Upper Limits to the Complex Growth Rate in Magneto-Thermal Instability in a Couple-Stress Fluid in a Porous Medium

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1 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 Upper Limits to the Complex Growth Rate in Magneto-Thermal Instability in a Couple-Stress Fluid in a orous Medium AJAIB S. BANYAL Department of Mathematics Govt. College Nadaun (Hamirpur (H INDIA 7733 E mail: ajaibbanyal@rediffmail.com ABSTRACT A layer of couple-stress fluid heated from below in a porous medium is considered in the presence of uniform vertical magnetic field. Following the linearized stability theory and normal mode analysis the paper through mathematical analysis of the governing equations of couple-stress fluid convection with a uniform vertical magnetic field in porous medium for any combination of perfectly conducting free and rigid boundaries of infinite horizontal extension at the top and bottom of the fluid established that the complex growth rate of oscillatory perturbations neutral or unstable must lie inside a semi-circle R ε 4 lp = Ep π εp F + π l in the right half of a complex -plane Where R is the thermal Rayleigh number F is the couple-stress parameter of the fluid l is the medium permeability ε p is the porosity of the porous medium is the thermal rantl number p and is the magnetic randtl number which prescribes the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in the couple-stress fluid heated from below in the presence of uniform vertical magnetic field in a porous medium. The result is important since the exact solutions of the problem investigated in closed form are not obtainable for any arbitrary combinations of perfectly conducting dynamically free and rigid boundaries. Keywords: Thermal convection; Couple-Stress Fluid; Magnetic field; ES; Chandrasekhar number. MSC No.: 76A5 76E6 76E5; 76E7.. INTRODUCTION Stability of a dynamical system is closest to real life in the sense that realization of a dynamical system depends upon its stability. Right from the conceptualizations of turbulence instability of fluid flows is being regarded at its root. A detailed account of the theoretical and experimental study of the onset of thermal instability (Bénard Convection in Newtonian fluids under varying assumptions of hydrodynamics and hydromagnetics has been given by Chandrasekhar (98 and the Boussinesq approximation has been used throughout which states that the density changes are disregarded in all other terms in the equation of motion except in the external force term. The formation and derivation of the basic equations of a layer of fluid heated from below in a porous medium using the Boussinesq approximation has been given in a treatise by Joseph (976. When a fluid permeates through an isotropic and homogeneous porous medium the gross effect is represented by Darcy s law. The study of layer of fluid heated from below in porous media is motivated both theoretically and by its practical applications in engineering. Among the applications in engineering disciplines one can name the food processing industry the chemical processing industry solidification and the centrifugal casting of metals. The development of geothermal power resources has increased general interest in the properties of convection in a porous medium. Stommel and Fedorov (967 and Linden (974 have remarked that the length scales characteristic of doublediffusive convecting layers in the ocean may be sufficiently large so that the Earth s rotation might be important in their formation. Moreover the rotation of the Earth distorts the boundaries of a hexagonal convection cell in a fluid through porous medium and this distortion plays an important role in the extraction of energy in geothermal regions. The forced convection in a fluid saturated porous medium channel has been studied by Nield et al (996. An extensive and updated account of convection in porous media has been given by Nield and Bejan (999. The effect of a magnetic field on the stability of such a flow is of interest in geophysics particularly in the study of the earth s core where the earth s mantle which consists of conducting fluid behaves like a porous medium that can become conductively unstable as result of differential diffusion. Another application of the results of flow through a porous medium in the presence of magnetic field is in the study of the stability of convective geothermal flow. A good account of the effect of rotation and magnetic field on the layer of fluid heated from below has been given in a treatise by Chandrasekhar (98. MHD finds vital applications in MHD generators MHD flow-meters and pumps for pumping liquid metals in metallurgy geophysics MHD couplers and bearings and physiological processes such magnetic 85

2 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 therapy. With the growing importance of non-newtonian fluids in modern technology and industries investigations of such fluids are desirable. The presence of small amounts of additives in a lubricant can improve bearing performance by increasing the lubricant viscosity and thus producing an increase in the load capacity. These additives in a lubricant also reduce the coefficient of friction and increase the temperature range in which the bearing can operate. Darcy s law governs the flow of a Newtonian fluid through an isotropic and homogeneous porous medium. However to be mathematically compatible and physically consistent with the Navier-Stokes equations Brinkman (949 heuristically proposed the introduction of the term µ q k µ q ε (now known as Brinkman term in addition to the Darcian term. But the main effect is through the Darcian term; Brinkman term contributes very little effect for flow through a porous medium. Therefore Darcy s law is proposed heuristically to govern the flow of this non-newtonian couple-stress fluid through porous medium. A number of theories of the micro continuum have been postulated and applied (Stokes (966; Lai et al (978; Walicka (994. The theory due to Stokes (966 allows for polar effects such as the presence of couple stresses and body couples. Stokes s (966 theory has been applied to the study of some simple lubrication problems (see e.g. Sinha et al (98; Bujurke and Jayaraman (98; Lin (996. According to the theory of Stokes (966 couple-stresses are found to appear in noticeable magnitudes in fluids with very large molecules. Since the long chain hyaluronic acid molecules are found as additives in synovial fluid Walicki and Walicka (999 modeled synovial fluid as couple stress fluid in human joints. The study is motivated by a model of synovial fluid. The synovial fluid is natural lubricant of joints of the vertebrates. The detailed description of the joints lubrication has very important practical implications; practically all diseases of joints are caused by or connected with a malfunction of the lubrication. The external efficiency of the physiological joint lubrication is caused by more mechanisms. The synovial fluid is caused by the content of the hyaluronic acid a fluid of high viscosity near to a gel. A layer of such fluid heated from below in a porous medium under the action of magnetic field and rotation may find applications in physiological processes. MHD finds applications in physiological processes such as magnetic therapy; rotation and heating may find applications in physiotherapy. The use of magnetic field is being made for the clinical purposes in detection and cure of certain diseases with the help of magnetic field devices. Sharma and Thakur ( have studied the thermal convection in couple-stress fluid in porous medium in hydromagnetics. Sharma and Sharma ( have studied the couple-stress fluid heated from below in porous medium. Kumar and Kumar ( have studied the combined effect of dust particles magnetic field and rotation on couple-stress fluid heated from below and for the case of stationary convection found that dust particles have destabilizing effect on the system where as the rotation is found to have stabilizing effect on the system however couple-stress and magnetic field are found to have both stabilizing and destabilizing effects under certain conditions. Sunil et al. ( have studied the global stability for thermal convection in a couplestress fluid heated from below and found couple-stress fluids are thermally more stable than the ordinary viscous fluids. ellow and Southwell (94 proved the validity of ES for the classical Rayleigh-Bénard convection problem. Banerjee et al (98 gave a new scheme for combining the governing equations of thermohaline convection which is shown to lead to the bounds for the complex growth rate of the arbitrary oscillatory perturbations neutral or unstable for all combinations of dynamically rigid or free boundaries and Banerjee and Banerjee (984 established a criterion on characterization of non-oscillatory motions in hydrodynamics which was further extended by Gupta et al. (986. However no such result existed for non-newtonian fluid configurations in general and for couple-stress fluid configurations in particular. Banyal ( have characterized the non-oscillatory motions in couple-stress fluid. Keeping in mind the importance of couple-stress fluids and magnetic field in porous media as stated above the present paper is an attempt to prescribe the upper limits to the complex growth rate of arbitrary oscillatory motions of growing amplitude in a layer of incompressible couple-stress fluid in a porous medium heated from below in the presence of uniform vertical magnetic field opposite to force field of gravity when the bounding surfaces are of infinite horizontal extension at the top and bottom of the fluid and are perfectly conducting with any combination of dynamically free and rigid boundaries. The result is important since the exact solutions of the problem investigated in closed form are not obtainable for any arbitrary combination of perfectly conducting dynamically free and rigid boundaries.. FORMULATION OF THE ROBLEM AND ERTURBATION EQUATIONS Here we consider an infinite horizontal incompressible electrically conducting couple-stress fluid layer of 86

3 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 T thickness d heated from below so that the temperature and density at the bottom surface z = are ρ and ρ T at the upper surface z = d are d and d respectively and that a uniform adverse temperature gradient dt β = ( H is maintained. The fluid is acted upon by a uniform vertical magnetic field H. This fluid layer is flowing through an isotropic and homogeneous porous medium of porosity ε and of medium k permeability. µ and q ( u v w Let ρ p T η e denote respectively the fluid density pressure temperature resistivity magnetic permeability and filter velocity of the fluid respectively Then the momentum balance mass balance and energy balance equation of couple-stress fluid and Maxwell s equations through porous medium governing the flow of couple-stress fluid in the presence of uniform vertical magnetic field (Stokes(955; Joseph (976; Chandrasekhar (98 are given by ' q p δρ µ + q. q = + g + ν ε t ε ρo ρ k ρ q µ + e ( H H 4πρ. q = ( dt E dt + ( q. T = κ T (3. H = (4 dh ε = ( H. q+ εη H dt d = + ε q. dt t Where ρscs E = ε + ( ε ρcv (5 stands for the convective derivatives. Here ρ is a constant while s c s ρ and c v stands for the density and heat capacity of the solid (porous matrix material and the fluid respectively ε is the medium porosity and r( x y z. The equation of state is [ ( T ] ρ = ρ α T (6 g( g Where the suffix zero refer to the values at the reference level z =. Here is acceleration due to gravity and α is the coefficient of thermal expansion. In writing the equation ( we made use of the Boussinesq approximation which states that the density variations are ignored in all terms in the equation of ' motion except the external force term. The kinematic viscosityν µ couple-stress viscosity magnetic µ thermal diffusivity κ and electrical resistivity η and the coefficient of thermal expansion permeability e α are all assumed to be constants. The basic motionless solution is o ( 87

4 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 q = ( ρ = ρ ( + αβ p=p(z (7 Here we use the linearized stability theory and the normal mode analysis method. Assume small z perturbations around the basic solution and let δρ p T = βz + T δ θ q ( u v w respectively the perturbations in density ρ pressure p temperature T velocity ( H q h = ( hx hy hz and ( denote and the magnetic field H =. The change in density δρ caused mainly by the perturbation θ in temperature is given by ρ + δρ = ρ[ α( T + θ T ] = ρ αρθ δρ = αρ θ i.e.. (8 Then the linearized perturbation equations of the couple-sress fluid reduces to q ε t ' µ = δp gαθ ν ρ k ρ q+ 4 µ e h H πρ (9. q = ( E θ = βw + κ θ t (. h = ( ε t h = H. q+ εη h. (3 3. NORMAL MODE ANALYSIS Analyzing the disturbances into two-dimensional waves and considering disturbances characterized by a particular wave number we assume that the erturbation quantities are of the form [ θ h ] = W ( z ( z Where w z [ K( ] z k x ky ( exp ik x ik y nt x + y + (4 are the wave numbers along the x- and y-directions respectively ( k = k x + k y resultant wave number n is the growth rate which is in general a complex constant and K(z W( z ( z is the are the functions of z only. Using (4 equations (9-(3 Within the framework of Boussinesq approximations in the non-dimensional form transform to F + W = Ra + QDD ( a K ε l l (5 and p K = DW Ep = W ( Where we have introduced new coordinates x ' y' z' D = d/' (6 (7 and = (x/d y/d z/d in new units of length d and. For convenience the dashes are dropped hereafter. Also we have substituted nd a = kd = ν 88

5 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 ν p = κ p ν = is the thermal randtl number; η is the magnetic randtl number; dimensionless medium permeability gαβd 4 R = κν ' µ /( ρd F = ν is the thermal Rayleigh number and W =W = βd κ k l = d is the is the dimensionless couple-stress viscosity parameter; µ eh d Q = 4πρ νηε Hd K = K εη is the Chandrasekhar number. Also we D =dd ( and dropped have Substituted and for convenience. Now consider the case for any combination of the horizontal boundaries as rigid-rigid or rigid-free or freerigid or free-free at z= and z= as the case may be and are perfectly conducting. The boundaries are maintained at constant temperature thus the perturbations in the temperature are zero at the boundaries. The appropriate boundary conditions with respect to which equations (5-(7 must possess a solution are W = = on both the horizontal boundaries (8 DW= on a rigid boundary (9 D W = on a dynamically free boundary ( K = on both the boundaries as the regions outside the fluid are perfectly conducting ( Equations (5-(7 and appropriately adequate boundary conditions from (8-( pose an eigenvalue problem for and we wish to Characterize i when r. 4. MATHEMATICAL ANALYSIS We prove the following theorems: and Theorem : If R F Q r W K then the necessary condition for the existence of nontrivial solution ( i of equations (5 - (7 and the boundary conditions (8 ( and any combination of (9 and ( is that R ε lp Ep 4π εpf + π l. roof: Multiplying equation (5 by W (the complex conjugate of W throughout and integrating the resulting equation over the vertical range of z we get F + W ( D a W W ( D a W l ( = Ra ε W + QW DD a K l ( Taking complex conjugate on both sides of equation (7 we get Ep = W Therefore using (3 we get W = Ep (3 (4 Also taking complex conjugate on both sides of equation (6 we get [ D a p ] K = DW (5 Therefore using (5 and using boundary condition (8 we get W DD ( a K = DW K = K p K (6 89

6 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 Substituting (4 and (6 in the right hand side of equation ( we get F + W ( D a W W ( D a W l ε l = Ra Ep + QK p (7 Integrating the terms on both sides of equation (7 for an appropriate number of times by making use of the appropriate boundary conditions (8 -( we get F { + } DW a W D W a DW a W ε l l K = Ra { D + a + Ep } ( DK a K Q D K a DK a K Qp +. (8 ( And equating the real and imaginary parts on both sides of equation (8 and cancelling i throughout from imaginary part we get r F { DW a W } D W a DW a W Ra { D a } l = + ε l 4 Q D K + a DK + a K + r Ra Ep ( + Qp DK a K (9 and { DW + a W } = Ra Ep + Qp ( DK + a K ε (3 Equation (3 implies that ( DK a K Qp + Ra Ep (3 is negative definite and also a Q { DK + a K } W εp (3 We first note that since W and K satisfy W ( = = W( ( = = ( K ( = = K( and in addition to satisfying to governing equations and hence we have from the Rayleigh-Ritz inequality Schlutz (973 DW π W ; and Banerjee et al. (99 have proved that DK π D W π DW D K π K DK (33 Further multiplying equation (7 and its complex conjugate (3 and integrating by parts each term on right hand side of the resulting equation for an appropriate number of times and making use of boundary conditions on ( = = ( namely along with ( we get 9

7 ISSN (aper ISSN 5-9 (Online Vol.5 No Ep r ( D + a + E p = W since r i And E p therefore the equation (34 gives W W (34 (35 (36 It is easily seen upon using the boundary conditions (8 that ( D + a = = Real part of ( a D ( D a ( D a ( D a ( D a Upon utilizing the inequality (35 and (36 inequality (37 gives ( D + a (37 (Utilizing Cauchy-Schwartz-inequality W Ep (38 Now R l ε F and r thus upon utilizing (3 and the inequalities (3( (33 and (38 the equation (9 gives F ( π R I + a π + a + a W l εp Ep (39 Where r I = + ( DW + a W + Qa ( DK + a K ε l is positive definite. Therefore we must have R Ep π εp which implies that R 4π F π + Ep εp l (π + a a F l ( π +a Since the minimum value of is Hence if r and i then And this completes the proof of the theorem. a 4π at a = π R Ep ε lp 4π εpf + π l.. (4 9

8 ISSN (aper ISSN 5-9 (Online Vol.5 No CONCLUSIONS The inequality (4 for r and i can be written as R ε 4 lp r + i Ep π εp F + l π The essential content of the theorem from the point of view of linear stability theory is that for the configuration of couple-stress fluid of infinite horizontal extension heated form below having top and bottom bounding surfaces are of infinite horizontal extension at the top and bottom of the fluid and are perfectly conducting with any arbitrary combination of dynamically free and rigid boundaries in the presence of uniform vertical magnetic field parallel to the force field of gravity the complex growth rate of an arbitrary oscillatory motions of growing amplitude lies inside a semi-circle in the right half of the r i - plane whose Centre is at the origin and radius R ε lp is equal to Ep 4π εpf + π l Where R is the thermal Rayleigh number F is the couple-stress parameter of the fluid l is the medium permeability ε p is the porosity of the porous medium is the thermal p randtl number and is the magnetic randtl number and it provided an important improvement in the domain of bounds of perturbation to Banyal and Khanna (. The result is important since the exact solutions of the problem investigated in closed form are not obtainable for any arbitrary combinations of perfectly conducting dynamically free and rigid boundaries. 6. REFRENCES Banerjee M. B. and Banerjee B. 984 A characterization of non-oscillatory motions in magnetohydronamics. Ind. J. ure & Appl Maths. 5(4: Banerjee M.B. Gupta J.R. and rakash J. 99 On thermohaline convection of Veronis type J. Math. Anal. Appl. Vol.79 No. pp Banerjee M.B. Katoch D.C. Dube G.S. and Banerjee K. 98 Bounds for growth rate of perturbation in thermohaline convection. roc. R. Soc. A Banyal A.S. The necessary condition for the onset of stationary convection in couple-stress fluid Int. J. of Fluid Mech. Research Vol. 38 No.5 pp Banyal A. S. and Khanna M. Bounds for the Complex Growth Rate of a erturbation in a Couple-Stress Fluid in the resence of Magnetic Field in a orous Medium Int. J. Advances in Engg. And Technology (accepted for publication. Brinkman H.C. 949 problems of fluid flow through swarms of particles and through macromolecules in solution research(londonvol. p.9. Bujurke N.M. and Jayaraman G. 98 The influence of couple-stresses in squeeze films Int. J. Mech Sci. Vol. 4 pp Chandrasekhar S. 98 Hydrodynamic and Hydromagnetic Stability Dover ublications NewYork. Gupta J.R. Sood S.K. and Bhardwaj U.D. 986 On the characterization of nonoscillatory motions in rotatory hydromagnetic thermohaline convection Indian J. pure appl.math.7( pp-. Joseph D.D. 976 Stability of fluid motions vol.ii Springer-Verlag berlin. Kumar V. and Kumar S. On a couple-stress fluid heated from below in hydromagnetics Appl. Appl. Math Vol. 5( pp Lai W.M. KueiS.C. and MowV.C. 978 Rheological equtions for synovial fluids J. of Biomemechanical eng. vol. pp Lin J.R. 996 Couple-stress effect on the squeeze film characteristics of hemispherical bearing with reference tosynovial joints Appl. Mech.engg. vol. pp Linden.F. 974 Salt fingers in a steady shear flow Geophys. Fluid Dynamics v.6pp. -7 Nield D.A. and Bejan A. 999 Convection in porous medium Springer and Verlag Newyark. Nield D.A. and Junqueira S.L.M. and LageJ.L.996 forced convection in a fluid saturated porous medium ahannel with isothermal or isoflux boundaries J. fluid Mech. vol.3 pp. -4. ellow A. and Southwell R.V. 94 On the maintained convective motion in a fluid heated from below. roc. Roy. Soc. London A Schultz M.H. 973 Spline Analysis rentice Hall Englewood Cliffs New Jersy. Sharma R.C. and Thakur K. D. Couple stress-fluids heated from below in hydromagnetics Czech. J. hys. Vol. 5 pp

9 ISSN (aper ISSN 5-9 (Online Vol.5 No.3 5 Sharma R.C. and Sharma S. On couple-stress fluid heated from below in porous medium Indian J. hys Vol. 75B pp Sinha. Singh C. and rasad K.R. Couple-stresses in journal bearing lubricantsand the effect of convection Wear vol.67 pp Stokes V.K. 966 Couple-stress in fluids hys. Fluids Vol. 9 pp Stommel H. and Fedorov K.N. 967 Small scale structure in temperature and salinity near Timor and Mindano Tellus vol. 9 pp Sunil Devi R. and Mahajan A. Global stability for thermal convection in a couple stress fluid Int. comm.. Heat and Mass Transfer 38pp Walicka A. Micropolar flow in a slot between rotating surfaces of revolution Zielona Gora TU ress 994. Walicki E. and Walicka A. 999Inertial effect in the squeeze film of couple-stress fluids in biological bearings Int. J. Appl. Mech. Engg. Vol. 4 93

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