Transp Porous Med (2009) 77:79–86 DOI 10.1007/s11242-008-9263-3

Centrifugal Acceleration Induced Convection in a Magnetic Fluid Saturated Anisotropic Rotating Porous Medium S. Saravanan

Received: 25 March 2008 / Accepted: 29 June 2008 / Published online: 19 July 2008 © Springer Science+Business Media B.V. 2008

Abstract A theoretical investigation is made to study the influence of magnetic field on the onset of convection induced by centrifugal acceleration in a magnetic fluid filled porous medium. The layer is assumed to exhibit anisotropy in mechanical as well as thermal sense. Numerical solutions are obtained using the Galerkin method for the eigenvalue problem arising from the linear stability theory. It is found that the magnetic field has a destabilizing effect and can be suitably adjusted depending on the anisotropy parameters to enhance convection. The effect of anisotropies of magnetic fluid filled porous media is shown to be qualitatively different from that of ordinary fluid filled porous media. This phenomenon may be helpful to increase the efficiency of suitable heat transfer devices. Keywords

Magnetic fluid · Centrifugal force · Free convection · Anisotropy

Nomenclature Latin Symbols B¯ C H¯ H¯ o iˆ K K˜ p Kr k L M

Magnetic flux density Specific heat capacity Magnetic field Uniform external magnetic field Unit vector in the x-direction −(∂ M/∂ T )Ho ,To , pyromagnetic coefficient Permeability Anisotropy parameter of permeability Wavenumber Thickness of the layer Magnetization

S. Saravanan (B) Department of Mathematics, Bharathiar University, Coimbatore 641 046, Tamil Nadu, India e-mail: [email protected]

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M1 p Pr Rm Rc T t v¯ Greek Symbols α β χ ε λ˜ λr µ µ ρ σ ω ∇ Subscripts b S

x ,z o

S. Saravanan

Nonlinearity in magnetization Pressure Prandtl number Magnetic Rayleigh number Centrifugal Rayleigh number Temperature Time Filtration velocity Thermal expansion coefficient Basic temperature gradient (∂ M/∂ H )Ho ,To , magnetic susceptibility Porosity Thermal conductivity Anisotropy parameter of conductivity Dynamic viscosity Magnetic permeability of the vacuum Density Complex growth rate Rotation rate Del operator Basic state Solid matrix x, z directions Reference state

1 Introduction Magnetic fluids differ from ordinary fluids by showing magnetic as well as flow properties. Convection can also originate in these fluids from the temperature dependence of their magnetization. This property makes them useful in space research, where the role of gravity can be replaced by a magnetic body force. Moreover, magnetic forces can be used to create circulation in small passages where natural convection is either absent or ineffective. Theoretical studies on convective instabilities in magnetic fluids including external rotational effects are many (see Auernhammer and Brand 2000; Desaive et al. 2004; Sunil et al. 2004). All these works deal with rotation in the presence of gravity. However, there is a separate class of flows associated with centrifuges in process industries that need separate attention. Here, in the noninertial frame of reference, which is rotating with the same angular velocity as that of the centrifuge, centrifugal acceleration arises as a dominating quantity suppressing gravity to a sufficiently large extent. In order to model heat transfer phenomena in such situations, the onset of convection in an ordinary fluid filled porous medium subjected to the centrifugal acceleration alone was first studied by Vadasz (1994). Flows of this type have been reviewed in Vadasz (1998) and an experimental verification of the resulting temperature and flow fields has been presented in Vadasz and Heerah (1998). Saravanan and Yamaguchi (2005a), Saravanan and Yamaguchi (2005b) investigated the same problem in the case of magnetic fluids. This analogous study on magnetic fluids was motivated by the recent fabrication of ‘magnetic wood’, a new type of wood permeated by a magnetic fluid,

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by Oka and his group (see Oka et al. 2002), which has promising applications. This paper determines stability characteristics for setting up centrifugal convection in a magnetic fluid filled porous medium, which is anisotropic in both mechanical and thermal sense. Though the extent of anisotropy of a porous medium can be reduced by artificial means, fabricating an ideal isotropic porous medium continues to be a challenging one. It is well known that the preferential orientation of the porous grains at the pore scale results in mechanical anisotropy. However, one of the surprising results is that the overall thermal conductivity strongly depends on the internal heat transfer across the pore surfaces and thus behaves as a structural property rather than an intrinsic thermal property of the material (see Tzou 1995). It should be noted that polydispersity of magnetic fluids appears naturally since the particles in all commercially manufactured magnetic fluids form chain-like aggregates called ‘micelles’ (see Zubarev 2002). In the presence of strong external body force gradients, these aggregates make the volume fraction within the magnetic fluid a nonuniform one and in turn alter the overall thermal conductivity (Li et al. 2005). Hence, we focus on the onset of centrifugal convection in a magnetic fluid filled porous medium with anisotropic permeability and thermal conductivity tensors.

2 Mathematical Analysis A magnetic fluid saturated Darcian anisotropic porous layer −L/2 < x < L/2 subject to a constant rotation rate ω about a vertical axis is considered (Fig. 1). The layer is heated on one side (right boundary) and cooled on the opposite side (left boundary). A constant uniform external magnetic field H o is applied parallel to the x-direction. Free convection occurs as a result of the centrifugal body force, while the gravitational force is neglected. The layer is assumed to be narrow in the y-direction so that the y-component of centrifugal acceleration may be neglected.

Fig. 1 Physical schematic

z w 2x

Insulated Ho cold

hot

Insulated x = -L/2

x x = L/2

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The conservation equations in the rotating frame of reference under the Boussinesq approximation are ∇ · v¯ = 0 (1) µ 2 ˆ v¯ = −∇ p + µ0 M∇ H − ρω x i (2) K˜ p 
  ∂ M dT ∂M dH ∂T + (1 − ε) (ρC) S + µ0 T = ∇ · λ˜ · ∇T (3) ε ρCV,H − µ0 H ∂T dt ∂t ∂ T dt     where K˜ p = K x iˆiˆ + K z jˆ jˆ + kˆ kˆ is the permeability, λ˜ = λx iˆiˆ + λz jˆ jˆ + kˆ kˆ is the thermal conductivity and ε is the porosity. The components of the conductivity are expressed in terms of the conductivities of the fluid and solid phases as λi = ελi +(1−ε)(λi )S , i = x, z. Since the applied magnetic field is static and the magnetic fluid is not electrically conducting, the electric field vector vanishes. The appropriate Maxwell’s equations are ∇ · B¯ = 0 ∇ × H¯ = 0

(4)

¯ B¯ = µ0 ( H¯ + M)

(6)

(5)

The relaxation time of the magnetization of the magnetic fluid is assumed to be so small that its dynamics can be disregarded in the analysis of hydrodynamic phenomena. Thus, the magnetization is aligned with the magnetic field and depends on the magnitude of the magnetic field and temperature. The density and magnetic equations of state are ρ = ρo [1 − α(T − To )]

(7)

M = Mo + χ(H − Ho ) − K (T − To )

(8)

Following Saravanan and Yamaguchi (2005a), Saravanan and Yamaguchi (2005b), we study the stability of the following quiescent basic state

  K x βx ˆ K x βx ˆ ¯ ¯ v¯b = 0, Tb = To + βx, Hb = Ho + i, Mb = Mo − i (9) 1+χ 1+χ using the method of small perturbations. The amplitudes of perturbations are governed by the system of non-dimensional coupled ordinary differential equations 
 1 2 2 (10) K r k 2 + Rm K r k 2 D = 0 (D − K r k )U − Rm + Rc x + 2 [(D 2 − λr k 2 ) − Pr σ ] − U = 0

(11)

(D − M1 k ) − D = 0

(12)

2

2

where U (x), (x) and (x) are the respective amplitudes of x-component of perturbed filtration velocity, temperature and magnetization, k = (ky2 + kz2 )1/2 is the wave number and D ≡ d/d x. The non-dimensional parameters appearing in (11–12) are Rm = ρC1 µ0 K 2 β 2 K x L 2 /λx µN (1 + χ) the magnetic Rayleigh number, Rc = ρC1 αβω2 K x L 3 / λx ν N the centrifugal Rayleigh number, Pr = ρC1 ν/λx the Prandtl number, M1 = (1 + M0 /H0 )/(1 + χ) the nonlinearity in magnetization, K r = K z /K x the anisotropy parameter of permeability and λr = λz /λx the anisotropy parameter of conductivity. The vanishing velocity, temperature and magnetic potential perturbations at the rigid sidewalls serve as the boundary conditions:

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Centrifugal Acceleration Induced Convection in a Magnetic Fluid

U = =  = 0 at x = ±

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1 2

(13)

The governing system (10–13) can be expressed into a self-adjoint form and hence the onset of the stationary mode (σ = 0) alone is discussed for this setup. The Galerkin method is used to solve the eigenvalue problem. Accordingly, we choose 
1 x i−1 , i = 1, 2, 3, . . . (14) U i = θ i = i = x 2 − 4 as the linearly independent trial functions for U, and , respectively. The results reported here were obtained using 12 term approximations as further increase in the number of terms produced changes only in the fourth decimal.

3 Discussion The effect of magnetic field on the onset of centrifugal convection in a magnetic fluid saturated anisotropic porous medium is investigated with the objective of understanding control over convection. In the case of ordinary fluids, i.e. when Rm = 0 and M1 → ∞ (10–12) reduce to a single equation with λr /K r as the only parameter representing the anisotropy of the porous medium, as considered by Govender (2006).As a particular case, when Rm = 0, M1 = 100 and λr /K r = 5, we obtained the stability results for an ordinary fluid, Rcc = 201.4858 and kc = 2.1415, close to those reported by Govender (2006), which may serve as a partial verification of the numerical results. We fixed M1 as 10 throughout the paper as its effect has already been discussed by Saravanan and Yamaguchi (2005a). The marginal curves of Rc obtained by solving (10–12) are displayed in Fig. 2 for different values of λr and K r keeping λr /K r = 1. All these curves attain a unique minimum at a particular Rc called the critical centrifugal Rayleigh number, Rcc , accompanied by a critical wave number kc . It is seen that an increase in λr and K r shifts the marginal curves downwards and to a lower wave number region. Thus, a combined increase in the anisotropy parameters enhances the onset of convection and increases the cell size at the secondary state. This is clear from the streamlines and isotherms that are exhibited at the threshold. The possible physical rationale behind the change in convection cell size may be given as follows. In an isotropic case (λr = K r = 1), the quiescent layer is replaced by a series of fully developed counter rotating cells at the threshold, as shown in Fig. 2, for sufficiently higher values of Rm (see Saravanan and Yamaguchi 2005a). Hence, the combined influence of mechanical and thermal anisotropies alters the extent of these cells in the z-direction. It is well known that at the onset of instability, the favoured flow pattern tend to arrange themselves such that either the tangential permeability along the streamlines is as large as possible or the tangential thermal conductivity is as small as possible. When λr = K r = 10, the permeability and thermal conductivity are maximum along the z-direction. This offers least resistance to the flow in the z-direction and immediately dissipates temperature disturbances and flattens the corresponding isotherms in that direction. Hence, the three-cell pattern observed for the isotropic case fuses to form an elongated single cell mode. When λr = K r = 0.1, the opposite effect is expected and hence it degenerates into a multi-cell pattern. Moreover, a closer watch of the observed phenomena shows that the effect of λr on the setting up of secondary state is negligible when compared to that of K r , which may be due to higher Rm.

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S. Saravanan 30

27 0.1 0.3

24

1 10

Rc 21

100

18

15

12 0.1

1.0

10.0

k

Fig. 2 Marginal curves of Rc for different values of λr (=K r ) when Rm = 30, M1 = 10 and λr /K r = 1 with streamlines and isotherms at the threshold

Figure 3 shows the stability characteristics at the threshold against Rm for different values of λr and K r . In all cases, an increase in Rm produces an additional body force (Kelvin force) acting towards the hot boundary. This in turn combines with the existing centrifugal force and augments heat transfer across the layer via convection mechanism. Thus, Rcc decreases and becomes zero at a finite Rm depending on λr and K r . The magnetic body force alone can induce instability above this Rm. We also observe that λr /K r increases Rcc and hence suppresses the instability. Moreover, it is noticed that Rcc is determined by λr /K r alone in the lower Rm region, whereas it depends independently on the values λr and K r in the higher Rm region. Thus, the role of anisotropy in magnetic fluid saturated porous medium is qualitatively different from that in ordinary fluid saturated porous medium. We see that larger values of λr and K r with fixed λr /K r decreases both Rcc and kc . In other words, their effect is to destabilize the basic state accompanied by an increase in the cell size. This effect becomes significant for higher values of λr /K r . It should be noted that this problem is analogous to the Rayleigh–Bénard problem of a fluid layer subject only to the gravity and hence similar trends in results may be expected in the corresponding gravitational convection setup.

4 Conclusion Linear stability analysis made to understand the onset of centrifugal convection in magnetic fluid saturated anisotropic porous layer leads to the following conclusions: the magnetic mechanism produces an additive effect on setting up of the secondary state, thus lowering Rcc . It starts dominating the centrifugal mechanism and becomes the only cause for instability if the magnetic field strength is increased beyond a particular level, which depends on both anisotropy ratios. Larger values of λr and K r with fixed λr /K r enhance the onset condition and increase the cell size at the threshold. This knowledge may be helpful to sustain centrifugally driven flows in suitable applications.

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(a) a

1000

b c

100

Rcc 10

i

1

h g

d,e,f

0.1 40

10

50 60 70 80

200

100

Rm

(b)

11

d

10 9 8 7

b

6

kc

g

5 4

a

e

i

3 c

2

h f

1 0

0

100

200 0

25

50 0

10

20

Rm

Fig. 3 Critical stability characteristics for various combinations of λr , K r and λr /K r when M1 = 10: a—10, 0.1, 100; b—1, 0.1, 10; c—10, 1, 10; d—0.1, 0.1, 1; e—1, 1, 1; f—10, 10, 1; g—0.1, 1, 0.1; h—1, 10, 0.1; i—0.1, 10, 0.01

Acknowledgements This work was supported by UGC, India through DRS SAP in Fluid Dynamics. The author thanks SERC-DST, India for its financial sypport through the award of a Research Project.

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