Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium

International Journal of Engineering Science 44 (2006) 683–698 www.elsevier.com/locate/ijengsci

Effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium Sunil

a,*

, Anu Sharma a, Pavan Kumar Bharti a, R.G. Shandil

b

a

b

Department of Applied Sciences, National Institute of Technology, Hamirpur 177 005, India Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla 171 005, India Received 12 May 2006; accepted 12 May 2006

Abstract This paper deals with the theoretical investigation of the effect of rotation on a layer of micropolar ferromagnetic fluid heated from below saturating a porous medium subjected to a transverse uniform magnetic field. For a flat fluid layer contained between two free boundaries, an exact solution is obtained using a linear stability analysis theory and normal mode analysis method. For the case of stationary convection, the effect of various parameters like medium permeability, rotation, non-buoyancy magnetization, coupling parameter, spin diffusion parameter and micropolar heat conduction has been analyzed. The critical magnetic thermal Rayleigh number for the onset of instability are also determined numerically for sufficiently large values of magnetic parameter M1 and results are depicted graphically. The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and rotation. The oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. The sufficient conditions for the non-existence of overstability are also obtained.  2006 Elsevier Ltd. All rights reserved.

1. Introduction Micropolar fluids are fluids with internal structures in which coupling between the spin of each particle and the macroscopic velocity field is taken into account. They represent fluids consisting of rigid, randomly oriented or spherical particles suspended in viscous medium, where the deformation of fluid particles is ignored (e.g. polymeric suspension, animal blood, liquid crystals). Micropolar fluid theory was introduced by Eringen [1] in order to describe some physical systems which do not satisfy the Navier–Stokes equations. The equations governing the micropolar fluid involve a spin vector and microinertia tensor in addition to the velocity vector. This theory can be used to explain the flow of colloidal fluids, liquid crystals, animal bloods etc. The generalization of the theory including thermal effects has been developed by Kazakia and Ariman [2] and Eringen [3]. The theory of thermomicropolar convection began with Datta and Sastry [4] and interestingly *

Corresponding author. Tel.: +91 1972 22 4890; fax: +91 1972 22 3834. E-mail addresses: [email protected], [email protected] (Sunil).

0020-7225/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2006.05.003

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Sunil et al. / International Journal of Engineering Science 44 (2006) 683–698

continued by Ahmadi [5], Labon and Perez-Garcia [6], Bhattacharyya and Jena [7], Payne and Straughan [8], Sharma and Kumar [9,10] and Sharma and Gupta [11]. The above works give a good understanding of thermal convection in micropolar fluids. The numerical solution of thermal instability of a rotating micropolar fluid has been discussed by Sastry and Rao [12]. In many situations involving suspensions, as in the magnetic fluid case, it might be pertinent to demand an Eringen micropolar description. This was suggested, in fact, by Rosensweig [13] in his monograph. Zahn and Greer [14] have considered interesting possibilities in a planer micropolar ferromagnetic fluid flow with an AC magnetic field. They examined a simpler case where the applied magnetic fields along and transverse to the duct axis are spatially uniform and varying sinusoidally with time. In the uniform magnetic field the magnetization characteristic depends on particle spin but does not depend on fluid velocity. Micropolar ferromagnetic fluid stabilities have become an important field of research these days. A particular stability problem is Rayleigh–Be´nard instability in a horizontal thin layer of fluid heated from below. A detailed account of thermal convection in a horizontal thin layer of Newtonian fluid heated from below has been given by Chandrasekhar [15]. The thermal convection in Newtonian ferromagnetic fluids has been studied by many authors [16–23]. The mechanism of controlling convection in a ferromagnetic fluid is important in material processing in space because of its applications to the possibility of producing various new materials. Abraham [24] has considered the Rayleigh–Be´nard convection in a micropolar ferromagnetic fluid layer permeated by a uniform, vertical magnetic field with free–free, isothermal, spin-vanishing, magnetic boundaries. She observed that the micropolar ferromagnetic fluid layer heated from below is more stable as compared with the classical Newtonian ferromagnetic fluid. The analog of problem in micropolar ferromagnetic fluid reported above, does not seem to have been addressed in a porous medium. Thermal convection in porous media is also of interest in geophysical systems, electrochemistry, metallurgy, chemical technology, geophysics and bio-mechanics. A comprehensive review of the literature concerning convection in porous medium may be found in the book by Nield and Bejan [25]. The effect of rotation on thermal convection in micropolar fluids is important in certain chemical engineering and biochemical situations. Qin and Kaloni [26] have considered a thermal instability problem in a rotating micropolar fluid. They found that, depending upon the values of various micropolar parameters and the low values of the Taylor number, the rotation has a stabilizing effect. The effect of rotation on thermal convection in micropolar fluids has also been studied by Sharma and Kumar [27], whereas the effect of rotation on thermal convection in micropolar fluids in porous medium has been considered by Sharma and Kumar [28]. More recently, Sunil et al. [29,30] have studied the effect of rotation on the thermal convection problems in ferromagnetic fluid saturating a porous medium. In view of the recent increase in the number of non-isothermal situations wherein magnetic fluid are put to use in place of classical fluids, we intend to extend our work to the problem of thermal convection in Eringen’s micropolar fluid to the micropolar ferromagnetic fluid saturating a porous medium in the presence of rotation. This problem, to the best of our knowledge, has not been investigated yet and we believe that it has possible applications in the microgravity environment. 2. Mathematical formulation of the problem Here we consider an infinite, horizontal layer of thickness d of an electrically non-conducting incompressible thin micropolar ferromagnetic fluid heated from below saturating a porous medium. The temperature T at the and top surfaces z ¼  12 d are T0 and T1, respectively and a uniform temperature gradient dT   bottom   b ¼ dz is maintained (see Fig. 1). Both the boundaries are taken to be free and perfect conductors of heat. The gravity field g = (0, 0, g) and uniform vertical magnetic field intensity H = (0, 0, H0) pervade the system. The whole system is assumed to rotate with angular velocity X = (0, 0, X) along the vertical axis, which is taken as z-axis. This fluid layer is assumed to be flowing through an isotropic and homogeneous porous medium of porosity e and medium permeability k1, where the porosity is defined as the fraction of the total volume of the medium that is occupied by void space. Thus 1  e is the fraction that is occupied by solid. For an isotropic medium the surface porosity (i.e. the fraction of void area to total area of a typical cross section) will normally be equal to e. The mathematical equations governing the motion of micropolar ferromagnetic fluids saturating a porous medium for the above model are as follows:

Sunil et al. / International Journal of Engineering Science 44 (2006) 683–698

685

z-axis

Ω = (0, 0, Ω) d T1 2 -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - -- g = (0,0, − g ) Incompressible -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - rotating micropolar H = (0,0, H ) -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - -0 O ferromagnetic fluid z = 0 -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - -- - - - - - - - - - - - - - - - saturating a porous - - - - - - - -- - - - -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - -- - medium -- - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - - - - - - d -------------------------------------z=− y-axis 2 z=

x-axis

T0

Heated from below

Fig. 1. Geometrical configuration.

The continuity equation for an incompressible fluid is r  q ¼ 0:

ð1Þ

The momentum and internal angular momentum equations for a Darey model are   q0 o 1 1 2q þ ðq  rÞ q ¼ rp þ qg þ r  ðHBÞ  ðf þ gÞq þ 2fðr  xÞ þ 0 ðq  XÞ; k1 e ot e e     o 1 1 þ ðq  rÞ x ¼ 2f r  q  2x þ l0 ðM  HÞ þ ðk0 þ g0 Þrðr  xÞ þ g0 r2 x: q0 I ot e e

ð2Þ ð3Þ

The temperature equation for an incompressible micropolar ferromagnetic fluid is "   #   oM DT oT oM DH q0 C V ;H  l0 H  þ ð1  eÞqs C s þ l0 T  oT V ;H Dt ot oT V ;H Dt ¼ K 1 r2 T þ dðr  xÞ  rT :

ð4Þ

The density equation of state is q ¼ q0 ½1  aðT  T a Þ;

ð5Þ

where q, q0, q, x, t, p, g, f, k 0 , g 0 , d, I, l0, B, CV,H, Ta, M, K1 and a are the fluid density, reference density, filter velocity, microrotation, time, pressure, shear kinematic viscosity coefficient, coupling viscosity coefficient or vortex viscosity, bulk spin viscosity coefficient, shear spin viscosity coefficient, micropolar heat conduction coefficient, moment of inertia (microinertia constant), magnetic permeability, magnetic induction, heat capac1Þ ity at constant volume and magnetic field, average temperature given by T a ¼ ðT 0 þT , magnetization, thermal 2 conductivity and thermal expansion coefficient, respectively. The effect of rotation contributes two terms: (a) 2 2 centrifugal force  q20 gradjX  rj and (b) Coriolis force 2qe 0 ðq  XÞ. In Eq. (2), p ¼ pf  12 q0 jX  rj is the reduced pressure, whereas pf stands for fluid pressure. The partial derivatives of M are material properties which can be evaluated once the magnetic equation of state, such as (9) below, is known. Maxwell’s equations, simplified for a non-conducting fluid with no displacement currents, become r  B ¼ 0;

r  H ¼ 0;

ð6a; bÞ

where the magnetic induction is given by B ¼ l0 ðH þ MÞ:

ð7Þ

We assume that the magnetization is aligned with the magnetic field, but allow a dependence on the magnitude of the magnetic field as well as the temperature

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M¼

H MðH ; T Þ: H

ð8Þ

The magnetic equation of state is linearized about the magnetic field, H0, and an average temperature, Ta, to become M ¼ M 0 þ vðH  H 0 Þ  K 2 ðT  T a Þ;

ð9Þ

where the magnetic susceptibility and the pyromagnetic coefficient are defined by     oM oM v¼ ; K2 ¼  : oH H 0 ;T a oT H 0 ;T a ^ ^ Here H0 is the uniform magnetic field of the fluid layer when placed in an external magnetic field H ¼ H ext 0 k, k is unit vector in the z-direction, H = jHj, M = jMj and M0 = M(H0, Ta). The basic state is assumed to be quiescent state and is given by q ¼ qb ¼ ð0; 0; 0Þ; b¼

x ¼ xb ¼ ð0; 0; 0Þ; q ¼ qb ðzÞ; p ¼ pb ðzÞ; T ¼ T b ðzÞ ¼ T a  bz;     K 2 bz ^ K 2 bz ^ Hb ¼ H 0  k; Mb ¼ M 0 þ k and H 0 þ M 0 ¼ H ext 0 ; 1þv 1þv

T1  T0 ; d

ð10Þ

where the subscript b denotes the basic state. 3. The perturbation equations We shall analyze the stability of the basic state by introducing the following perturbations: q ¼ qb þ q 0 ;

x ¼ xb þ x0 ; 0

H ¼ Hb ðzÞ þ H ;

q ¼ qb þ q0 ;

p ¼ pb ðzÞ þ p0 ;

T ¼ T b ðzÞ þ h;

0

M ¼ Mb ðzÞ þ M ;

ð11Þ

where q 0 = (u, v, w), x 0 = (x1, x2, x3), p 0 , h, H 0 and M 0 are perturbations in velocity q, spin (microrotation) x, pressure p, temperature T, magnetic field intensity H and magnetization M. The change in density q 0 , caused mainly by the perturbation h in temperature, is given by q0 ¼ q0 ah:

ð12Þ

Then the linearized perturbation equations of the micropolar ferromagnetic fluid become q0 ou op0 oH 0 1 2q ¼ þ l0 ðM 0 þ H 0 Þ 1  ðf þ gÞu þ 2fX01 þ 0 Xv; k1 e ot ox oz e q0 ov op0 oH 02 1 2q ¼ þ l0 ðM 0 þ H 0 Þ  ðf þ gÞv þ 2fX02  0 Xu; k1 e ot oy oz e 0 0 q0 ow op oH 1 l K 2 bh ¼ þ l0 ðM 0 þ H 0 Þ 3  ðf þ gÞw þ 2fX03  l0 K 2 bH 03 þ 0 2 þ gaq0 h; k1 1þv e ot oz oz ou ov ow þ þ ¼ 0; ox oy oz     oh o oU0 l0 T 0 K 22 b 2 qC 1  l0 T 0 K 2 e w  dbX03 ; ¼ K 1 r h þ qC 2 b  ot ot oz ð1 þ vÞ where qC1 = eq0CV,H + (1  e)qsCs + el0K2H0, qC2 = q0CV,H + l0K2H0. Eqs. (8) and (9) yield ) H 03 þ M 03 ¼ ð1 þ vÞH 03  K 2 h;

; 0 H 0i þ M 0i ¼ 1 þ M H 0i ði ¼ 1; 2Þ H0

ð13Þ ð14Þ ð15Þ ð16Þ ð17Þ

ð18Þ

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687

where we have assumed K2(Tb  Ta)  (1 + v)H0, X0 ¼ ðX01 ; X02 ; X03 Þ ¼ ðr  x0 Þ. Thus the analysis is restricted to physical situation in which the magnetization induced by temperature variation is small compared to that induced by the external magnetic field. Eq. (6b) means we can write H 0 = $U 0 , where U 0 is the perturbed magnetic potential. Eliminating u, v, p 0 between Eqs. (13)–(15), using Eq. (16), and taking curl once on (3) and considering only kth component, we obtain    0 q0 o f þ g l K 2b 2q of 2 2 oU þ ð19Þ r w ¼ l0 K 2 b r1 þ q0 gaðr21 hÞ þ 0 2 ðr21 hÞ þ 2fr2 X03  0 X ; k1 ð1 þ vÞ oz e ot oz e   oX0 1 q0 I 3 ¼ 2f r2 w þ 2X03 þ g0 r2 X03 : ð20Þ e ot The vertical component of the vorticity equation is q0 of1 2q0 X ow f þ g  ¼ f; e oz k1 1 e ot where f1 ¼ ov  ou stands for the z-component of the vorticity. ox oy From (18), we have   o2 U0 M0 oh ð1 þ vÞ 2 þ 1 þ r21 U0  K 2 ¼ 0: oz oz H0

ð21Þ

ð22Þ

4. Normal mode analysis method Analyzing the disturbances into normal modes, we assume that the perturbation quantities are of the form ðw; h; f1 ; U0 ; X03 Þ ¼ ½W ðz; tÞ; Hðz; tÞ; Zðz; tÞ; Uðz; tÞ; X3 ðz; tÞ exp iðk x x þ k y yÞ;

ð23Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kx, ky are the wave numbers along the x- and y-directions respectively and k ¼ ðk 2x þ k 2y Þ is the resultant wave number. Following the normal mode analysis, the linearized perturbation dimensionless equations   1 o 1 þ N1 1=2 þ ð24Þ ðD2  a2 ÞW  ¼ aR1=2 ½M 1 DU  ð1 þ M 1 ÞT   þ 2N 1 ðD2  a2 ÞX3  T A DZ  ; e ot k 1   1 o 1 þ N1  1=2 þ ð25Þ Z ¼ T A DW  ; e ot k 1

oX 1 I 0 3 ¼ 2N 1 ðD2  a2 ÞW  þ 2X3 þ N 3 ðD2  a2 ÞX3 ; ð26Þ e ot oT  o ð27Þ P 0r   eP r M 2  ðDU Þ ¼ ðD2  a2 ÞT  þ aR1=2 ð1  M 2 ÞW   aR1=2 N 5 X3 ; ot ot D2 U  a2 M 3 U  DT  ¼ 0; ð28Þ where the following non-dimensional parameters are introduced: ð1 þ vÞK 1 aR1=2 gabd 4 qC 2 K 1 aR1=2  H; U; R ¼ ; T ¼ mK 1 qC 2 bmd K 2 qC 2 bmd 2 z o k1 m m l0 K 22 b ; a ¼ kd; z ¼ ; D ¼  ; k 1 ¼ 2 ; P r ¼ qC 2 ; P 0r ¼ qC 1 ; M 1 ¼ d oz K1 K1 ð1 þ vÞaq0 g d

0 1þM H0 l0 T 0 K 22 f g0 d I ; N1 ¼ ; N3 ¼ 2 ; N5 ¼ M2 ¼ ; M3 ¼ ; I0 ¼ 2 ; 2 g ð1 þ vÞ ð1 þ vÞqC 2 gd qC 2 d d  3 2 2 2 X3 d 2Xd d ; TA ¼ and Z  ¼ Z: X3 ¼ m me m t ¼

mt ; d2

W ¼

Wd ; m

U ¼

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Sunil et al. / International Journal of Engineering Science 44 (2006) 683–698

5. Exact solution for free boundaries Here the simplest boundary conditions chosen, namely free–free, no-spin, isothermal with infinite magnetic susceptibility v in the perturbed field keep the problem analytically tractable and serve the purpose of providing a qualitative insight into the problem. The case of two free boundaries is of little physical interest, but it is mathematically important because one can derive an exact solution, whose properties guide our analysis. Thus the exact solution of the system (24)–(28) subject to the boundary conditions W  ¼ D2 W  ¼ T  ¼ X3 ¼ DU ¼ 0

at z ¼

1 2

ð29Þ

is written in the form 

W ¼ A1 e

rt



cos pz ;



T ¼ B1 e

rt



cos pz ;



DU ¼ C 1 e

rt



cos pz ;



U ¼



 C 1 rt e sin pz ; p



X3 ¼ D1 ert cos pz ;

ð30Þ

where A1, B1, C1, D1 are constants and r is the growth rate which is, in general, a complex constant. Substituting Eqs. (30) in Eqs. (24)–(28) and dropping asterisks for convenience, we get following equations " # 2     r 1 þ N1 r 1 þ N1 r 1 þ N1 2 2 2 1=2 1=2 þ þ þ ðp þ a Þ þ p T A A1 þ ðaR M 1 Þ C 1  aR ð1 þ M 1 Þ B1 e e e k1 k1 k1   r 1 þ N1 þ  2N 1 ðp2 þ a2 Þ ð31Þ D1 ¼ 0; e k1 

2N 1 2 ðp þ a2 ÞA1 þ fI 0 r þ 4N 1 þ N 3 ðp2 þ a2 ÞgD1 ¼ 0; e

ð1  M 2 ÞaR1=2 A1  ðp2 þ a2 þ P 0r rÞB1 þ ðeP r M 2 rÞC 1  aR1=2 N 5 D1 ¼ 0; 2

2

2

 p B1 þ ðp þ a M 3 ÞC 1 ¼ 0:

ð32Þ ð33Þ ð34Þ

For existence of non-trivial solutions of the above equations, the determinant of the coefficients of A1, B1, C1 and D1 in Eqs. (31)–(34) must vanish. This determinant on simplification yields T r4i  iU r3i  V r2i þ iW ri þ X ¼ 0: Here

 T ¼b

 L2 I 1 ; e2

  b2 bL2 I 1 4N 1 0 2ð1 þ N ½ðL I þ L N Þ þ Þ þ ; 1 1 2 3 1 e2 e P‘ e 

  0 b2 2ð1 þ N 1 Þ 4N 1 3 L1 N 3 0 ðI 1 L1 þ N 3 L2 Þ þ 2 ðL1  N 1 L2 Þ V ¼b þ P‘ e2 e e      ð1 þ N 1 ÞL2 8N 1 ð1 þ N 1 ÞI 1 xR1 ð1  M 2 ÞL3 þb þ  T A1 L2 ;  I1 P‘ e P‘ e 

   0 ð1 þ N 1 Þ 4N 1 3 2L1 N 3 ð1 þ N 1 Þ 2 ð1 þ N 1 Þ 0 ð2L1  N 1 L2 Þ W ¼b ðI 1 L1 þ N 3 L2 Þ þ þb eP ‘ P‘ P‘ e "  # 4N 1 ð1 þ N 1 Þ2 L2 xR1 L3 2N 1 N 05 0 0 þb N 3 ð1  M 2 Þ  þ T A1 ðI 1 L1 þ N 3 L2 Þ  e e P 2‘     I 1 ð1 þ N 1 Þ 4N 1 þ  xR1 ð1  M 2 ÞL3  4N 1 T A1 L2 ; P‘ e U¼

ð35Þ

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689

" #   L1 ð1 þ N 1 Þ N 03 ð1 þ N 1 Þ 4N 21 4N 1 ð1 þ N 1 Þ2 2 0 þ b L1 X ¼b  þ N 3 T A1 P‘ P‘ e P 2‘       xR1 ð1 þ N 1 ÞL3 2N 1 N 05 4N 1 L3 0 b N 3 ð1  M 2 Þ  ;  4N 1 T A1 L1  xR1 ð1  M 2 Þð1 þ N 1 Þ P‘ e P‘ 3



where R TA a2 r ; T ¼ ; x ¼ ; iri ¼ 2 ; I 1 ¼ p2 I 0 ; N 03 ¼ p2 N 3 ; N 05 ¼ p2 N 5 ; P ‘ ¼ p2 k 1 ; A 1 p4 p p4 p2 b ¼ 1 þ x; L1 ¼ ð1 þ xM 3 Þ; L2 ¼ ½ðP 0r  eP r M 2 Þ þ xP 0r M 3  and L3 ¼ ½1 þ xM 3 ð1 þ M 1 Þ:

R1 ¼

6. The case of stationary convection When the instability sets in as stationary convection (and M2 ffi 0), the marginal state will be characterized by ri = 0, then the Rayleigh number is given by ( " #  2 ) ð1 þ N 1 Þð4N 1 þ bN 03 Þ P‘ 4N 21 b2 bð1 þ xM 3 Þ b þ T A1  P‘ 1 þ N1 e  R1 ¼ ; ð36Þ 0  2N 1 N 5 xf1 þ xM 3 ð1 þ M 1 Þg 4N 1 þ b N 03  e which expresses the modified Rayleigh number R1 as a function of the dimensionless wave number x, buoyancy magnetization parameter M1, the non-buoyancy magnetization parameter M3, medium permeability parameter P‘ (Darcy number), Taylor number T A1 , coupling parameter N1 (coupling between vorticity and spin effects), spin diffusion (couple stress) parameter N 03 and micropolar heat conduction parameter N 05 (coupling between spin and heat fluxes). The parameters N1 and N 03 measure the micropolar viscous effect and micropolar diffusion effect, respectively. The classical results in respect of Newtonian fluids can be obtained as the limiting case of present study. Setting N1 = 0, and keeping N 03 arbitrary in Eq. (36), we get R1 ¼

ð1 þ xÞð1 þ xM 3 Þð1 þ x þ T A1 P 2‘ Þ ; xP ‘ f1 þ xM 3 ð1 þ M 1 Þg

ð37Þ

which is the expression for the Rayleigh number of rotating ferromagnetic fluids in porous medium. Setting M3 = 0 in Eq. (37), we get R1 ¼

ð1 þ xÞð1 þ x þ T A1 P 2‘ Þ ; xP ‘

ð38Þ

the classical Rayleigh–Be´nard result in porous medium for the rotating Newtonian fluid case. Before we investigate the effects of various parameters and a discussion of the results depicted by figures, we first make some comments on the parameters N1, N 03 and N 05 arising due to the suspended particles. Assuming the Clausius–Duhem inequality, Eringen [31] presented certain thermodynamic restrictions which lead to non-negativeness of N1, N 03 and N 05 . It is obvious that couple stress comes into play at small values of N 03 . This supports the condition that 0 6 N1 6 1 and that N 03 is small positive real number. The parameter N 05 has to be finite because the increasing of concentration has to practically stop somewhere and hence it has to be a positive, finite real number. This typical order of magnitudes of N1, N 03 and N 05 mentioned above applies to fluid systems encountered in material processing under microgravity in space. The range of the values for the other parameters is as in the ferroconvection problem in porous medium involving Newtonian ferromagnetic fluid [22,23,29,30]. To investigate the effect of medium permeability, rotation, non-buoyancy magnetization, coupling param1 eter, spin diffusion parameter and micropolar heat conduction parameter, we examine the behaviour of dR , dP ‘ dR1 dR1 dR1 dR1 dR1 , , , and analytically. Eq. (36) gives dT A dM 3 dN 1 dN 0 dN 0 1

3

5

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Sunil et al. / International Journal of Engineering Science 44 (2006) 683–698 2

dR1 bL1 ½ðbN 03 þ 4N 1 Þfbð1 þ N 1 Þ  T A1 P 2‘ g    ; ¼ 2N 1 N 05 dP ‘ xP 2‘ ð1 þ N 1 ÞL3 4N 1 þ b N 03  e which is always negative if T A1 <

1 P 2‘

N 03 >

and

2N 1 N 05 : e

ð39Þ

ð40Þ

This shows that the medium permeability has a destabilizing effect when conditions (40) hold. In the absence of micropolar viscous effect (coupling parameter) and rotation, the medium permeability always has a destabilizing effect on the system. Thus in micropolar ferromagnetic fluid heated from below saturating a porous medium, there is a competition between the destabilizing role of medium permeability and stabilizing role of micropolar viscous effect and rotation, but there is complete destabilization in the above inequalities given by (40). Thus the destabilizing behaviour of medium permeability is virtually unaffected by magnetization parameters but is significantly affected by micropolar parameters i.e. N1, N 03 , N 05 and Taylor number T A1 . Eq. (36) also gives bL1 P ‘ ðbN 03 þ 4N 1 Þ    ; 2N 1 N 05 xð1 þ N 1 ÞL3 4N 1 þ b N 03  e

dR1 ¼ dT A1

ð41Þ

which is always negative if N 03 >

2N 1 N 05 : e

ð42Þ

This shows that the rotation has a stabilizing effect when condition (42) holds. In the absence of micropolar viscous effect (coupling parameter), the rotation always has a stabilizing effect on the system. Eq. (36) also yields  n 

o ðN 0 b þ 4N ÞP 1 ‘ T A1 bM 1 b P1‘ fð1 þ N 1 ÞN 03 b þ 4N 1 g þ 4N 21 P1‘  be þ 3 dR1 ð1 þ N 1 Þ   ¼ ; ð43Þ 0  2N N dM 3 1 5 2 0 f1 þ xM 3 ð1 þ M 1 Þg 4N 1 þ b N 3  e which is always negative if 1 b > P‘ e

and

N 03 >

2N 1 N 05 : e

ð44Þ

This shows that the non-buoyancy magnetization has a destabilizing effect when conditions (44) hold. In the absence of micropolar viscous effect, the non-buoyancy magnetization always has a destabilizing effect on the dR1 system. Here we also observe that in a non-porous medium, dM is always negative if N 03 > 2N 1 N 05 . Thus 3 the medium permeability and porosity have significant role in developing condition for the destabilizing behaviour of non-buoyancy magnetization. It can easily be derived from (36) that dR1 bL1 ¼ dN 1

h

N 03 b3 P‘

n

2N 05 e



N0

þ ð1þN3

1Þ

2

2

bð1 þ N 1 Þ  T A1 P 2‘

o

þ 8bN 1



1 P‘



o i T A1 P ‘  N N0 2N 1 þ b N 03  1e 5 þ ð1þN 4bN 1 N 03 þ 2e ðbN 05  2eÞf4N 21 þ bN 03 þ ð2N 1 þ 1Þg 2 1Þ ; h

i2 2N N 0 xL3 4N 1 þ b N 03  1e 5

 be

n

ð45Þ

which is always positive if 1 b > ; P‘ e

N 03 >

N 1 N 05 ; e

T A1 <

1 P 2‘

and

N 05 > 2e:

ð46Þ

This shows that coupling parameter has a stabilizing effect when conditions (46) hold. In the absence of rotadR1 tion and in a non-porous medium, (45) yields that dN is always positive, implying thereby the stabilizing effect 1

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of coupling parameter. Thus in rotating micropolar ferromagnetic fluid heated from below saturating a porous medium, there is a competition between the destabilizing role of medium permeability and stabilizing role of micropolar viscous effect and rotation, but there is complete stabilization in the above inequalities given by (46). Thus the stabilizing behaviour of coupling parameter is virtually unaffected by magnetization parameters but is significantly affected by micropolar parameters i.e. N1, N 03 , N 05 , Taylor number T A1 and porosity e. It follows from Eq. (36) that   N0 P2 2b3 N 1 L1 N 05 b þ N 1 bðN 05  2P ‘ Þ þ T A1 5 ‘ dR1 1 þ N1 ¼ ; ð47Þ   0 2 dN 03 2N N 1 5 0 exP ‘ L3 4N 1 þ b N 3  e which is always negative if N 05 > 2P ‘ : This shows that the spin diffusion has a destabilizing effect when condition (48) holds. Eq. (36) also yields     ð1 þ N 1 ÞN 03 b 4N 1 1 b ðbN 03 þ 4N 1 ÞP ‘ 2 2 2b L1 N 1 b þ þ 4N 1  þ T A1 dR1 P‘ P‘ e P‘ 1 þ N1 ¼ ;   0 2 dN 05 2N N 1 5 0 exL3 4N 1 þ b N 3  e

ð48Þ

ð49Þ

which is always positive if 1 b > : P‘ e

ð50Þ

This shows that the micropolar heat conduction has a stabilizing effect when condition (50) holds. Here we dR1 also observe that in a non-porous medium, dN 0 is always positive, implying thereby the stabilizing effect of 5 micropolar heat conduction. For sufficiently large values of M1 [16], we obtain the results for the magnetic mechanism operating in a porous medium ( " #  2 ) ð1 þ N 1 Þð4N 1 þ bN 03 Þ P‘ 4N 21 b2 bð1 þ xM 3 Þ b þ T A1  P‘ 1 þ N1 e  Rm ¼ R1 M 1 ¼ ; ð51Þ 0  2N 1 N 5 x2 M 3 4N 1 þ b N 03  e where Rm is the magnetic thermal Rayleigh number. As a function of x, Rm given by Eq. (51) attains its minimum when P 5 x5 þ P 4 x4 þ P 3 x3 þ P 2 x2 þ P 1 x þ P 0 ¼ 0:

ð52Þ

The coefficients P0, . . . , P5, being quite lengthy, has not been written here and are evaluated during numerical calculations. The values of critical wave number for the onset of instability are determined numerically using Newton– Raphson method by the condition dRdxm ¼ 0. With x1 determined as a solution of Eq. (52), Eq. (51) will give the required critical magnetic thermal Rayleigh number Nc. The critical magnetic thermal Rayleigh number (Nc), depends on M3, P‘, T A1 and micropolar parameters N1, N 03 and N 05 . Values of Nc determined for various values of M3, P‘, T A1 and micropolar parameters N1, N 03 and N 05 are illustrated in Figs. 2–7. Figs. 2–4 represent the plots of critical magnetic thermal Rayleigh number Nc versus P‘ for various values of T A1 , Nc versus T A1 for various values of P‘ and Nc versus M3 for various values of P‘, respectively, in the presence and absence of coupling parameter N1. From Fig. 2, the destabilizing effect of the medium

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Fig. 2. Marginal instability curve for variation of Nc versus P‘ for e = 0.5, N 03 ¼ 2, N 05 ¼ 0:5 and M3 = 5.

Fig. 3. Marginal instability curve for variation of Nc versus T A1 for e = 0.5, N 03 ¼ 2, N 05 ¼ 0:5 and M3 = 5.

Fig. 4. Marginal instability curve for variation of Nc versus M3 for e = 0.5, N 03 ¼ 2, N 05 ¼ 0:5 and T A1 ¼ 100.

Sunil et al. / International Journal of Engineering Science 44 (2006) 683–698

Fig. 5. Marginal instability curve for variation of Nc versus N1 for M3 = 5, e = 0.5, N 03 ¼ 2, N 05 ¼ 0:5 and T A1 ¼ 100.

Fig. 6. Marginal instability curve for variation of Nc versus N 03 for M3 = 5, e = 0.5, N1 = 0.2, N 05 ¼ 0:5 and T A1 ¼ 100.

Fig. 7. Marginal instability curve for variation of Nc versus N 05 for M3 = 5, e = 0.5, N1 = 0.2, N 03 ¼ 2 and T A1 ¼ 100.

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permeability is depicted for lower value of the Taylor number, whereas, for sufficiently higher values of the Taylor number, medium permeability may have a destabilizing or a stabilizing effect. This can also observe in Fig. 3. Figs. 3 and 4 indicate that the non-buoyancy magnetization have a destabilizing effect whereas rotation has stabilizing effect on the system. Also, it is observed from Figs. 2 and 4 that higher values of Nc are needed for onset of convection in the presence of N1, hence justifying the stabilizing effect of coupling parameter. Figs. 5–7 represent the plots of critical magnetic thermal Rayleigh number Nc versus N1, N 03 and N 05 for various values of P‘. Figs. 5 and 7 indicate that the coupling parameter and micropolar heat conduction parameter has a stabilizing effect, whereas Fig. 5 indicates that the spin diffusion (couple stress) parameter have a destabilizing effect on the system. 7. Principle of exchange of stabilities Here we examine the possibility of oscillatory modes, if any, on stability problem due to the presence of magnetization parameter, medium permeability, rotation and micropolar parameters. Equating the imaginary parts of Eq. (35), we obtain    1 2L2 N 1 ð1 þ N 1 Þ ri  2 ðI 1 L1 þ N 03 L2 Þb2 þ þ I1 b r2i e P‘ e e  0     2 2N 3 ð1 þ N 1 ÞL1 3 ð1 þ N 1 Þ 4N 1 ð1 þ N 1 Þ  0 þ ð2L1  N 1 L2 Þ þ N 3 L2 þ I 1 L1 b b þ P‘ P‘ eP ‘ e !   2 xR1 L3 2N 1 N 05 4N 1 ð1 þ N 1 Þ L2 0  N 3 ð1  M 2 Þ þ þ þ T A1 ðN 3 L2 þ I 1 L1 Þ b e e P‘     I 1 ð1 þ N 1 Þ 4N 1 þ 4N 1 T A1 L2  xR1 ð1  M 2 ÞL3 þ ¼ 0: ð53Þ P‘ e It is evident from Eq. (53) that ri may be either zero or non-zero, meaning that the modes may be either nonoscillatory or oscillatory. Limiting case: In the absence of rotation ðT A1 ¼ 0Þ, the vanishing of the determinant of the coefficients of A1, B1, C1 and D1 in Eqs. (31)–(34) yields (after simplification) ð54Þ iU r3i  V r2i þ iW ri þ X ¼ 0;   L2 I 1 where U ¼ b , e   b2 I 1 4N 1 ; V ¼ ½ðL1 I 1 þ L2 N 03 Þ þ bL2 ð1 þ N 1 Þ þ e P‘ e     

 L1 N 03 ð1 þ N 1 Þ 4N 1 ð1 þ N 1 Þ W ¼ b3 þ b2 ðL1  N 1 L2 Þ þ b ðI 1 L1 þ N 03 L2 Þ þ 4N 1 L2  xR1 ð1  M 2 ÞL3 I 1 ; e e P‘ P‘       0 2 ð1 þ N 1 ÞN 3 4N 1 2N 1 N 05 3 2 ð1 þ N 1 Þ 0 þb  xR1 ð1  M 2 Þ4N 1 L3 :  N 1 L1  bxR1 L3 ð1  M 2 ÞN 3  X ¼ b L1 P‘ e e P‘ Equating the imaginary parts of Eq. (54), we obtain       b I 1 4N 1 þ bN 03 1 þ N1 4N 2 b þ ri  L2 I 1 r2i þ b bL1 ð1 þ N 1 Þ þ ð4N 1 þ bN 03 Þ  1 L2  xR1 ð1  M 2 ÞL3 I 1 ¼ 0: P‘ e P‘ e e

ð55Þ It is evident from Eq. (55) that ri may be either zero or non-zero, meaning that the modes may be either nonoscillatory or oscillatory. In the absence of micropolar viscous effect (N1 = 0) and microinertia (I1 = 0), we obtain the result as

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 bL1 L2 ¼ 0: þ ri e P‘

695



ð56Þ

Here the quantity inside the bracket is positive definite because the typical values of M2 are +106 [16]. Hence ri ¼ 0;

ð57Þ

which implies that the oscillatory modes are not allowed and the principle of exchange of stabilities is satisfied for micropolar ferromagnetic fluid heated from below saturating a porous medium, in the absence of micropolar viscous effect and microinertia. Thus from Eq. (53), we conclude that the oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. Thus, it is important to note that, the Taylor number T A1 gives significant contribution in developing the oscillatory modes in the stability analysis. 8. The case of overstability The present section is devoted to find the possibility that the observed instability may really be overstability. Since we wish to determine the Rayleigh number for the onset of instability through state of pure oscillations, it suffices to find conditions for which (35) will admit of solutions with ri real. Equating real and imaginary parts of (35) and eliminating R1 between them, we obtain A2 c21 þ A1 c1 þ A0 ¼ 0; where, c1 ¼

ð58Þ

r2i ,



  2  I 1 2N 1 N 05 L2 I 1 L2 ð1 þ N 1 Þð1  M 2 Þ 2 þ I 1 L1 ð1  M 2 Þ b þ A2 ¼ 3 b; ð59Þ e2 P ‘ e e  0    

N L1 2N 1 N 05 4N 1 ðL1  L2 N 1 Þ 2N 1 N 05 0 N 03 ð1  M 2 Þ  N ð1  M Þ  A1 ¼ b4 33 þ b3 2 3 e3 e e e 

0 0 0 0 N 3 ð1 þ N 1 ÞL2 4N N 4N N L ð1  M Þ 2N ð1 þ N ÞN 1 1 1 2 1 1 0 0 5 3 5 þ N 3 ð1  M 2 Þ  þ ðN 3 L2  I 1 L1 Þ þ e3 e2 P ‘ e e3 P ‘ "

16N 21 ð1  M 2 Þ 8N 1 L2 ð1 þ N 1 Þ N 1 N 05 0 þ b2 ðL  N L Þ þ N ð1  M Þ  1 1 2 2 3 e3 e2 P ‘ e # "     2 I 1 ð1 þ N 1 Þ 2N 1 N 05 L2 2N 21 L2 ð1  M 2 Þ 7 þ I 1 L1 ð1  M 2 Þ þ 8N 1 þ þb P‘ e e eP 2‘ #   3 2N 1 L2 N 1 ð1  M 2 Þ I 2 L2 ð1  M 2 Þð1 þ N 1 Þ þ 2  T A1 I 1 N 05 þ 1 P‘ e P 3‘     L2 bL1 I 2 N 1 L2 ð1  M 2 Þ  þ T A1 I 21 ð1  M 2 Þ : ð60Þ þ 1 P‘ P‘ e The coefficient A0 being quite lengthy and not needed in the discussion of overstability, has not been written here. Since ri is real for overstability, both values of c1 ð¼ r2i Þ are positive. The sum of the roots of Eq. (58) is A1  A2 , and if this is to be negative, then A1 > 0 (because A2 > 0). It is clear from (60) that A1 is positive if N 03 ð1  M 2 Þ >

4N 1 N 05 ; e

N 03 L2 > I 1 L1 ;

L1 > N 1 L2 ;

L2 bL1 > P‘ e

and

N 1 ð1  M 2 Þ > T A1 I 1 N 05 P‘

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i.e. if 4N 1 N 05 I 1 ð1 þ xM 3 Þ ; N 03 > 0 ; > ðP r  eP r M 2 Þ eð1  M 2 Þ N 1 ð1  M 2 Þ ; T A1 < I 1 N 05 P ‘

N 03

1 > P 0r ; N1

P 0r

>

  eP r M 2 1þx þ P‘ e 1 þ xM 3

and

which implies that  

4N 1 N 05 I 1 ð1 þ xM 3 Þ 1 eP r M 2 1þx ; 0 > P 0r > þ P‘ N 03 > max and ; N1 e eð1  M 2 Þ ðP r  eP r M 2 Þ 1 þ xM 3 N 1 ð1  M 2 Þ : T A1 < I 1 N 05 P ‘ n o n 1þxo 4N 1 N 05 I 1 ð1þxM 3 Þ eP r M 2 0 1 2Þ Thus, for N 03 > max eð1M ; > P > þ P , overstability cannot , and T A1 < N I11ð1M 0 ‘ r N1 e 1þxM 3 N0 P‘ 2 Þ ðP eP r M 2 Þ r

5

occur and the principle of the exchange of stabilities is valid. Hence the above conditions are the sufficient conditions for the non-existence of overstability, the violation of which does not necessarily imply the occurrence of overstability. 9. Discussion of results and conclusions Thermal convection in a micropolar ferromagnetic fluid layer heated from below saturating a porous medium subjected to a transverse uniform magnetic field and under the simultaneous action of the rotation of the system has been considered. We have investigated the effect of various parameters like medium permeability, non-buoyancy magnetization, rotation, coupling parameter, spin diffusion parameter and micropolar heat conduction on the onset of convection. The principal conclusions from the analysis of this paper are as under: (i) For the case of stationary convection, the medium permeability has destabilizing effect in the absence of micropolar viscous effect and rotation. The destabilizing behaviour of medium permeability is virtually unaffected by magnetization parameters but is significantly affected by micropolar parameters and rotation. The rotation has a stabilizing effect under certain condition. In the absence of micropolar viscous effect (coupling parameter), the rotation always has a stabilizing effect on the system. The non-buoyancy magnetization and coupling parameter has a destabilizing effect and stabilizing effect, respectively, under certain conditions. In the absence of micropolar viscous effect, the non-buoyancy magnetization always has a destabilizing effect, whereas in the absence of rotation and in a non-porous medium, coupling parameter has a stabilizing effect on the system. The spin diffusion parameter has also a destabilizing effect under certain condition whereas the micropolar heat conduction has a stabilizing effect under certain condition. Here we also observe that in a non-porous medium, the stabilizing effect of micropolar heat conduction always has been reported. (ii) The critical magnetic thermal Rayleigh numbers for the onset of instability are also determined numerically for sufficiently large values of buoyancy magnetization parameter M1 and the results are depicted graphically. The effects of governing parameters on the stability of the system are discussed below: • Figs. 2 and 3 represent the plots of critical magnetic thermal Rayleigh number Nc versus P‘ for various values of T A1 and Nc versus T A1 for various values of P‘ in the presence and absence of coupling parameter N1. Fig. 2 illustrates that as P‘ increases, Nc decreases for small values of T A1 , whereas for the higher values of T A1 , Nc decreases for lower values of P‘ and then increases for higher values of P‘. This behaviour can also be observed in Fig. 3. In order to investigate our results, we must review the results and its physical explanations. It is well known that the rotation introduces vorticity into the fluid in case of Newtonian fluid [15]. Then the fluid moves in the horizontal planes with higher velocities. On account of this motion, the velocity of the fluid perpendicular to the planes reduces, and hence delays the onset of convection. When the fluid layer is assumed to be flowing through an isotropic and homogeneous porous medium, free from rotation or small rate of rotation, then the medium permeability has a destabilizing effect. As medium permeability increases, the void space increases

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and as a result of this, the flow quantities perpendicular to the planes will clearly be increased. Thus increase in heat transfer is responsible for early onset of convection. Thus increasing P‘ lead to decrease in Nc. In case of high rotation, the motion of the fluid prevails essentially in the horizontal planes. This motion is increased as medium permeability increases. Thus the component of the velocity perpendicular to the horizontal planes reduces, leading to delay in the onset of convection. Hence medium permeability has stabilizing effect in case of high rotation. • Fig. 4 represents the plot of critical magnetic thermal Rayleigh number Nc versus M3 for various values of P‘ in the presence and absence of coupling parameter N1. This figure indicates that the nonbuoyancy magnetization has a destabilizing effect. Also, it is observed from Fig. 4 that only small value of N1, onset of convection is delayed. This shows that higher values of Nc are needed for onset of convection in the presence of N1, hence justifying the stabilizing effect of coupling parameter. • Fig. 5 represents the plot of critical magnetic thermal Rayleigh number Nc versus N1, for various values of P‘. It is observed from Fig. 4 that the coupling parameter has a stabilizing effect. Clearly Nc increases with increasing N1. As N1 increases, concentration of microelements also increases, and as a result of this a greater part of the energy of the system is consumed by these elements in developing gyrational velocities in the fluid, leading to delay in the onset of convection. • Fig. 6 represents the plot of critical magnetic thermal Rayleigh number Nc versus N 03 for various values of P‘. Fig. 5 indicates that the spin diffusion (couple stress) parameter have a destabilizing effect on the system. Here we observe that Nc decreases with increasing N 03 . As N 03 increases, the couple stress of the fluid increases, which causes the microrotation to decrease; rendering the system prone to instability. Nevertheless, the above phenomenon is true in porous or non-porous medium. • Fig. 7 represents the plot of critical magnetic thermal Rayleigh number Nc versus N 05 for various values of P‘. Fig. 6 indicates that the micropolar heat conduction parameter has a stabilizing effect. When N 05 increases, the heat induced into the fluid due to microelements is also increased, thus inducing the heat transfer from the bottom to the top. The decrease in heat transfer is responsible for delaying the onset of convection. Thus increasing N 05 lead to increase in Nc. In other words, N 05 stabilizes the flow. (iii) The principle of exchange of stabilities is found to hold true for the micropolar ferromagnetic fluid saturating a porous medium heated from below in the absence of micropolar viscous effect, microinertia and rotation. Thus oscillatory modes are introduced due to the presence of the micropolar viscous effect, microinertia and rotation, which were non-existent in their absence. n o n 1þxo 4N 1 N 05 I 1 ð1þxM 3 Þ eP r M 2 0 1 2Þ (iv) If N 03 > max eð1M ; þ P , overstability cannot and T A1 < N I11ð1M 0 eP M Þ , N > P r > ‘ e Þ 1þxM N 05 P ‘ ðP r 2 2 1 3 r occur and the principle of the exchange of stabilities is valid. Hence the above conditions are the sufficient conditions for the non-existence of overstability, the violation of which does notnnecessarily imply o the occurrence of overstability. In the absence of rotation, the conditions N 03 > max

4N 1 N 05 ; I 1 ð1þxM 3 Þ eð1M 2 Þ ðP 0r eP r M 2 Þ

of overstability. Thus the rotational effect and P10 > N 1 are the sufficient conditions n for the non-existence r o eP r M 2 0 1þx 2Þ . In the absence of micropolar contribute two conditions i.e. P r > 1þxM 3 þ P ‘ e and T A1 < N I11ð1M N 05 P ‘ and magnetic parameters (i.e. non-magnetic fluid) in a non-porous medium the above conditions, as expected reduce to m > qK0 C1 v , i.e. kinematic viscosity is greater than thermal diffusivity, which is in good agreement with the results obtained earlier. Thus the presence of coupling between vorticity and spin effects (micropolar viscous effect), microinertia and rotation may bring overstability in the system. Thus from the above analysis, we conclude that the micropolar parameters and rotation have a profound influence on the onset of convection in porous medium. It is hoped that the present work will serve as a vehicle for understanding more complex problems involving the various physical effects investigated in the present problem. References [1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [2] Y. Kazakia, T. Ariman, Heat-conducting micropolar fluids, Rheol. Acta 10 (1971) 319–325.

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