Effect of Uniform Horizontal Magnetic Field on Thermal Instability in A Rotating Micropolar Fluid Saturating A Porous Medium

Transcription

1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue Ver. III (Jan. - Fe. 06), Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar Fluid Saturating orous Medium Bhupander Singh, Department of Mathematics,Meerut College, Meerut U.. INDI-5ooo stract: This paper deals with the theoretical investigation of thermal instaility of a thin layer of electrically non-conducting, incompressile, rotating micropolar fluid saturating a porous medium in the presence of horizontal magnetic heated from elow. dispersion relation is otained for a flat fluid layer contained etween two free oundaries using a linear staility analysis theory and normal mode analysis method. The influence of various parameters like medium permeaility, rotation, coupling parameter, micropolar coefficient, magnetic field and micropolar heat conduction parameter has een analyzed on the onset of stationary convection and results are depicted graphically. The principle of exchange (ES) is found valid. The sufficient condition for non-existence of overstaility are also otained. Keywords: Thermal Instaility; Micropolar Fluid; Horizontal Magnetic Fluid; Rotation; orous Medium I. Introduction Eringen[] was the first who introduced a continuum theory of micro-fluids in 964 which takes into account the local motions and deformations of the sustructure of the fluid. Compared to the classical Newtonian fluids, micropolar fluids are characterized y two supplementary variales such as the spin and micro-inertia tensor. Microrotation takes place due to spin whereas the micro-inertia tensor descries the distriutions of atoms and molecules inside the fluid elements. Moreover, the stress tensor in no longer symmetric. Liquid crystals, colloidal fluids, lood etc. are examples of micropolar fluids. Thermal effects in microfluids have also een discussed y Eringen[] and Kazakia et. al[0]. Sastry and Rao[9] discussed the staility of a hot rotating micropolar fluid layer etween two rigid oundaries without taking into account the rotation effect in the angular momentum equation. On the other hand Bhattacharyya and as[8] took into account the rotation effect in oth momentum equations with free oundaries. Sastry and Rao[9] noticed that when the fluid is heated from elow, the rotation of the system delays the onset of staility. Further, for small or moderate values of Taylor numer T, the system is more stale as the micropolar parameter increases. Y. Qin and.n. Kaloni[] studied the effect of rotation on thermal convection in micropolar fluid and found that the effect of rotation in a stationary convection is stailizing and that higher the values of the micropolar parameters, the more the stailizing effect. For over-staility to e possile, r (andtl numer) must e less than / ( k), where k is a coupling parameter. Sharma and Kumar[3] studied the effect of rotation on thermal convection in micropolar fluid in non-porous medium and found that the presence of coupling etween thermal and micropolar effects and uniform rotation may introduce oscillatory modes in the system. Sharma and Kumar[4] studied the effect of rotation on thermal convection in micropolar fluid in porous medium and found that the presence of rotation and coupling etween spin and heat fluxes may ring over-staility in the system and the medium permeaility also rings in over-staility in the system. The permeaility has a stailizing effect on stationary convection. Sharma and Kumar[5, 6] also studied the effect of magnetic field on the micropolar fluids heated from elow in a non-porous and porous medium, they found that in the presence of various coupling parameters, the magnetic field has a stailizing effect whereas the medium permeaility has destailizing effect on stationary convection. Reena and Rana[7] have studied the thermal convection of rotating micropolar fluid in hydromagnetic saturating a porous medium, they took the vertical magnetic field and found that the oscillatory modes are introduced due to the presence of rotation and magnetic field. For stationary convection, the medium permeaility has destailizing effect under the condition 8 H and the magnetic field has a stailizing effect under the condition andtl numer. and H 8, where is the magnetic DOI: / age

2 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. II. Mathematical Formulation In this prolem, we consider an infinite, horizontal, electrically non-conducting, incompressile micropolar fluid layer of thickness d. This layer is heated from elow such that the lower oundary is held at constant temperature T T and the upper oundary is held at fixed temperature 0 T T so that T 0 T, therefore a uniform temperature gradient dt is maintained. The physical structure of the prolem is one of infinite dz extent in x and y directions ounded y the planes z 0 and z d. The fluid layer is assumed to e flowing through an isotropic and homogeneous porous medium of poricity and the medium permeaility, which is acted upon y a uniform rotation (0, 0, 0) and gravity field g (0, 0, g). uniform magnetic field H ( H0, 0, 0) is applied along x-axis. The magnetic Raynold numer is assumed to very small so that the induced magnetic field can e neglected in comparison to the applied field. We also assumed that oth the oundaries are free and no external couples and the heat sources are present. Fig. The equations governing the motion of micropolar rotating fluids saturating a porous medium following Boussinesq approximation are as follows : The equation of continuity for an incompressile fluid is. q 0...() The equation of momentum, following Darcy law, is given y o q (. ) p g eˆ q q t z q N o e ( q Ω) ( H) H...() 4 The equation of internal momentum is given y o i N ( q. ) N ( ) (. N) N q N...(3) t The equation of energy is given y T ocv scs ( ) o Cv ( q. ) T T T ( N ). T...(4) t and the equation of state is o [ ( T T0 )]...(5) Where q, N, p,, o, s,,, e,, j,,,, T, t, T,,, To, Cv, Cs and e ˆz denote respectively filter velocity, microrotation, pressure, fluid density, reference density, density of solid matrix, fluid viscosity, coupling viscosity coefficient, magnetic permeaility, medium permeaility, micro-inertia coefficient, micropolar viscosity coefficients, temperature, time, thermal conductivity, micropolar heat conduction coefficient, coefficient of thermal expansion, reference temperature, specific heat at constant volume, specific heat of solid matrix and unit vector along z-direction. The Maxwell s equations ecome H ( q H) m H...(6) t and. H 0...(7) where m is the magnetic viscosity. III. Basic State Of The olem The asic state of the prolem is taken as q q (0, 0, 0), N N (0, 0, 0), Ω Ω (0, 0, 0), p p( z), ( z) and HH H0 (, 0, 0) DOI: / age

3 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. Using this asic state, equations () to (7) yield dp g 0 dz...(8) T z T 0...(9) and o ( z)...(0) IV. Dispersion Relation Making use of the following perturations q q q, N N N,, T T, H H h and following non dimensional variales and dropping the stars, * * * x x d, y y d, z z d, * T q q, N T N, o t d t *, d * d d, T * * p p, Ω Ω d od * h H o h, where T T is the thermal diffusivity, and using normal mode analysis ocv [ w, z, z,, mz, hz ] [ W( z), X( z), G( z), ( z), M( z), B( z)] exp[ ikx iky t] Where is the staility parameter which is, in general, a complex constant and a k k is the wave numer and eliminating X, G,, M and B we have ( K) ( D a K ) W Ra E r ( D a ) W δ δ C( D a ) K ( D a ) W K K ( D a ) j C ( D a ) K W 0 K ( K ) j C ( D a ) K ( D a ) K 0 kx Q ( D a ) D W kx Q ( D a ) ( D a ) W m m.() V. Boundary Conditions Here, we consider the case when oth oundaries are free as well as eing perfect conductor of heat, while the adjoining medium is perfectly conducting, then we have w w 0, N 0 at z 0 and z and ( N ) z 0 at z 0,...() z The oundary conditions () now ecome W D W 0 DX M, 0, DB 0 at z 0 and lso DM 0, B 0 on a perfectly conducting 3 oundary also D 0, D G 0, D M 0, D B 0, 3 D X 0 at z 0 and z x y...(3) From equation (3), we oserve that ( n) D W 0, n is a positive integer at z 0, Therefore the proper solution W characterizing the lowest mode is W W 0 sin z, Where W 0 is a constant. Sustituting for W in equation (), we have E ( K ) [ C K ] r r m K r r k x Q C K j ( ) r K C K m K j m r j K r r k x Q C K j m r δ K Ra j C K ( ) r K C K m K j m K r r k x Q C K j m Where a 4 0 E r r C K E m j...(4) DOI: / age

4 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. VI. Stationary Convection For stationary marginal state we put 0 in (4), we get ( ) r K K C K kx Q [ C K ] m m ( ) r K K C K kx Q [ C K ] m m K δ ( ) [ ] r K 40 Ra C K K C k [ ( )[ ] r k x Q C K m C K m m m When Q 0 (in the asence of magnetic field), we have K K ( K)( C K) ( K)( C K) 40 ( C K) R a K δ K C K ( K)( C K) When 0 0 (In the asence of rotation) we have K ( K )( C K ) R a δk ( C K)..(5)..(6)...(7) When 0 (In the asence of coupling etween spin and heat fluxes), equation (7) reduces to the expression otained y Sharma and Kumar (997). From (5), we get m ( K) ( ) K k x Q ( ) 4 0 ( ) R m a ( δ ) ( K) ( ) K k x Q ( ) Where K C...(8) To investigate the effects of medium permeaility K, rotation 0, coupling parameter K, micro coefficient, magnetic field Q, and micropolar heat conduction parameter δ, we examine the ehaviour of dr dr dr dr dr dr,,,,, analytically. d d0 dk d dq dδ From (8), we otain m ( K) ( ) K k x Q ( ) ( K) ( ) ( K) ( ) 3 m ( K) ( ) K k x Q ( ) r 4 0( ) dr dk a ( δ ) ( m K) ( ) K k x Q ( ) Thus, the medium permeaility has a destailizing effect when δ and 0 max.{, } Q. mkx...(9) In the asence of micropolar heat conduction parameter ( δ 0) and rotation ( 0 0), equation (9) reduces to dr ( K)( ) ( K) dk a ( ) a DOI: / age

5 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. Which is always negative, thus in this case the medium permeaility always has a destailizing effect on the system for stationary convection in porous medium whatever the strength of magnetic field is applied. In the asence of rotation ( 0 0), equation (9) reduces to dr ( K)( ) dk a ( ) dr 0 dk if dr δ and 0 if δ d Thus, the medium permeaility may have dual role in the asence of rotation whatever the strength of magnetic field is applied. In the asence of magnetic field ( Q 0), equation (9) reduces to ( K)( ) ( K)( ) K 4 0( ) K dr K dk a ( δ ) ( ) ( K) K K Thus, in the asence of magnetic field the medium permeaility has destailizing effect when From (8), we have max.{ δ, } and 0 ( ) dr 8 0 ( ) d 0 a ( δ ) m ( K)( ) K k x Q( )...(0) Thus, the rotation has destailizing effect if max.{ δ, k}. Whatever the magnetic field is applied. In the asence of micropolar heat conduction parameter ( δ 0), equation (0) reduces to dr 8 0 ( ) d0 m a ( K)( ) K k x Q( ) Thus, the rotation has stailizing effect when k, whatever the strength of magnetic field is applied in the asence of micropolar heat conduction parameter. From (8), we have m dr mk x( ) ( K) ( ) K k x Q ( ) 4 0 ( ) K dq a ( δ ) m ( K) ( ) K k x Q ( ) r..() Thus, the magnetic field has a stailizing effect when max. { δ, k} and 0 Q. k In the asence of micropolar heat conduction parameter ( δ 0) and rotation ( 0 0), we have dr dq mk x 0, a which is always positive, therefore, the magnetic field has a stailizing effect, whatever the strength of magnetic field is applied. From (8), we have m x DOI: / age

6 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. dr dk m ( K) ( ) K k x Q ( ) ( ) K 4 0( ) a ( δ ) m ( K) ( ) k k x Q( ) r Thus, the coupling parameter has stailizing effect for all 0 In the asence of rotation ( 0 0), equation () reduces to ( ) dr K dk a ( ) K if max., K...() δ, and 0 Q. mkx Thus the coupling parameter has a stailizing effect if max.{ δ, }. Whatever the strength of magnetic field is applied. In the asence of magnetic field ( Q 0), the equation () reduces to ( ) ( K)( ) K 4 ( ) K 0 dr dk a ( δ ) ( K)( ) K k Thus, in the asence of magnetic field, the coupling parameter has a stailizing effect if max.{ δ, } and 0 ( ) From (8),we have m ( δ ) ( K) K k x Q m ( K) ( ) K k x Q ( ) 4 0( ) 4 0 ( ) ( 4 δ δ ) m ( K) ( ) K k x Q ( ) 3 dr m ( δ ) ( K) ( ) K k x Q ( ) r d K...(3) a m ( δ ) ( K) ( ) K k x Q ( ) Thus, the micropolar coefficient has a stailizing effect if ( K) δ, K and 0 Q K mkx In this asence of rotation ( 0 0), equation (3) reduces to m ( δ ) ( K) K k x Q m ( δ ) ( K)( ) K k x Q( ) dr d a ( δ ) Thus, the micropolar coefficient has a stailizing effect in the asence of rotation if δ and ( K), whatever the strength of magnetic field is applied. K In the asence of micropolar heat conduction parameter ( δ 0), equation (3) reduces to DOI: / age

7 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. 4 K ( K) ( ) K m kx Q ( ) 4 0 ( ) dr r d a ( ) ( K) ( ) K m K k x Q( ) Thus, the micropolar coefficient has destailizing effect in the asence of micropolar heat conduction parameter ( δ 0) if ( K) and o Q. K k In the asence of rotation ( 0 0) and micropolar heat conduction parameter ( δ 0) 4 dr K d a ( ) DOI: / age m x, equation (3) reduces to Which is always negative, thus in the asence of rotation and micropolar heat conduction parameter, the micropolar coefficient has destailizing effect whatever the strength of magnetic field is applied. From (8), we have m ( K)( ) K k x Q ( ) r dr 4 0 ( ) d m a ( δ ) ( K)( ) K k x Q( ) Thus, the micrpolar heat conduction parameter has a stailizing effect if VII. Oscillatory Convection Equation (4) can e rewritten as [ ] ( )( l )( ) K( ) ( l ) m m 3 δ Ra ( m ) l ( )( l )( m ) k ( m ) 3 ( l ) Where ( K). K 4 0( )( l ) ( m )...(4) j K Kx Q m K E, ( K), l,, 3, K C C C ut i i in (4) and separating real and imaginary parts, we have ( ) i 4 0 ( 5 6) 5 6 i R a ( 3 4 ) and i( ) 4 0 i( 5 6) 56 R a ( 3 4 ) Where ( )( ) K ( l) l 3 m i i 3 ( l ) i m ( ) i ( 3 l Km ) i lm i δ 3 lm i δ δ 4 li m i l m i 5 ( ) lm i 6 m ( ) i li m ( ) l i Eliminating R etween (5) and (6), we get ( 3 4 ) ( ) i 4 0 ( ( 3 4 ) i( ) 4 0 i( 5 6) 5 6 i...(5)...(6)

8 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. fter putting the expressions for,, 3, 4, 5 and 6, we otain a0 i a i ai a3i a4i a5 0 or i 0 Where δ m a0 l m l m ( m l) l m m l...(7) δ l m 4 4 l m ( m lm ) l m ( m ) l m ( l m ) 4 δ δ a l m ( m l) l m ( m l) m l l m ( m l) δ 3 l m ( m l) lm ( m l) l m lm ( m l) l m δ m l l δ m 3l m δ l m ( m l)( l ) l m ( m l) l m ( l ) m l δ l m ( l ) m ( m l) lm ( K) l m ( l ) m ( m l)( l ) ( m l)( m Km δ l m m ( m l) ( m l)( l ) m l m δ m l lm 3( m l) ( l ) m 3 l m m ( l m Km) l m 3 l m ( l m Km ) ( l ) m l l m l l m l l m 3 ( ) ( ) ( ) l m ( m l)( m 3 l) ( l ) m ( l m Km δ lm ( l ) m l m 4 m ( m l)( l) δ lm 3( l ) m ( m l) 3 l m m l ( m 3 l) 3 3 m l m Km 3 l m ( m 3l ) l m 3 l m m 3 l l 3 3 ( ) 3 ( )( ) ( ) ( ) 3 3 lm m l m lm l m 3 l m ( m ( m 3 l) 4 0 l 3 m 3 ( m l) l m l δ m 8 0 l m ( m l) 4 0l m l m ( l ) 8 0 l m 6 0 l m Similarly a, a3, a4, a 5 are defined as the coefficients of i, i, i and i in equation (7). Let i r, then equation (7) ecomes a0r ar ar a3r a4r a5 0...(8) Since r, which is always positive, so that the sum of roots of equation (8) is positive ut this is impossile i a if a0 0 and a 0, ecause the sum of roots of equation (8) is 0 a. Thus, a0 0 and a 0 are the sufficient conditions for the non-existence of overstaility. Clearly, a0 0 and DOI: / age

9 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. E 4 0 if rk a δ, E δ 6 K, 0 m, ( K) 3 E /3 0l min, 0 E 0δ 0 ( m ) 4 0 r and max., min r Q, 3kx m 0kx m 3l kx m 3 kx m...(9) Hence for the conditions given in (9), overstaility cannot occur and the principle of exchange of staility (ES) is valid. VIII. Conclusions: For stationary Convection :. The critical Rayleigh numer increases as the medium permeaility decreases when 0 Q k m x max.{ δ, } and, thus the medium permeaility has a destailizing effect (see figure ). In the asence of rotation and micropolar heat conduction, the medium permeaility has destailizing effect whatever the strength of magnetic field is applied(see figure 3). In the asence of rotation, the critical Rayleigh numer increases as the medium permeaility decreases when δ and the critical Rayleigh numer increases as the medium permeaility increases when δ, thus, in the asence of rotation the medium permeaility may have dual role, whatever the strength of magnetic field is applied (see figure 4). In the asence of magnetic field, the medium permeaility has destailizing effects according to the conditions max.{ δ, }, 0 (see figure 5). ( ). When δ, the critical Rayleigh numer increases with the decreasing of rotation, thus the max.{, } rotation has destailizing effect when max.{ δ, }, whatever the magnetic field is applied (see figure 6 ) In the asence of micropolar heat conduction parameter, the rotation has stailizing effect when (see figure 7). 3. When max.{ δ, } and 0 Q, the critical Rayleigh numer increases with the increasing of k m x Chandrashekhar numer, thus the magnetic field has a stailizing effect when max.{ δ, } and 0 Q (see figure 8).In the asence of micropolar heat conduction parameter and rotation, the k m x magnetic field also has a stailizing effect without any condition (see figure 9). 4. When max.{, } and 0 Q, the coupling parameter has a stailizing effect (see figure 0). In k m x the asence of magnetic field, the coupling parameter has a stailizing effect when max.{ δ, } and 0 (see figure ).In the asence of rotation, the coupling parameter has a stailizing effect ( ) when max.{ δ, } ( see figure ). 5. When ( K) δ, K and 0 Q, the critical Rayleigh numer increases with the K k m x increasing of micropolar coefficient, thus, the micropolar coefficient has a stailizing effect under the aove conditions (see figure 3). In the asence of rotation, the micropolar coefficient has a stailizing effect when ( K) δ and, whatever the strength of magnetic field is applied. In the asence of K micropolar heat conduction parameter, the micropolar coefficient has destailizing effect when K DOI: / age

10 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. ( K) 0 and r K Q. In the asence of rotation and micropolar heat conduction parameter, the K k m x micropolar coefficient has destailizing effect without any condition. 6. When ( K), the critical Rayleigh numer increases as the micropolar heat conduction parameter K increases, thus, the micropolar heat conduction parameter has a stailizing effect if ( K). K For oscillatory Convection : The sufficient conditions for the non-existence of overstaility are given y conditions (9). References []..C. Eringen, Theory of thermo-microfluids, J. Math. nal. ppl., vol. 38, pp. 480, (97). []..C. Eringen, Int. J. Eng. Sci., 89, (964). [3]. R.C. Sharma and. Kumar, Effect of rotation on thermal convection in micropolar fluids, Int. J. Engg. Sci., vol. 3(3), pp , (994). [4]. R.C. Sharma and. Kumar, Effect of rotation on thermal convection in micropolar fluids in porous medium, Int. J. ure ppl. Math. 9, 95 (998). [5]. R.C. Sharma and. Kumar, On micropolar fluids heated from elow in hydromagnetics, J. Non-Equilirium Thermodynamics, vol. 0, pp , (995). [6]. R.C. Sharma and. Kumar, On micropolar fluids heated from elow in hydromagnetic in porous medium, Czech. J. hys, 47, 647 (997). [7]. Reena and U.S. Rana, Thermal convection of rotating micropolar fluid in hydromagnetic saturating a porous medium, IJE Transaction, 375, (008). [8]. S.. Battacharyya and M. as, Int. J. Eng. Sci. 3, 37 (985). [9]. V.U.K. Shastry and V.R. Rao, Int. J. Eng. Sci. 5, 449 (983). [0]. Y. Kazakia and T. riman, Heat conducting micropolar fluids, Rheol. cta, 0, 39 (97). []. Y. Qin and.n. Kaloni, thermal instaility prolem in a rotating micropolar fluid, Int. J. Eng. Sci. 30, 7 (99). List of Figures Fig.: Marginal instaility curve for the variation of R v/s K (Medium permeaility) for =0.5, r =, m =4, k x =0.05, δ =0.05, K=, =0.5, a=. Fig. 3: Marginal instaility curve for the variation of R v/s K (Medium permeaility) for =0.5, =0.5, K=, δ =0, Ω 0 =0, r =, m =4, k x =0.05, a= DOI: / age

11 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. Fig. 4: Marginal instaility curve for the variation of R v/s K (Medium permeaility) for r =, m =4, K=, k x =0.05, Ω 0 =0. δ =0.05, Q=45000, Fig. 5: Marginal instaility curve for the variation of R vs K (Medium permeaility) for δ =0.05, Q=0, r =, m =4, K=, k x =0.05, Ω 0 =0.5, =0.5, =0.5. Fig. 6: Marginal instaility curve for the variation of R vs Ω 0 (Rotation) for =0.5, =0.5, K=, k x =0.05, r =, m =4, δ =0.05, K =0.03, Q= DOI: / age

12 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. Fig. 7: Marginal instaility curve for the variation of R vs Rotation ( Ω 0 ) for =0.05, =0.5, r =, m =4, K=, K =0.03, δ =0, Q=45000, k x =0.05. Fig. 8: Marginal instaility curve for the variation of R vs Q for the =0.05, =0.5, r =, m =4, K=, k x =0.05, K =0.03, δ =0.05, Ω 0 =0. Fig. 9: Marginal instaility curve for the variation of R v/s Q for the =0.5, =0.5, r =, m =4, K=, K =0.03, δ =0, Ω 0 =0, k x =0.05. DOI: / age

13 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. Fig.0: Marginal instaility curve for the variation of R v/s K for =0.5, =0.5, K =0.03, k x =0.05, δ =0.05, r =, m =4, Ω 0 =0, Q= Fig.: Marginal instaility curve for the variation of R v/s K for =0.5, =0.5, K =0.03, k x =0.05, δ =0.05, r =, m =4, Ω 0 =0, Q= Fig. : Marginal instaility curve for the variation of R vs K for =0.5, =0.5, K =0.03, δ =0.05, 0 Ω =0.5, r =, m =4, k x =0.05, Q=0. DOI: / age

14 Effect of Uniform Horizontal Magnetic Field on Thermal Instaility in Rotating Micropolar. Fig. 3: Marginal instaility curve for the variation of R v/s δ (Micropolar heat conduction parameter) for =0.5, =0.5, r =, m =4, K=, K =0.03, k x =0.05, Q=45000, Ω 0 =0. DOI: / age