STABILITY OF FERROFLUID FLOW IN ROTATING ...

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MAGNETOHYDRODYNAMICS Vol. 42 (2006), No. 1, pp. 41–55

STABILITY OF FERROFLUID FLOW IN ROTATING POROUS CYLINDERS WITH RADIAL FLOW J. Singh, R. Bajaj Centre for Advanced Study in Mathematics, Panjab University, Chandigarh, India-160014

Stability of a ferroﬂuid ﬂow in an annular region between two co-axially rotating porous cylinders with a radial ﬂow in the presence of an axially applied magnetic ﬁeld has been investigated numerically. Magnetic ﬁeld perturbations have also been considered in the ferroﬂuid in the annulus, and it has been found that for all values of the radial Reynolds number Re, the ﬂow is stabilized by the applied magnetic ﬁeld. The stability characteristics have been found to depend heavily upon the radius ratio ξ of the cylinders.

Introduction. The problem of a ﬂuid ﬂow in an annular space between two co-axially rotating cylinders originated from the early experiments of Maurice Couette in 1890 (see P. Chossat and G. Iooss [1]). He carried out experiments on a Couette-Viscometer to measure viscosity, which is related to the torque exerted by the ﬂuid on the inner cylinder, which was kept stationary, while the outer one was rotated with a constant angular velocity Ω2 . The Couette-Viscometer was suitable only for a speciﬁc range of Ω2 , in which the ﬂow was nearly laminar. Arnulph Mallock (see [1]) conducted experiments on the same apparatus, but he allowed the inner cylinder to rotate with a constant angular velocity Ω1 and ﬁxed the outer cylinder. He found that the Couette ﬂow loses its stability when Ω1 exceeds a certain critical value. Taylor [2] carried out a complete linear stability analysis of the basic Couette ﬂow, both theoretically and experimentally. His theoretical ﬁndings show that the Couette ﬂow loses its stability when Ω1 exceeds a certain critical value and instability sets in as a regular, periodic pattern of horizontal toroidal vortices, equally spaced along the vertical axis of the cylinders. The new ﬂow is called a Taylor-vortex ﬂow. The theoretical results obtained by Taylor were in excellent agreement with his experimental observations. After Taylor, this problem has been studied extensively by Chossat and Iooss [1], Chandrasekhar [3], Di Prima [4], Coles [5], Krueger et al. [6], etc. An attractive feature of this experiment is the large number of diﬀerent ﬂows, which can be observed. The stability of the Taylor–Couette ﬂow is aﬀected when an additional ﬂow is superimposed on it. The instability of a viscous ﬂow in between two co-axially rotating permeable cylinders, with superposition of a radial ﬂow, has been studied by S.P. Mishra et al. [7] and K. Min et al. [8]. This type of ﬂow has applications in dynamic ﬁltration using a rotating ﬁlter. Rotating ﬁlters are used to separate plasma from blood and in numerous other biological ﬁltrations. In such devices, a suspension is contained in between a rotating inner porous cylinder and a stationary outer cylinder. As a result, the concentrate is retained in an annulus, while the ﬁltrate passes radially through the porous wall of the inner cylinder. Taylor-vortices, which appear as a result of instability, wash the surface of the inner cylinder and hence prevent ﬁlling of pores of the ﬁlter with the concentrate [8]. S.P. Mishra et al. [7] investigated the stability of an elasticoviscous 41

ﬂuid between rotating permeable cylinders with a radial ﬂow. Using a perturbation method, they obtained an approximate solution of the ﬂuid velocity ﬁeld. K. Min et al. [8] have considered the linear stability of a viscous ﬂow between rotating porous cylinders with a radial ﬂow. They have found that a radially inward ﬂow and a strong radial outward ﬂow have a stabilizing eﬀect, whereas the weak radial outﬂow has a destabilizing eﬀect on the axisymmetric Taylor-vortex ﬂow. They have considered the both cylinders to be permeable because in case of only inner porous cylinder, no analytic stable solution exists. Their analysis shows that the radially inward ﬂow shifts the vortices towards the inner cylinder, which has a positive eﬀect on the Taylor-vortices in a dynamic ﬁlter device. Besides, some investigations in this ﬁeld have been performed by Johnson et al. [9] and Wereley et al. [10]. The stability of a dissipative ﬂow of a conducting ﬂuid between rotating permeable cylinders with the superposition of a radial ﬂow in hydromagnetics has been studied by Chang et al. [11]. They have discussed an asymptotic stability behaviour for large values of the radial Reynolds number. Lee et al. [12] investigated experimentally the reverse osmosis (RO), which uses the Taylor–Couette instability to reduce ﬂux decline related to concentration polarization and membrane fouling. Their theoretical results were conﬁrmed by their experimental observations. They have found that with the increase of rotational speed and transmembrane pressure the ﬂux and the rejection are altered. A wide range of applications of the Taylor–Couette ﬂow and a rich variety of dynamic patterns that appear as instabilities have made theoreticians to investigate this ﬂow under more general physical considerations such as the eﬀect of magnetic ﬁeld on the Couette ﬂow stability, when the ﬂuid under consideration is electrically conducting [2], and when the ﬂuid is a magnetically polarizable ferroﬂuid [13, 14]. The ferroﬂuid ﬂow in rotating cylinders with an axial magnetic ﬁeld has given rise to industrial applications such as the production of ferroﬂuid seals for rotating shafts. The stability of the Taylor–Couette ﬂow in ferrohydrodynamics has been studied by Niklas et al. [15], Chang et al. [16] and Singh and Bajaj [17]. They have found that the basic ﬂow is stabilized with the increase of the applied magnetic ﬁeld magnitude. The stabilizing action of the magnetic ﬁeld on the Taylor–Couette instability motivated us to investigate the eﬀect of an axial magnetic ﬁeld on the stability of a ferroﬂuid ﬂow in rotating porous cylinders, with superposition of a radial ﬂow. We have investigated this stability problem, accounting for axisymmetric disturbances and magnetic ﬁeld changes in the annulus in the non-equilibrium state of the ﬂow. 1. Formulation. We consider a ﬂow of a viscous, incompressible, Newtonian ferroﬂuid of uniform density ρ in between two co-axial porous cylinders of radii r1 and r2 (r1 < r2 ), respectively, rotating about the vertical axis with uniform angular velocities Ω1 and Ω2 , respectively. The cylinders are assumed to be inﬁnitely long. A constant vertical magnetic ﬁeld h0 = (0, 0, h0 ) in the cylindrical polar co-ordinates is applied to this system. The governing equations for this system are given by: ∂u 1 µ0 µ0 m · ∇h + ∇ × (m × h ) , + u · ∇u = − ∇p + ν∇2 u + ∂t ρ ρ 2ρ ∂m 1 + u · ∇m = − (∇ × u ) × m − α(m − m0 ) − βm × (m × h ) , ∂t 2 ∇ · u = 0, 42

∇ × h = 0,

∇ · (m + h ) = 0 ,

(1) (2) (3)

where u , h and m are the ﬂuid velocity, magnetic ﬁeld and ferroﬂuid magnetization, respectively, at any point (r , θ , z ) inside the ﬂuid at a given time t . p is the total pressure of the ferroﬂuid, ν is the kinematic viscosity of the ferroﬂuid kb Tb µ0 ,β= , where and µ0 is the magnetic permeability of free space. α = 3Vh η 6ϕρν kb , Tb , and Vh are the Boltzmann constant, temperature of the ﬂuid and the hydrodynamic volume of each ferrocolloid particle, respectively; ϕ is the volume fraction of ferromagnetic particles and m0 = (0, 0, m0 ) is the equilibrium magnetization of the ferroﬂuid, which is related to an equilibrium magnetic ﬁeld h0 by the h Langevin formula [13]–[14], m0 = nm(coth ψ − 1/ψ) 0 , where n is the volume |h0 | density of ferromagnetic particles in the ferroﬂuid, m is the magnetic moment of mh0 each ferromagnetic particle and ψ = µ0 is the magnetic ﬁeld parameter. kb Tb A radial ﬂow in the form of U(r ) = (a/r , 0, 0) is superimposed on the Couette ﬂow, where a is a constant determined from the boundary condition U(r1 ) = (u1 , 0, 0), u1 is radial velocity of the ﬂuid at the boundary of the inner cylinder. In equations (1)–(3), the velocity ﬁeld u satisﬁes the following boundary conditions: u = (u1 r1 /r , r Ωj , 0)

at

r = rj ,

for

j = 1, 2.

(4)

The magnetic induction ﬁeld b = µ0 (m + h ) and the magnetic ﬁeld h satisfy: · [m + h ] = 0, n

× [h ] = 0, n

at r = r1 , r2 ,

(5)

denotes an outward drawn unit normal to the curved surface of the outer where n cylinder and [m + h ] and [h ] denote the diﬀerence in m + h and h , respectively, across the boundaries. The basic state of this ﬂow is given by u = (Re ν/r , r Ω, 0),

h = h0 ,

m = m0 ,

p = p0 ,

(6)

where Re = u1 r1 /ν is the radial Reynolds number for the inner cylinder, r12 Ω1 (1 − Ω∗ ξ Re ) Ω1 (Ω∗ − ξ 2 ) , B = , Ω∗ = Ω2 /Ω1 , (1 − ξ Re+2 ) r2Re (1 − ξ Re+2 ) ξ = r1 /r2 and p0 (r ) = −ρν 2 Re/(2r2 ) + ρ r Ω2 dr .

Ω = ArRe + Br−2 , A =

Ω1 > 0,

Let R = r2 − r1 denote the width of the gap between the cylinders. We nondimensionalize the system of equations (1)–(3), using dimensionless variables deﬁned by r = r /R, θ = θ , z = z /R, and t = t ν/R2 . We consider small perturbations in the basic solution (6) so that the nonequilibrium state can be represented in the form u = (Re ν/r , r Ω, 0) + (ν/R) u , h = (0, 0, h0 ) + ν(2ρ/µ0 )1/2 R h , (7) m = (0, 0, m0 ) + ν(2ρ/µ0 )1/2 R m, p = p0 (r ) + ρν 2 R2 p . Upon substituting (7) in equations (1)–(3), the solution for a resulting dimensionless system after its linearization can be Fourier-analyzed in normal modes u = ur (r), uθ (r), uz (r) ei(ωt+kz) , (8) h = hr (r), hθ (r), hz (r) ei(ωt+kz) ,

(9) 43

[r1∗ , r2∗ ],

m = mr (r), mθ (r), mz (r) ei(ωt+kz) ,

(10)

p = P(r) ei(ωt+kz) ,

(11)

r1∗

r2∗

where r ∈ = r1 /R, = r2 /R, z ∈ (−∞, ∞), t ∈ [0, ∞); k is assumed to be real, and the parameter ω is assumed to be complex in general. Substituting equations (8)–(11) in the linearized dimensionless system, we obtain the following equations (DD∗ − k 2 )ur = DP + iω − Re/r2 + (Re/r)D ur − − (2R2 Ω/ν)uθ − ikHh0 mr + Hh0 Dmz , (12) (DD∗ − k 2 )uθ = iω + (Re/r)D∗ uθ + (R2 /ν)(rDΩ + 2Ω)ur − ikHh0 mθ , (13) (DD∗ − k 2 )uz = ikP + iω + (Re/r)D uz − − ikHh0 mz − H(m0 + h0 )(D∗ hr + ikhz ) , (14)

iω + R2 α/ν + R2 βm0 h0 /ν + (Re/r)D mr + R2 rDΩ/(2ν) mθ = = (Hm0 /2)(ikur − Duz ) + (βR2 m20 /ν)hr , (15)

iω + R2 α/ν + R2 βm0 h0 /ν + (Re/r)D mθ − R2 rDΩ/(2ν) mr =

iω + R2 α/ν + (Re/r)D mz = 0,

= (ikHm0 /2)uθ + (βR2 m20 /ν)hθ , (16) D∗ ur = −ikuz ,

hθ = 0,

Dhz = ikhr , (17)

∗

D (mr + hr ) + ik(mz + hz ) = 0 , (18) 1 d d , D∗ ≡ + and H = R(µ0 /2ρ)1/2 ν. Stability characteristics where D ≡ dr dr r have been expressed in terms of a standard Taylor number T = −4AΩ1 R4 /ν 2 . The system of equations (12)–(18) satisﬁes the following boundary conditions: at r = r1∗ , r2∗ .

ur = uθ = uz = mr + hr = hz = 0,

(19)

We take the eﬀect of radial ﬂow U(r ) on the ferroﬂuid magnetization m only through the perturbed ﬂuid velocity u = (ur , uθ , uz ) in the basic ﬂuid velocity. Under these considerations, the system of equations (12)–(18) has been reduced to a system of ten ﬁrst order ordinary diﬀerential equations, applying the following transformations: D ∗ u r = X2 ,

u r = X1 , ∗

D u θ = X6 , ∗

2

∗

D∗ (mr + hr ) = X4 ,

mr + h r = X 3 , 2

(DD − k )(mr + Hr ) = X7 ,

(DD − k )ur = X9

∗

∗

∗

∗

u θ = X5 ,

2

D (DD − k )(mr + Hr ) = X8 ,

2

and D (DD − k )ur = X10 .

A resulting system of the ten ﬁrst order ordinary diﬀerential equations is: D ∗ X1 = X2 , 2

DX2 = k X1 + X9 , ∗

D X3 = X4 , 2

DX4 = k X3 + X7 , ∗

D X5 = X6 , 44

(20) (21) (22) (23) (24)

DX6 =D∗ (R2 rΩ/ν)X1 + ikβR2 Hm20 h0 g2 δ/(2ν) X3 + +k 2 (1 + H2 m0 h0 g3 /2)X5 + (Re/r)X6 + (H2 m0 h0 g2 δ/2)X9 ,

(25)

D ∗ X7 = X8 , DX8 = − Hm0 ik 3 δδ1 g2 D∗ (R2 rΩ/ν) X1 + +δ1 k 2 f1 + βR2 H2 m30 h0 (kg2 δ)2 /(2ν) X3 + 2k 2 δ1 D(δ) X4 + +(δ1 Hm0 ik 3 /2) 2R2 Ωg1 δ/ν − f2 − k 2 H2 m0 h0 g2 g3 δ/2 X5 − −(δ1 Hm0 ik 3 /2) Re g2 δ/r + 2D(g2 δ) X6 + +k 2 1 + δ1 (δ − 1) − δδ1 H2 m20 g1 /2 X7 + +(δ1 Hm0 ik/2) f3 − (Hkg2 δ)2 m0 h0 /2 − 2Re g1 δ/r2 X9 + +(δ1 Hm0 ik/2) Re g1 δ/r + 2D(g1 δ) X10 ,

(26)

DX10

D∗ X9 = X10 ,

= δ1 δk H m0 (m0 + h0 )D (R rΩ/ν)g2 /2 X1 + +ikH(m0 + h0 )δ1 f1 + βm0 h0 (RHm0 kg2 δ)2 /(2ν) X3 + 2

2

∗

(27)

(28)

2

+2ikH(m0 + h0 )δ1 D(δ)X4 + +k 2 δ1 2R2 Ω/ν + (H2 m0 (m0 + h0 )/2)(f2 + k 2 H2 m0 h0 g2 g3 δ/2) X5 + (29) + δ1 k 2 H2 m0 (m0 + h0 )/2 2D(g2 δ) + Re g2 δ/r X6 + +δ1 Hik (m0 + h0 )(δ − 1) + m0 X7 + +δ1 k 2 − 2Re/r2 − H2 m0 (m0 + h0 )(f3 − m0 h0 (Hkg2 δ)2 )/2) X9 + +δ1 Re/r − H2 m0 (m0 + h0 )D(g1 δ)) X10 , where

g1 = ν/R2 )(iων/R2 + α + βm0 h0 (iων/R2 + α + βm0 h0 )2 + (rDΩ/2)2 , g2 = − νrDΩ/(2R2 ) (iων/R2 + α + βm0 h0 )2 + (rDΩ/2)2 ,

g3 =g1 + βR2 g22 m20 δ/ν ,

−1 δ = [1+βR2 m20 g1 /ν]−1 , δ1 = 1+H2 m0 (m0 +h0 )g1 δ/2 , f1 = D∗ D(δ)−2/rD(δ), f2 = D∗ D(g2 δ) − 2/rD(g2 δ),

and f3 = D∗ D(g1 δ) − 2/rD(g1 δ).

The boundary conditions (19) in terms of new variables can be restated as: X1 = X2 = X3 = X4 = X5 = 0,

at r = r1∗ , r2∗ .

(30)

The system of equations (20–30) leads to a two-point boundary value problem (BVP) that has been solved by a shooting method [17, 18]. The BVP is converted into a set of ﬁve problems (IVPs), where the boundary conditions (30) at one of the boundaries are taken as the initial conditions for a corresponding IVP. Five IVPs are then solved numerically, using the fourth order Runge–Kutta method, to obtain ﬁve linearly independent solutions so that any solution of the actual system (20–29) is a linear combination of these ﬁve functions, each of which is a function of r and dimensionless parameters: Ω∗ , ξ, ψ, Re, k, T and ω. As the solution is required to satisfy the remaining ﬁve boundary conditions at the second boundary, it will lead to a secular equation of the form: F (Ω∗ , ξ, ψ, Re, k, T, ω) = 0 .

(31)

Marginal state is obtained by setting ω = 0 in equation (31). For the given values of the parameters Ω∗ , ξ, ψ and Re, we have solved equation (31) numerically to obtain the critical Taylor number Tc and the corresponding axial wave number kc at the onset of instability. 45

2. Results. We have performed numerical analysis of equation (31) for a diester-based ferroﬂuid of magnetite, and its physical properties have been given in [13]. The radial Reynolds number Re for the inner cylinder is varied from −45 to +45 and the magnetic ﬁeld parameter ψ was varied from 0 to 100 in order to obtain the stability characteristics in terms of the critical Taylor number Tc at the marginal state. Fig. 1a,b respectively show the variation of the critical Taylor number Tc and the critical axial wave number kc with the radial Reynolds number Re for the inner porous cylinder at ﬁxed parametric values: Ω∗ = 0, ϕ = 0.2 and ξ = 0.95. The curves in each plot have been drawn for diﬀerent values of the magnetic ﬁeld parameter ψ = 0, 2 and 100, respectively. Curve ψ = 0 in Fig. 1a shows the variation of Tc with the radial Reynolds number Re in the absence of applied

(a)

10000 9000

Tc

8000 7000 6000

ψ = 102

5000

ψ=2

4000

ψ=0 3000

−40

−30

−20

−10

0

10

20

30

40

Re

(b)

3.45 3.40 3.35 3.30

kc

3.25 3.20

ψ=0

3.15 3.10

ψ=2

3.05 3.00 2.95

ψ = 102 −40

−30

−20

−10

0

10

20

30

40

Re

Fig. 1. Variation of (a) Tc and (b) kc with Re at ﬁxed values ξ = 0.95, Ω∗ = 0 and diﬀerent ψ.

46

(a)

Re = −20

6500 6000

Tc

5500 5000

Re = 0

4500

Re = 20

4000 3500 3000

0

10

20

30

40

50

60

70

80

90

100

90

100

ψ 3.20

(b) 3.15

kc

3.10

3.05

Re = −20 3.00

Re = 20 Re = 0 0

10

20

30

40

50

60

70

80

ψ

Fig. 2. Variation of (a) Tc and (b) kc with ψ at ﬁxed values, ξ = 0.95, Ω∗ = 0 and diﬀerent Re. magnetic ﬁeld. These results match with those obtained by Min et al. [8] for an ordinary ﬂuid. At a given value of the magnetic ﬁeld parameter ψ, Tc decreases, as the parameter Re is increased from −45 to 0. At further increase of Re, Tc decreases slightly, while at certain high values of the radial Reynolds number Re, Tc starts increasing. Thus, the superimposed radial inﬂow stabilizes the basic ﬂow and a weak superimposed radial outﬂow destabilizes the basic ﬂow at ﬁxed ψ. The high superimposed radial outﬂow, however, stabilizes the basic ﬂow. With the increase of the magnetic ﬁeld parameter ψ, the ﬂow gets stabilized. Fig. 1b shows that at a given value of the magnetic ﬁeld parameter ψ, the critical wave number kc is small at a weak superimposed radial ﬂow, if compared to a strong superimposed radial ﬂow. The curves show that kc decreases with the increase of the magnetic ﬁeld parameter ψ. Variations of the critical Taylor number Tc with the magnetic ﬁeld parameter ψ are illustrated in Fig. 2a. Three curves correspond to Re = −20, 0 and 20, 47

(a)

2.5

III

Tc × 104

2.0

V

IV 1.5

II 1.0

I

0.5

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

0.75

0.80

0.85

0.90

0.95

ξ (b)

4.8 4.6

III

4.4

kc

4.2

V

4.0

II 3.8 3.6

IV

3.4 3.2

I

3.0 0.50 0.55

0.60

0.65

0.70

ξ

Fig. 3. Variation of (a) Tc and (b) kc with ξ at ﬁxed values Ω∗ = 0. Curves I–V correspond to (Re, ψ) = (0, 0), (10, 0), (−10, 0), (10, 2) and (−10, 2), respectively.

respectively. The curve, corresponding to Re = 0, shows a variation of the critical Taylor number Tc with the magnetic ﬁeld parameter ψ in the absence of radial ﬂow, and the curve matches with the one obtained by Singh and Bajaj [17] for a ferroﬂuid. The curves in Fig. 2a show that Tc increases with the increase of the magnetic ﬁeld parameter ψ and becomes constant at high values of ψ. Nature of the curves in (ψ, Tc ) plane at imposing the radial ﬂow remains the same as that without the radial ﬂow. For a strong inﬂow (Re = −20), the curve shifts upwards, and for a strong outﬂow (Re = 20), the curve shifts downwards. The corresponding variation of kc with ψ is shown in Fig. 2b. Curves in Fig. 2b show that at a given value of Re, kc decreases sharply with the increase of ψ until it attains a minimum, then it starts increasing slowly with a further increase of ψ. We have found that the stability characteristics depend heavily upon the radius ratio parameter ξ. This dependence can be visualized by comparing curves in Fig. 3a. The curves show that Tc decreases with the increase of the parameter ξ 48

13000 12000 11000

f

Tc

10000 9000 8000

c

e

7000 6000

d b

5000

a

4000 3000

0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

−Ω∗

Fig. 4. Variation of Tc with −Ω∗ at ﬁxed values, ξ = 0.95. Diﬀerent curves a to f correspond to (R, ψ) = (10, 0), (0, 0), (−10, 0), (10, 2), (0, 2) and (−10, 2), respectively. from 0.50 to 0.95. Curves IV and V in Fig. 3 correspond to a variation of Tc with ξ at the applied magnetic ﬁeld ψ = 2 and Re = 10 and −10, respectively, which show that the magnetic ﬁeld has a stabilizing eﬀect on the ﬂow at all permissible values of the parameter ξ and the considered range of parameter Re. Curves I, II and III in Fig. 3a, drawn in the absence of applied magnetic ﬁeld (ψ = 0), are similar to those obtained by Min et al. [8]. The corresponding variation of kc with ξ is illustrated in Fig. 3b. Fig. 4 shows the variation of Tc with −Ω∗ at a ﬁxed value of ξ = 0.95, when the cylinders are counter-rotating. Six curves have been drawn at diﬀerent 5000

f

4500

e

Tc

4000

d c

3500

b a

3000

2500 0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

∗

Ω ∗

Fig. 5. Variation of Tc with Ω at ﬁxed values, ξ = 0.95. Diﬀerent curves a to f correspond to (Re, ψ) = (10, 0), (0, 0), (−10, 0), (10, 2), (0, 2) and (−10, 2), respectively.

49

3.35 3.30

c

3.25

kc

3.20

b a

3.15 3.10

f

3.05

e

3.00 0

0.15

0.10

0.15

0.20

0.25

0.30

0.35

d 0.40

0.45

0.50

∗

−Ω ∗

Fig. 6. Variation of kc with −Ω at ﬁxed values, ξ = 0.95. Diﬀerent curves a to f correspond to (Re, ψ) = (10, 0), (0, 0), (−10, 0), (10, 2), (0, 2) and (−10, 2), respectively.

values of (Re, ψ). With the increase of the relative angular velocity in the range 0 ≤ −Ω∗ ≤ 0.5, the instability appears at a higher Taylor number. When the cylinders are co-rotating (Ω∗ > 0), the corresponding variation of Tc with Ω∗ is presented in Fig. 5. At a given parametric value of (Re, ψ), the ﬂow is destabilized by Ω∗ increasing from 0 to 0.5. The corresponding variations of the critical axial wave number kc are presented in Fig. 6, when the cylinders are counter-rotating, and in Fig. 7, when the cylinders are co-rotating. 3.18 3.16

c

3.14

b a

kc

3.12 3.10 3.08 3.06 3.04

f d

3.02 0

0.05

e

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

Ω∗

Fig. 7. Variation of kc with Ω∗ at ﬁxed values, ξ = 0.95. Diﬀerent curves a to f correspond to (Re, ψ) = (10, 0), (0, 0), (−10, 0), (10, 2), (0, 2) and (−10, 2), respectively.

50

1.0

1.0

0.8

0.8

d

0.6

c b

0.4

b

0.6

a

uθ

ur

d c

a

0.4

0.2 0.2 0 0 −0.2 1.0

1.5

2.0

1.0

1.5

r

r 0.2

1.0

d

0

a

0.5 −0.2

b

br

uz

c 0

2.0

b

−0.4 −0.6

a

c

−0.5 −0.8

d

−1.0 1.0

−1.0 1.5

2.0

1.0

1.5

r

2.0

r

1.0

0.5

a hz

Fig. 8. Proﬁles for normalized functions ur , uθ , uz , br , hz drawn at ξ = 0.50, Ω∗ = −0.5, R = 0.01 m and ψ = 2. Dotted curves correspond to R = 0. Four curves a, b, c, d in each subplot have been drawn at Re = −1, 0, 2 and 10, respectively.

0

b c d

−0.5

−1.0 1.0

1.5

2.0

r

51

1.0

0.8

0.8

0.6

0.6

ur

uθ

1.0

0.4

0.4

0.2

0.2

0 19.0

19.2

19.4

19.6

0 19.0

19.8

19.2

19.4

r

19.6

19.8

19.6

19.8

r 0

1.0

−0.2 0.5

0

br

uz

−0.4

−0.5

−0.8

−1.0

−1.0 19.0

−0.6

19.2

19.4

19.6

19.8

r

19.0

19.2

19.4

r

1.0

hz

0.5

Fig. 9. Proﬁles for normalized functions ur , uθ , uz , br , hz drawn at ξ = 0.95, Ω∗ = −0.5, R = 0.01 m and ψ = 2. Four curves in each subplot have been drawn at Re = −1, 0, 2 and 10, respectively.

0

−0.5

−1.0 19.0

19.2

19.4

19.6

r

52

19.8

Fig. 10. The normalized velocity stream function F1 ∝ rur sin(kz) deﬁned by: −rur = ∂z F1 , ruz = ∂r F1 , drawn for 0 ≤ z ≤ 1 at Ω∗ = 0, Re = 45, ξ = 0.95 and ψ = 10.

z

1

0 r1∗

r2∗

r

The eﬀect of superimposed radial ﬂow on the normalized velocity proﬁles, radial component br of perturbations in magnetic induction, and the axial component hz of perturbations in a magnetic ﬁeld at the onset of instability can be seen from Fig. 8. The four curves a to d drawn in each subplot of Fig. 8 correspond to Re = −1, 0, 2 and 10, respectively. First three subplots in Fig. 8 show the eﬀect of Re on the velocity proﬁles at ξ = 0.50, Ω∗ = −0.5 and ψ = 2. Dotted curves correspond to no superimposed radial ﬂow, i.e., Re = 0. These curves show that a strong radial inﬂow shifts the vortices towards the inner cylinder, and a strong radial outﬂow shifts the vortices towards the outer cylinder. Similar eﬀects of Re on the normalized components br and hz occur and they can be seen from the last two subplots in Fig. 8. The normalized proﬁles for ur , uθ , uz , br , and hz are shown in Fig. 9 at ﬁxed values ξ = 0.95, Ω∗ = −0.5 and ψ = 2. Four curves, which correspond to four diﬀerent values of Re, as in Fig. 8, are close to each other. Comparison of 1

z

Fig. 11. The normalized velocity stream function F2 ∝ rbr cos(kz) deﬁned by: −rbr = ∂z F2 , rbz = ∂r F2 , for 0 ≤ z ≤ 1 at Ω∗ = 0, Re = 45, ξ = 0.95 and ψ = 10.

0

r1∗

r2∗

r

53

Fig. 8 and Fig. 9 shows that the normalized proﬁles obtained at ξ = 0.5 diﬀer considerably from those obtained at ξ = 0.95 for ﬁxed values of the rest of the parameters. This allows a conclusion that the radius ratio ξ aﬀects the proﬁles for perturbation variables at the onset of instability in a radial ﬂow superimposed on the Taylor–Couette ferroﬂuid ﬂow. The normalized velocity stream function F1 ∝ rur sin(kz) deﬁned by −rur = ∂z F1 ,

ruz = ∂r F1

is presented in Fig. 10 at Ω∗ = 0, Re = 45, ξ = 0.95 and ψ = 10. The cell pattern has been drawn for 0 ≤ z ≤ 1. The corresponding normalized function F2 ∝ rbr cos(kz) for the magnetic induction ﬁeld deﬁned by −rbr = ∂z F2 ,

rbz = ∂r F2

in a similar way as the velocity stream function F1 is shown in Fig. 11. The cells in the region for 0 ≤ z ≤ 0.5 and the cells in the region for 0.5 ≤ z ≤ 1 in the ﬁgure are oppositely oriented. 3. Conclusion. We have investigated the stability of an axisymmetric ferroﬂuid ﬂow with a superimposed radial ﬂow in between two uniformly rotating porous cylinders in the presence of an axial magnetic ﬁeld. The radially inward ﬂow has a stabilizing eﬀect on the axisymmetric Taylor-vortex ferroﬂuid ﬂow both in the absence of magnetic ﬁeld (ψ = 0) and in the presence of applied magnetic ﬁeld (ψ = 0). A weak superposed radial outﬂow has a destabilizing eﬀect and a strong superposed radial outﬂow has a stabilizing eﬀect on the Taylor-vortex ﬂow as in the absence as in the presence of the applied magnetic ﬁeld. The stability characteristics of the ﬂow, considered in this paper along with the radial Reynolds number Re, have been found to depend heavily upon the radius ratio ξ. Proﬁles for the perturbed variables have also been found to depend strongly on the parameter ξ. The angular velocity ratio Ω∗ has a destabilizing eﬀect on the ﬂow for all considered values of (Re, ψ). The applied magnetic ﬁeld has been found to stabilize the circular Couette ferroﬂuid ﬂow, with a superimposed radial ﬂow. The results obtained here for the ferroﬂuid in the absence of applied magnetic ﬁeld (ψ = 0) reduce to the results obtained by Min et al. [8] for ordinary ﬂuids. In the present analysis, we have considered the onset of instability against axisymmetric disturbances only. In general, the onset of instbility can be a nonaxisymmetric mode for the suﬃciently negative value of Ω∗ . The work is in progress. REFERENCES [1] P. Chossat, G. Iooss. The Couette–Taylor Problem 1991).

(Springer-Verlag,

[2] G.I. Taylor. Stability of a viscous liquid contained between two rotating cylinders. Phil. Trans. Roy. Soc. Lond., vol. A 223 (1923), pp. 289–343. [3] S. Chandrasekhar. Hydrodynamic and Hydromagnetic Stability (Oxford University Press, Oxford, 1966). [4] R.C. Di Prima. Stability of nonrotationally symmetric disturbances for viscous ﬂow between rotating cylinders. Phys. Fluids, vol. 4 (1961), no. 6, pp. 751–755. 54

[5] D. Coles. Transition in circular Couette ﬂow. Eur. J. Fluid Mech., vol. 21 (1965), no. 3 pp. 385–425. [6] E.R. Krueger, A. Gross, R.C. Di Prima. On the relative importance of Taylor-vortex and non-axisymmetric modes in ﬂow between rotating cylinders. J. Fluid Mech., vol. 24 (1966), no. 3 pp. 521–538. [7] S.P. Mishra, J.S. Roy. Flow of elasticoviscous liquid between rotating cylinders with suction and injection. Phys. Fluids, vol. 1 (1968), no. 10 pp. 2074–2081. [8] K. Min, R.M. Lueptow. Hydrodynamic stability of viscous ﬂow between rotating porous cylinders with radial ﬂow. Phys. Fluids, vol. 6 (1994), no. 1, pp. 144–151. [9] E.C. Johnson, R.M. Lueptow. Hydrodynamic stability of viscous ﬂow between rotating porous cylinders with radial and axial ﬂow. Phys. Fluids, vol. 9 (1997), no. 12, pp. 3687–3696. [10] S.T. Wereley, R.M. Lueptow. Inertial particle motion in a Taylor– Couette rotating ﬁlter. Phys. Fluids, vol. 11 (1999), no. 2, pp. 325–333. [11] T.S. Chang, W.K. Sartory. Hydromagnetic stability of dissipative ﬂow between rotating permeable cylinders. Part 1. Stationary critical modes. J. Fluid Mech., vol. 27 (1967), no. 1, pp. 65–79. [12] S. Lee, R. Lueptow. Experimental veriﬁcation of a model for rotating reverse osmosis. Desalination, , vol. 146 (2002), pp. 353–359. [13] R.E. Rosenswieg. Ferrohydrodynamics (Cambridge University Press, Cambridge, 1985). [14] V.G. Bashtovoy, B.M. Berkowsky, A.N. Vislovich. Introduction to Thermomechanics of Magnetic Fluids (Springer – Verlag, 1988). [15] M. Niklas, H.M. Krumbhaar, M.H. Lucke. Taylor vortex ﬂow of ferroﬂuids in the presence of general magnetic ﬁelds. J. Magn. Magn. Mat., vol. 81 (1989), pp. 29–38. [16] M.H. Chang, C.K. Chen, H.C. Weng. Stability of ferroﬂuid ﬂow between concentric rotating cylinders with an axial magnetic ﬁeld. Int. Jour. Eng. Sci., vol. 41 (2003), pp. 103–121. [17] J. Singh, R. Bajaj. Couette ﬂow in ferroﬂuids with magnetic ﬁeld. J. Magn. Magn. Mat., vol. 294 (2005), pp. 53–62. [18] D.L. Harris, W.H. Reid. On the stability of viscous ﬂow between rotating cylinders. J. Fluid Mech., vol. 20 (1964), no. 1, pp. 95–101. Received 24.10.2005.

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