

k1 chemical reaction constant, thermal diffusivity, D mass diffusion coefficient, n temperature exponent

REFERENCES [1]

H Schlichting and K Gersten, Boundary Layer Theory,8th ed., Springer, 1999

[2]

Soundalgekar, V. M ., Takhar H. S. and Vighnesam,N. V., 1988, Combined free and forced convection flow past a semi-infinite vertical plate with variable surface temperature, Nuclear Engineering and Design,110, 95-98

[3]

Wilks, Graham, 1973,Combined forced and free convection flow on vertical surfaces, Int. J. of Heat and M ass Transfer, 16, 1958-1964

[4]

Raju, M . S., Liu, X. Q. and Law, C. K., 1984, A formulation of combined forced and free convection past horizontal and vertical surfaces, Int. J. of Heat and M ass Transfer, 27(12), 2215-2224

[6]

[7]

[8]

M ukhopadhyay, S., 2009, Effect of radiation and variable fluid viscosity on flow and heat transfer along a symmetric wedge, Journal of Applied Fluid M echanics., 2(2), 29-34

magnetic parameter, dimensionless temperature, density,

Cw concentration at the wall,

[5]

[9]

Das, U. N., Deka, R. K. and Soundalgekar, V. M ., 1998, Effect of mass transfer on flow past an impulsively started infinite vertical plate with chemical reaction, The Bulletin, GUM A,5, 13-20 Bala A. Jyothi and Varma Vijaya Kumar, 2011, Unsteady MHD heat and mass transfer flow past a semi-infinite vertical porous moving plate with variable suction in the presence of heat generation and homogeneous chemical reaction, Int. J. of Appl. M ath. And M ech.,7(7), 20-24 M ostafa, A. A. M ahmoud, 2007, A note on variable viscosity and chemical reaction effects on mixed convection heat and mass transfer along a semi-infinite vertical plate, M athematical Problems in Engineering, 2007,1-7 M artin M ichael J. and Boyd Iain D., 2010, Falkner-Skan flow over a wedge with slip boundary conditions, Journal of Thermo physics and Heat Transfer.,24(2), 263-270

[12] Hsu Cheng-Hsing, Chen Chii-Sheng and Teng Jyh-Tong, 1997, Temperature and flow fields for the flow of a second grade fluid past a wedge, Int. J. Non-Linear M echanics, 32(5), 933-946 [13] M assoudi, M , 2001, Local non-similarity solutions for the flow of a non-Newtonian fluid over a wedge, Int. J. of Non-Linear M echanics, 36(6), 961-976 [14] M ureithi E.W. and M ason D. P., 2010, Local non-similarity solutions for a forced-free boundary layer flow with viscous dissipation, M athematical and Computational Applications, 15(4), 558-573 [15] Loganathan P., Puvi-arasu P. and Kandasamy R., 2010, Local non-similarity solution to impact of chemical reaction on MHD mixed convection heat and mass transfer flow over porous wedge in the presence of suction/injection, Applied M athematics and M echanics (Eng. ed.),31(12), 1517-1526 [16] Sattar M . A., 2011, A local similarity transformation for the unsteady two dimensional hydrodynamic boundary layer equations of a flow past a wedge, Int. J. of Appl. M ath and M ech., 7(1), 15-28 [17] Pop H. and Watanabe T., 1992, The effects of suction or injection in boundary layer flow and heat transfer on a continuous moving surface, Technische M echanik,13, 49-54 [18] Kandasami R., Hashim I., M uhaimin and Ruhaila, 2007, Effect of variable viscosity, heat and mass transfer on nonlinear mixed convection flow over a porous wedge with heat radiation in the presence of homogeneous chemical reaction, ARPN J.of Engineering and Applied Sciences, 2( 5), 44-53 [19] Kandasami R., Hashim I., M uhaimin and Seripah, 2007, Nonlinear M HD mixed convection flow and Heat and mass transfer of first order chemical reaction over a wedge with variable viscosity in the presence of suction or injection, Theoretical Applied M echanics, 34(2), 111-134 [20] Kandasamy R., Hashim I., Khamis A. B. and M uhaimin I., 2007, Combined heat and mass transfer in MHD free convection from a wedge with ohmic heating and viscous dissipation in the presence of suction or injection, Iranian Journal of Science and Technology, Trans. A, 31(A2), 152-162 [21] Williams III J. C., Rhyne T. B., 1980, Boundary layer development on a wedge impulsively set into motion, SIAM J Appl. M ath, 38(2), 215-224 [22] Kumari M ., Nath G., 1997, Development of mixed convection flow over a vertical plate due to an impulsive motion, Heat and M ass Transfer, 40(11), 823-828 [23] W M Deen, Analysis of Transport Phenomena, Oxford University Press, New York, 1998

Effect of variable viscosity on a nanofluid over a porous wedge Rudra Kanta Deka1, Mwblib Basumatary2 and Ashish Paul3,* 1

Department of Mathematics, Gauhati University, Guwahati-781014, Assam, India 2

Department of Mathematics, Lumding College, Lumding-782447, Assam, India

3

Department of Mathematics, Cotton College State University, Guwahati-781001, Assam, India *Email: [email protected]

Abstract: In this paper we have investigated the effects of variable viscosity on a water-based nano-fluid over a static porous wedge with variable temperature. Two different types of nanofluids; Copper-water and alumina-water nanofluids are considered. Dual solutions are obtained for negative pressure gradient. The physical aspects of nano-fluid flow and heat transfer characteristics are highlighted and discussed. It is observed that inclusion of nano-particles and temperature dependent viscosity delays the flow separation. Key words: Nano-fluid, Variable viscosity, Porous Wedge, Variable Temperature.

1.

Introduction: Studies on nanofluids have been the interests of scientists and engineers due to emerging technical and biomedical

applications. In general conventional fluids have poor heat transfer properties compared to solids. The thermal conductivity of fluids plays an important role on the heat transfer of fluids. Therefore, thermal conductivity of base fluids are increased by suspending nano-particles in fluids. The  term  “nanofluid”  was  first  introduced by Choi [1] by dispersing nano-particles in base fluids such as water, ethylene glycol and propylene glycol. There are several numerical and experimental studies on nano-fluids. For example, Abu-Nada and Oztop [2], Tiwari and Das [3] and Maïga et al. [4] investigated the heat transfer characteristics in nanofluids. They found that the presence of the nano-particles increases the thermal conductivity of the fluid appreciably and enhances the heat transfer characteristics. The comprehensive references on nano-fluids are cited in the book by Das et al. [5] as well as in papers by Daungthongsuk and Wongwises [6], Trisaksri and Wongwises [7], Wang and Mujumdar [8-10] and Kakaçand Pramuanjaroenkij [11]. Makinde and Aziz [12] and Khan and Pop [13] studied nanofluids over stretching sheet. Different works on nanofluids have been done by Yacob et al. [14], Nazar et al. [15] and Arifin et al. [16], Rohni et al. [17, 18, 19], Rosca et al. [20], Bachok et al. [21, 22], Tr𝑖̂mbitas et al. [23], Patrulescu et al. [24], Mahian et al. [25] and Loganathan and Vimala [26]. Bachok et al. [27]

investigated the flow of nanofluid over a moving porous surface, while Tr𝑖̂imbitas et al. [28] studied unsteady boundary layer flow of Cu-water nanofluid with heat flux at the surface of vertical plate. They found that dual solutions exist when the surface and the free stream move in opposite directions. In all these studies, fluid viscosity was assumed to be constant. It is well known that the viscosity of fluid change significantly, when temperature changes. Therefore, it is necessary to take this variation of viscosity with temperature into account. Many researchers investigated the effects of variable properties for fluid viscosity on flow and heat transfer over a surface, for example, Mukhopadhyay et al. [29] studied the flow of an incompressible viscous fluid past a continuously moving semi-infinite plate along with variable viscosity and variable temperature. Vajraveluet et al. [30] considered the effect of the temperature dependent fluid viscosity on nanofluids over a semi-infinite flat surface in the presence of viscous dissipation. The aim of this work is to study the effect of variable viscosity on the boundary layer flow of Cu-H20 and Al2O3H2O nanofluids over a heated porous wedge. The analyses are focused on the case of the occurrence of flow separation and other fluid characteristics. The governing partial differential equations are transformed into ordinary differential equation after a suitable transformation and solved the resulting equations by shooting method with Runge-KuttaFehlberg integration scheme. Effects of different parameters on the fluid characteristics are studied through figures and tables.

2.

Mathematical formulations: We consider a steady two-dimensional boundary layer flow past a porous wedge in two water-based nanofluids;

Cu-water and Al2O3-water nanofluids. It is also assumed that the base fluid and the nano-particles are in thermal equilibrium and no slip occurs between them. The thermophysical properties of the fluid and nano-particles are given in Table 1. We consider a Cartesian coordinate system (x, y) as shown in figure 1, where x and y are the coordinates measured along the surface of the wedge and normal to it respectively. Further, it is assumed that the temperature of the surface is 𝑇𝑤 and that of the ambient 𝑇∞ . The wedge is permeable with a lateral mass flux 𝑉𝑤 and the outer flow velocity 𝑈(𝑥) = 𝑈∞ 𝑥 𝑚 , where m is constant for a total angle

of the wedge. Under these assumptions and using the nanofluid model proposed by Tiwari

and Das [3], the conservation of mass and momentum equations for the nanofluid are respectively given by,

𝜕𝑢 𝜕𝑥

𝑢 𝑢

𝜕v

+

𝜕𝑢 𝜕𝑥 𝜕𝑇 𝜕𝑥

(1)

=0

𝜕𝑦

+v +v

𝜕𝑢 𝜕𝑦 𝜕𝑇 𝜕𝑦

=𝑈

𝑑𝑈 𝑑𝑥

= 𝛼𝑛𝑓

+

1

𝜕

𝜌 𝑛𝑓 𝜕𝑦

𝜇𝑛𝑓

𝜕𝑢

(2)

𝜕𝑦

𝜕2𝑇

(3)

𝜕𝑦 2

along with the boundary conditions at 𝑦 = 0

(4a)

𝑢 = 𝑈(𝑥) = 𝑈∞ 𝑥 𝑚 , 𝑇 → 𝑇∞ as 𝑦 → ∞

(4b)

𝑢 = 0, v = 𝑉𝑤 , 𝑇 = 𝑇𝑤 (𝑥)

where u and v are velocity components along x and y directions, respectively, while

nf

and

nf

are the thermal diffusivity

and viscosity of the nanofluid. The density and viscosity of the nanofluid are taken as, (5)

𝜌𝑛𝑓 = (1 − 𝜙)𝜌𝑓∞ + 𝜙𝜌𝑠 and 𝜇

𝑓 𝜇𝑛𝑓 = (1−𝜙) 2.5

(6)

where 𝜙 is the solid volume fraction of nano-particles. Here, 𝜇𝑓 is the coefficient of fluid viscosity varies as an inverse function of temperature given by Mukhopadhyay et al. [18] by the following relation: 1 𝜇𝑓

=

1

[1 + 𝛾(𝑇 − 𝑇∞ )] = 𝑎(𝑇 − 𝑇𝑟 )

𝜇 𝑓∞

(7)

where 𝑎=

𝛾 𝜇 𝑓∞

, 𝑇𝑟 = 𝑇∞ −

1

(8)

𝛾

In equation (8) above, both 𝑎and 𝑇𝑟 are constants and their values depend on the reference state and 𝛾, a thermal property of the fluid. In general, 𝑎 > 0 for liquids, and 𝑎 < 0 for gasesand 𝜇𝑓∞ is the coefficient of viscosity of base fluid far away from the surface of the wedge. Thermal diffusivity, heat capacitance and thermal conductivity of the nanofluid are defined as, 𝛼𝑛𝑓 = 𝜌𝐶𝑝

𝑛𝑓

𝐾𝑛𝑓 𝜌 𝐶𝑝

(9) 𝑛𝑓

= (1 − 𝜙) 𝜌𝐶𝑝

𝐾𝑛𝑓 = 𝐾𝑓∞

𝑓∞

+ 𝜙 𝜌𝐶𝑝

𝐾𝑠 +2𝐾𝑓 ∞ −2𝜙 𝐾𝑓 ∞ −𝐾𝑠 𝐾𝑠 +2𝐾𝑓 ∞ +𝜙 𝐾𝑓 ∞ −𝐾𝑠

𝑠

(10) (11)

Here suffices nf, f∞ and s represent respectively the thermo-physical properties of the nanofluids, base fluid and the nano-solid particles. Now we introduce the following similarity transformations: 𝜓 = 2𝜈𝑓∞ 𝑈𝑥 ⁄(𝑚 + 1)

1 ⁄2

𝑓(𝜂) = 2𝜈𝑓∞ 𝑈∞ ⁄(𝑚 + 1) 1 ⁄2

1 ⁄2 (𝑚 +1)⁄2

𝑥

𝑓(𝜂) (12)

1 ⁄2 (𝑚 −1)⁄2

𝜂 = 𝑦 𝑈(𝑚 + 1)⁄ 2𝜈𝑓∞ 𝑥 = 𝑦 𝑈∞ (𝑚 + 1)⁄ 2𝜈𝑓∞ 𝑥 𝜃(𝜂) = (𝑇 − 𝑇∞ )⁄(𝑇𝑤 − 𝑇∞ ) , (𝑇 − 𝑇𝑟 ) = (𝜃 − 𝜃𝑟 )(𝑇𝑤 − 𝑇∞ ) where 𝜂 is the similarity variable, 𝜓(𝑥, 𝑦) is the stream function defined by 𝑢 =

𝜕𝜓 𝜕𝑦

and v = −

𝜕𝜓 𝜕𝑥

. Also, 𝜈𝑓∞ is the

kinematic viscosity of the fluid far away from the surface and 𝜃𝑟 = (𝑇𝑟 − 𝑇∞ )⁄(𝑇𝑤 − 𝑇∞ ) = − 1⁄γ(𝑇𝑤 − 𝑇∞ ) is the variable viscosity parameter. The wall temperature of the wedge is assumed to have the form: 𝑇𝑤 (𝑥) = 𝑇∞ + 𝐴𝑥 𝑁

(13)

where A and N are constants. On substituting (12) and (13) into equations (2) and (3), we obtain the following non-linear ordinary differential equations, 𝑓 ′′′ − (1 − 𝜙)2.5 1 − 𝜙 + 𝜙

𝜃 ′′ − 𝑃𝑟 1 − 𝜙 + 𝜙

𝜌𝐶𝑝 𝜌𝐶𝑝

𝑠 𝑓∞

𝜌𝑠 𝜌𝑓∞

𝐾𝑓 ∞ 𝐾𝑛𝑓

1−

𝜃 𝜃𝑟

𝑓𝑓 ′′ + 𝛽 1 − 𝑓 ′

{(2 − 𝛽)𝑁𝑓 ′ 𝜃 − 𝑓𝜃 ′ } = 0

2

+

𝑓 ′′ 𝜃 ′ 𝜃 𝜃𝑟

𝜃𝑟 1−

=0

(14)

(15)

The corresponding boundary conditions are 𝑓 = 𝑓𝑤 , 𝑓 ′ = 0, 𝜃 = 1 at 𝜂 = 0 𝑓 ′ = 1, 𝜃 = 0 as 𝜂 → ∞ where primes denote the differentiation with respect to 𝜂 and 𝛽 =

(16) 2𝑚 1+𝑚

is the Hartree pressure gradient parameter. The

parameter 𝑓𝑤 is defined by, 𝑓𝑤 = −𝑉𝑤 2𝑥 ⁄ (𝑚 + 1)𝜈𝑓∞ 𝑈

1 ⁄2

(17)

Here, 𝑉𝑤 > 0 (i.e.𝑓𝑤 < 0) indicates injection and 𝑉𝑤 < 0 (i.e.𝑓𝑤 > 0) for suction, while 𝑉𝑤 = 0 (i.e. 𝑓𝑤 = 0) is for impermeable surface. Also, 𝑃𝑟 =

𝜈𝑓∞ 𝛼𝑓∞

is the Prandtl number of the base fluid. From equation (7), 𝜇𝑓 can be readily put

into the form, 𝜇𝑓 =

𝜇 𝑓∞ 𝜃 𝜃𝑟

1−

(18)

which clearly shows that, as 𝜃𝑟 → ∞ (or as 𝛾 = 0), this leads to 𝜇𝑓 →𝜇𝑓∞ (constant), i.e. the viscosity variation in the boundary layer is negligible. Also, according to the definition of variable viscosity parameter, for a given reference temperature 𝑇𝑟 , variation of 𝜃𝑟 indicates the variation of the temperature difference ∆𝑇(= 𝑇𝑤 − 𝑇∞ ). The effect of variable viscosity is important if 𝜃𝑟 is small. Since the viscosity of liquids decreases with increasing temperature, while it increases for gases, 𝜃𝑟 is negative for liquids and positive for gases. From equations (6) and (18), the dynamic viscosity and Prandtl number of the nanofluid can be written as, 𝜇𝑛𝑓 =

𝜇 𝑓∞

(19)

𝜃 𝜃𝑟

(1−𝜙)2.5 1−

(𝑃𝑟)𝑛𝑓 =

𝜇 𝑛𝑓 𝜌 𝐶𝑝

𝑛𝑓

(20)

𝜌 𝑛𝑓 𝐾𝑛𝑓

The equations (19) and (20) shows the variations of viscosity and Prandtl number of the nanofluid. It is observed that for larger values of | 𝜃𝑟 | i.e. as | 𝜃𝑟 |→∞, the viscosity of nanofluid approaches the viscosity of ambient fluid at 𝜙 = 0 (pure water). This means, in the absence of nano-particles, the viscosity becomes constant. Also, it can readily be seen that for 𝜙 ≠ 0 (0 < 𝜙 ≤ 0.2) i.e. in the presence of nano-particles, the viscosity of the nanofluid is greater than the ambient one. On the other hand, for the smaller value of | 𝜃𝑟 |, the viscosity of nanofluid within the range 0 < 𝜙 ≤ 0.2 is less than the ambient one. Also, taking account of Table 1, it can be verified that that as volume fraction increases Prandtl number of the nanofluid decreases and the decrease is more in Cu-water than Al2O3-water nanofluid. Also, an increase in | 𝜃𝑟 | increases the Prandtl number of the nanofluids. The physical quantities of most interest are the local skin–friction coefficient and the local Nusselt number, which are defined respectively by the following relations: 𝐶𝑓 =

𝜏𝑤 𝜌𝑓∞ 𝑈2

and 𝑁𝑢 =

𝑥𝑞 𝑤

(21)

𝐾𝑓 ∞ (𝑇𝑤 −𝑇∞ )

Now the surface shear stress 𝜏𝑤 and the surface heat flux 𝑞𝑤 are given by 𝜏𝑤 = 𝜇𝑛𝑓

𝜕𝑢 𝜕𝑦 𝑦=0

and 𝑞𝑤 = −𝐾𝑛𝑓

𝜕𝑇 𝜕𝑦 𝑦=0

(22)

Using equations (7), (12) and (22) the quantities (21) can be written as follows:

where 𝑅𝑒 =

𝑥𝑈 𝜈𝑓∞

[2𝑅𝑒⁄(𝑚 + 1)]1⁄2 𝐶𝑓 = [1⁄{(1 − 𝜙)2.5 (1 − 1/𝜃𝑟 )}]𝑓 ′′ (0)

(23)

[(𝑚 + 1)𝑅𝑒⁄2]−1⁄2 𝑁𝑢 = − 𝐾𝑛𝑓 /𝐾𝑓∞ 𝜃 ′ (0)

(24)

is the local Reynolds number.

3.

Results and discussions: The system of non-linear ordinary differential equations (14) and (15) with boundary conditions (16) have been

solved numerically by using shooting technique along with Runge-Kutta-Fehlberg integration scheme. The velocity profiles 𝑓 ′ (𝜂), temperature profiles 𝜃(𝜂), skin friction coefficient and local Nusselt number are calculated for various values of the parameters 𝜃𝑟 , 𝛽, 𝑓𝑤 , 𝜙 and N for Cu-water nanofluid and Al2O3-water nanofluid. The numerical results thus obtained are represented in figures. In our computation, value of Pr is taken as 6.2 (water) and 𝜙 is from 0 to 0.2. Also, we consider the values of 𝑓𝑤 within the range – 0.3 ≤ 𝑓𝑤 ≤ 0.3. When 𝜃𝑟 → ∞, the present study reduces to constant base fluid viscosity. For this particular case, we compare the values of skin friction coefficient and local Nusselt number with those obtained by Yacob et al. [14] for different values m and 𝜙 with N = 0 and 𝑓𝑤 = 0. The data in the Table 2 show excellent agreement. Figures 2 and 3 show the velocity and temperature profiles respectively for Cu-water nanofluid for first and second solutions with various values of pressure gradient parameter (𝛽) keeping the other parameters fixed (𝑓𝑤 = 0.3, 𝜃𝑟 = −0.5, Pr = 6.2, N = 0.5, 𝜙 = 0.1). From the first solutions, it is clear that an increase in 𝛽 leads to an increase in the normal flow near the surface of the wedge, while the temperature profiles decrease. The physical reason is that as 𝛽 increases, the flow acceleration increases, thereby thinning the boundary layer. Also, the shear stress at the surface increases as the temperature of the fluid along the surface of the wedge reduces. For accelerated flows (𝛽 > 0), we obtain velocity profiles without a point of inflection. On the other hand, for decelerated flows (𝛽 < 0), we obtain velocity profile with a point of inflection. The value of 𝛽= -0.395330 corresponds to the separation of velocity profile with 𝑓 ′′ (0) = 0.000009. The second solution proves the existence of dual solutions. Also, the results show that as the adverse pressure gradient parameter (−𝛽) is decreased, the magnitude of the reverse flow velocity decreases and the boundarylayer thickness increases. The term normal flow means that the flow velocity has a unique direction, and the reverse flow means that the velocity is both positive and negative in the interval of integration. The effect of N on the velocity and temperature profiles with fixed values of 𝛽= 1/6, 𝑓𝑤 = 0.3, 𝜃𝑟 = −0.5 and Pr = 6.2 is shown in figures 4 and 5 respectively. It is clear from figure 4 that an increase in N decreases the velocity of the fluids. However, the presence of Cu nano-particles leads to more thinning of the momentum boundary layer thickness and increases the velocity. From figure 5, it is seen that the temperature of pure water (𝜙 = 0) and Cu-water nano-fluid (𝜙 = 0.1) decreases with an increase in the value of N and the existence of Cu nano-particles increases the thermal

boundary layer thickness. This is because the thermal conductivity of copper (Cu) is higher than that of pure water (see Table 1). Figure 6 presents velocity and temperature profiles for Cu-water nanofluid and Al2O3-water nanofluid for various values of 𝑓𝑤 with N = 0.5, 𝜃𝑟 = −0.5, 𝜙 = 0.1, 𝛽=1/6. It is seen that all these profiles satisfies the boundary conditions (16), asymptotically. An increase in 𝑓𝑤 causes a significant increase in velocity and decrease in temperature profiles throughout the boundary layer for both nanofluids. So, suction accelerates the fluid motion and reduces the temperature of the fluid. We also found that the velocity in Al2O3-water nanofluid is relatively less than Cu-water nano-fluid. It is noted that though the thermal conductivity of Cu is higher than Al2O3, it is seen that the thermal boundary layer thickness for a Cu-water nanofluid is thinner than Al2O3-water nanofluid. Figure 7 illustrates the influence of variable viscosity parameter 𝜃𝑟 on the velocity and temperature profiles for Cu-water nano-fluid, when N = 0.5, 𝑓𝑤 = 0.3, 𝜙 = 0.1, Pr = 6.2 and 𝛽=1/6. From this figure, it is observed that the velocity profiles increase, whereas the temperature profiles decrease with the decreasing values of | 𝜃𝑟 |. This is due to the fact that when 𝛾 is fixed, smaller | 𝜃𝑟 | implies higher temperature difference between the wall and the ambient fluid. Thus the viscosity of the nanofluids decreases, which results in the reduction of the boundary layer thickness. It should be noticed that for larger values of | 𝜃𝑟 |, the velocity and temperature profiles are close to the case of constant viscosity (| 𝜃𝑟 |→∞). Thus, for larger values of | 𝜃𝑟 |,the variable viscosity effect can be neglected. The velocity gradient 𝑓 ′′ (0) against 𝛽 for various values of 𝑓𝑤 with 𝜃𝑟 = −0.5, 𝜙 = 0.1, N = 0.5 for Cu-water nano-fluid and Al2O3-water nanofluid is shown in figure 8. Also, for various values of 𝜙 with 𝑓𝑤 = 0.3, N = 0.5 when 𝜃𝑟 = −0.5 and 𝜃𝑟 → ∞ for the Cu-water nano-fluid is shown in figure 9. These figures show that there is only one solution, when 𝛽 ≥ 0, two solutions when 𝛽𝑐 ≤ 𝛽 < 0 and no solution when 𝛽 < 𝛽𝑐 . Here 𝛽𝑐 is the critical value of 𝛽 such that 𝑓 ′′ (0) ≈ 0. The values of 𝛽𝑐 are different for various values of parameters, 𝜃𝑟 , 𝑓𝑤 , 𝜙 and N. In the range, 𝛽𝑐 ≤ 𝛽 < 0, two values of 𝑓 ′′ (0) are seen; one is 𝑓 ′′ (0) > 0 and the other is 𝑓 ′′ (0) < 0. Physically, 𝑓 ′′ (0) > 0 means, the fluid exerts a drag force on the wedge, while 𝑓 ′′ (0) < 0 means the opposite and 𝑓 ′′ (0) ≈ 0 at 𝛽𝑐 means there is no wall shear stress. From both these figures, we also observe that the value of 𝑓 ′′ (0) > 0 is greatly increased as 𝛽 increases. Further, it is seen from figure 8 that, the velocity gradient 𝑓 ′′ (0) almost vanishes and the flow separates completely for the Cu-water nano-fluid and Al2O3-water nano-fluid respectively at 𝛽𝑐 = -0.395330 and -0.361996, when 𝑓𝑤 = 0.3; again at 𝛽𝑐 = -0.233830 and -0.233069 when 𝑓𝑤 = 0. The same result come out at 𝛽𝑐 = -0.082178 and -

0.105432 when 𝑓𝑤 = −0.3. These show that the critical values of adverse pressure gradient parameter, {−𝛽𝑐 } increase as the value of 𝑓𝑤 increases, and in the presence of suction, the value of {−𝛽𝑐 } is greater in the Cu-water nano-fluid than the Al2O3-water nanofluid. But in the presence of injection, the value of {−𝛽𝑐 } is found smaller in the Cu-water nanofluid than the Al2O3-water nanofluid. Thus, separation occurs earlier in the Al2O3-water than the Cu-water for 𝑓𝑤 ≥ 0, but it is just opposite for 𝑓𝑤 < 0. It also reveals that suction delays separation of flow. Also, we find that when 𝜃𝑟 = −0.5, the values of 𝛽𝑐 for the Cu-water nano-fluid are -0.354445, -0.395330 and -0.412158, and when 𝜃𝑟 → −∞ these values are 0.324414, -0.348948 and -0.355143 for 𝜙 = 0.0, 0.1 and 0.2 respectively. Therefore, the values of {−𝛽𝑐 } increase as 𝜙 increase for both the cases. Further, the value of {−𝛽𝑐 } for 𝜃𝑟 = −0.5 is greater than that of 𝜃𝑟 → −∞ (constant viscosity case), when there is variation in 𝜙. Hence, the inclusion of nano-particles and temperature dependent viscosity delays the flow separation. Figures 10, 11, 12 and 13 illustrate the variations of the skin friction coefficient [2Re/(m+1)]1/2Cf and local Nusselt number [(m+1)Re/2]-1/2Nu with the nano-particle volume fraction parameter ϕ for Cu-water nano-fluid and Al2O3-water nano-fluid for different values of N and

r

respectively. It is observed from these figures 10 and 11 that, for a particular

nano-particle, both the skin friction coefficient and the local Nusselt number increase with increasing the values of N. From figure 12, the skin friction coefficient is found to increase with increasing | r| while, in figure 13, the local Nusselt number decreases with increasing | r|. Further, it is observed from figures 10 and 12 that the inclusion of nano-particles into the base fluid increases the skin friction coefficient. It is also seen that the skin friction coefficient increases as the nano-particle volume fraction increases and that the increment is more pronounced in the case of a Cu-water nano-fluid as compared to an Al2O3-water nano-fluid. Also from figures 11 and 13, it is seen that the local Nusselt number increases with the increase in . In addition, it is noted that the lowest heat transfer rate is obtained for the Al2O3 nano-particles due to the domination of conduction mode of heat transfer. This is because Al2O3 has the lowest thermal conductivity compared to Cu, as presented in Table 1.

4.

Conclusions: In this study we have investigated the boundary layer flow and heat transfer of two water-based nanofluids over a

static porous wedge taking variable viscosity and surface temperature into account. The governing partial differential equations are transformed into a system of ordinary differential equation by using suitable similarity variables, and then

solved numerically using Runge–Kutta–Fehlberg  method  with  Newton’s  shooting  technique.  Dual  solutions are obtained for negative pressure gradient. It has been observed that in the presence of nano-particles, suction and viscosity delay the flow separation. The separation is seen earlier in Al2O3-water nanofluid than Cu-water nanofluid for the case of suction, but opposite for injection. Both the skin friction coefficient and rate of heat transfer coefficient increase with the increases in the nano-particle volume fraction 𝜙. An increase in pressure gradient parameter leads to an increase in the velocity, while temperature decreases. The velocity profiles increase, whereas the temperature profiles decrease with the decreasing values of variable viscosity parameter | 𝜃𝑟 |. Finally, it is concluded that the inclusion of nano-particles in fluid and the change in fluid viscosity with temperature are significantly enhances the rate of heat transfer.

Acknowledgement One of the authors (Rudra Kanta Deka) acknowledges the support of UGC, New Delhi vide Grant No. 42-10/2013 (SR).

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Table Captions: Table 1: Thermo-physical properties of the base fluid and the nan-oparticles (Yacob et al. [14]). Table 2: Comparison of results for [2𝑅𝑒⁄(𝑚 + 1)]1⁄2 𝐶𝑓 and [(𝑚 + 1)𝑅𝑒⁄2]−1⁄2 𝑁𝑢 when 𝜃𝑟 → ∞.

Figure Captions: Fig. 1: Sketch of the porous wedge and boundary layer flow description. Fig. 2: Velocity profiles of Cu-water nanofluid that show the existence of two solutions Fig. 3: Temperature profiles of Cu-water nanofluid that show the existence of two solutions Fig. 4: Velocity profiles for different N Fig. 5: Temperature profiles for different N Fig. 6: Velocity and temperature profiles for different 𝑓𝑤 Fig. 7: Velocity and temperature profiles for different 𝜃𝑟 Fig. 8: Velocity gradient f (0) against β for different fw Fig. 9: Velocity gradient f (0) against β  for different Fig. 10: Variation of the skin friction coefficient againstϕfor different N. Fig. 11: Variation of the local Nusselt number against ϕ for different N. Fig. 12: Variation of the skin friction coefficient againstϕfor different values of r. Fig. 13: Variation of the local Nusselt number against ϕ for different

r.

Table 1 Thermo-physical properties of the base fluid and the nan-oparticles (Yacob et al. (2011)). Physical properties

Fluid (water)

Cu

Al2O3

Cp(J/kgK)

4179

385

765

ρ(KG/m3)

997.1

8933

3970

k(W/mK)

0.613

400

40

α×10-7(m2/s)

1.47

1163.1

131.7

Table 2 Comparison of results for [2𝑅𝑒⁄(𝑚 + 1)]1⁄2 𝐶𝑓 and 𝜙

m

[(𝑚 + 1)𝑅𝑒⁄2]−1⁄2 𝑁𝑢 when 𝜃𝑟 → ∞.

[2𝑅𝑒⁄(𝑚 + 1)]1⁄2 𝐶𝑓 and [(𝑚 + 1)𝑅𝑒⁄2]−1⁄2 𝑁𝑢 Yacob et al. (2011)

Present work

0.1

0

0.7179 (1.11)

0.7179 (1.1101)

0.2

0

0.9992 (1.3342)

0.9992 (1.3342)

0.1

0.5

1.5881 (1.3472)

1.5882 (1.3473)

0.2

0.5

2.2105 (1.6048)

2.2106 (1.6048)

0.1

1

1.8843 (1.4043)

1.8843 (1.4043)

0.2

1

2.6226 (1.6692)

2.6227 (1.6692)

() results are for [(𝑚 + 1)𝑅𝑒⁄2]−1⁄2 𝑁𝑢

Fig. 1: Sketch of the porous wedge and boundary layer flow description

Fig. 2: Velocity profiles of Cu-water nanofluid that show the existence of two solutions

Fig. 3: Temperature profiles of Cu-water nanofluid that show the existence of two solutions

Fig. 4: Velocity profiles for different N

Fig. 5: Temperature profiles for different N

Fig. 6: Velocity and temperature profiles for different 𝑓𝑤

Fig. 7: Velocity and temperature profiles for different 𝜃𝑟

Fig. 8: Velocity gradient f (0) against β for different fw

Fig. 9: Velocity gradient f (0) against β for different

Fig. 10: Variation of the skin friction coefficient against ϕ for different N.

Fig. 11: Variation of the local Nusselt number against ϕ for different N.

Fig. 12: Variation of the skin friction coefficient against ϕ for different values of

r.

Fig. 13: Variation of the local Nusselt number against ϕ for different r.