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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.1265-1271 On The Region For Linear Growth Rate Of Perturbation In Magneto-Thermal Convection In A Couple-Stress Fluid In A Porous Medium Ajaib S. Banyal1* and Daleep K. Sharma2 1* Department of Mathematics, Govt. College Nadaun, Dist. Hamirpur, (HP) INDIA 2 Department of Mathematics, Rajiv Gandhi G. C. Kotshera, Shimla (HP), INDIAABSTRACT A layer of couple-stress fluid heated thermal instability (Bénard Convection) infrom below in a porous medium is considered in Newtonian fluids, under varying assumptions ofthe presence of uniform vertical magnetic field. hydrodynamics and hydromagnetics, has been givenFollowing the linearized stability theory and by Chandrasekhar 1 and the Boussinesqnormal mode analysis, the paper through approximation has been used throughout, whichmathematical analysis of the governing equations states that the density changes are disregarded in allof couple-stress fluid convection with a uniform other terms in the equation of motion, except in thevertical magnetic field in porous medium, for any external force term. The formation and derivation ofcombination of perfectly conducting free and the basic equations of a layer of fluid heated fromrigid boundaries of infinite horizontal extension below in a porous medium, using the Boussinesqat the top and bottom of the fluid, established approximation, has been given in a treatise bythat the complex growth rate of oscillatoryperturbations, neutral or unstable, must lie Joseph 2 . When a fluid permeates through aninside a semi-circle isotropic and homogeneous porous medium, the gross effect is represented by Darcy’s law. The 2 study of layer of fluid heated from below in porous R Pl p 2 media is motivated both theoretically and by its 2 , Ep1 p 2 (1 4 F ) Pl 2 2 practical applications in engineering. Among the applications in engineering disciplines one can namein the right half of a complex -plane, Where R the food processing industry, the chemicalis the thermal Rayleigh number, F is the couple- processing industry, solidification, and thestress parameter of the fluid, Pl is the medium centrifugal casting of metals. The development ofpermeability, is the porosity of the porous geothermal power resources has increased general interest in the properties of convection in a porousmedium, p1 is the thermal Prantl number and p 2is the magnetic Prandtl number, which medium. Stommel and Fedorov 3 and Linden 4 prescribes the upper limits to the complex have remarked that the length scales characteristicgrowth rate of arbitrary oscillatory motions of of double-diffusive convecting layers in the oceangrowing amplitude in the couple-stress fluid may be sufficiently large so that the Earth’s rotationheated from below in the presence of uniform might be important in their formation. Moreover, thevertical magnetic field in a porous medium. The rotation of the Earth distorts the boundaries of aresult is important since the exact solutions of the hexagonal convection cell in a fluid through porousproblem investigated in closed form, are not medium, and this distortion plays an important roleobtainable for any arbitrary combinations of in the extraction of energy in geothermal regions.perfectly conducting dynamically free and rigid The forced convection in a fluid saturated porousboundaries. medium channel has been studied by Nield et al 5 . An extensive and updated account of convection inKey Words: Thermal convection; Couple-Stress porous media has been given by Nield andFluid; Magnetic field; PES; Chandrasekhar number.MSC 2000 No.: 76A05, 76E06, 76E15; 76E07. Bejan 6 . The effect of a magnetic field on the1. INTRODUCTION stability of such a flow is of interest in geophysics, Right from the conceptualizations of particularly in the study of the earth’s core, where the earth’s mantle, which consist of conductingturbulence, instability of fluid flows is being fluid, behaves like a porous medium that canregarded at its root. A detailed account of thetheoretical and experimental study of the onset of become conductively unstable as result of differential diffusion. Another application of the 1265 | P a g e
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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.results of flow through a porous medium in the diseases of joints are caused by or connected with apresence of magnetic field is in the study of the malfunction of the lubrication. The externalstability of convective geothermal flow. A good efficiency of the physiological joint lubrication isaccount of the effect of rotation and magnetic field caused by more mechanisms. The synovial fluid ison the layer of fluid heated from below has been caused by the content of the hyaluronic acid, a fluid given in a treatise by Chandrasekhar 1 . of high viscosity, near to a gel. A layer of such fluid heated from below in a porous medium under the MHD finds vital applications in MHD action of magnetic field and rotation may findgenerators, MHD flow-meters and pumps for applications in physiological processes. MHD findspumping liquid metals in metallurgy, geophysics, applications in physiological processes such asMHD couplers and bearings, and physiological magnetic therapy; rotation and heating may findprocesses such magnetic therapy. With the growing applications in physiotherapy. The use of magneticimportance of non-Newtonian fluids in modern field is being made for the clinical purposes intechnology and industries, investigations of such detection and cure of certain diseases with the helpfluids are desirable. The presence of small amounts of magnetic field devices. of additives in a lubricant can improve bearingperformance by increasing the lubricant viscosity Sharma and Thakur 15 have studied theand thus producing an increase in the load capacity. thermal convection in couple-stress fluid in porousThese additives in a lubricant also reduce the medium in hydromagnetics. Sharma andcoefficient of friction and increase the temperature Sharma 16 have studied the couple-stress fluidrange in which the bearing can operate. heated from below in porous medium. Kumar and Darcy’s law governs the flow of aNewtonian fluid through an isotropic and Kumar 17 have studied the combined effect ofhomogeneous porous medium. However, to be dust particles, magnetic field and rotation on couple-mathematically compatible and physically stress fluid heated from below and for the case ofconsistent with the Navier-Stokes equations, stationary convection, found that dust particles haveBrinkman 7 heuristically proposed the destabilizing effect on the system, where as the rotation is found to have stabilizing effect on the 2introduction of the term q, (now known as system, however couple-stress and magnetic field are found to have both stabilizing and destabilizingBrinkman term) in addition to the Darcian term effects under certain conditions. Sunil et al. 18 have studied the global stability for thermal q . But the main effect is through the k convection in a couple-stress fluid heated from 1 below and found couple-stress fluids are thermallyDarcian term; Brinkman term contributes very little more stable than the ordinary viscous fluids.effect for flow through a porous medium. Therefore, Pellow and Southwell 19 proved theDarcy’s law is proposed heuristically to govern the validity of PES for the classical Rayleigh-Bénardflow of this non-Newtonian couple-stress fluidthrough porous medium. A number of theories of convection problem. Banerjee et al 20 gave a newthe micro continuum have been postulated and scheme for combining the governing equations of applied (Stokes 8 ; Lai et al 9 ; Walicka 10 ). thermohaline convection, which is shown to lead to the bounds for the complex growth rate of theThe theory due to Stokes 8 allows for polar effects arbitrary oscillatory perturbations, neutral orsuch as the presence of couple stresses and body unstable for all combinations of dynamically rigid or couples. Stokes’s 8 theory has been applied to the free boundaries and, Banerjee and Banerjee 21 study of some simple lubrication problems (see e.g. established a criterion on characterization of non- Sinha et al 11 ; Bujurke and Jayaraman 12 ; oscillatory motions in hydrodynamics which was Lin 13 ). According to the theory of Stokes 8 , further extended by Gupta et al. 22 . However no such result existed for non-Newtonian fluidcouple-stresses are found to appear in noticeable configurations, in general and for couple-stress fluid magnitudes in fluids with very large molecules.Since the long chain hyaluronic acid molecules are configurations, in particular. Banyal 23 havefound as additives in synovial fluid, Walicki and characterized the non-oscillatory motions in couple- Walicka 14 modeled synovial fluid as couple stress fluid. Keeping in mind the importance of couple-stress fluid in human joints. The study is motivated stress fluids and magnetic field in porous media, asby a model of synovial fluid. The synovial fluid is stated above,, the present paper is an attempt tonatural lubricant of joints of the vertebrates. The prescribe the upper limits to the complex growthdetailed description of the joints lubrication has very rate of arbitrary oscillatory motions of growingimportant practical implications; practically all amplitude, in a layer of incompressible couple-stress 1266 | P a g e
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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp. fluid in a porous medium heated from below, in the presence of uniform vertical magnetic field, . H 0 , (4) opposite to force field of gravity, when the bounding surfaces are of infinite horizontal extension, at the dH top and bottom of the fluid and are perfectly ( H .) q H , 2 (5) conducting with any combination of dynamically dt free and rigid boundaries. d The result is important since the exact Where 1 q . , stands for the solutions of the problem investigated in closed form, are not obtainable, for any arbitrary combination of dt t perfectly conducting dynamically free and rigid convective derivatives. Here boundaries. c E (1 ) s s , is a constant, c 2. FORMULATION OF THE PROBLEM 0 v AND PERTURBATION EQUATIONS Here we consider an infinite, horizontal, while s , c s and 0 , c v , stands for the density incompressible electrically conducting couple-stress and heat capacity of the solid (porous matrix) fluid layer, of thickness d, heated from below so material and the fluid, respectively, is the that, the temperature and density at the bottom surface z = 0 are T 0 and 0 , at the upper surface z medium porosity and r ( x, y, z ) . = d are T d and d respectively, and that a uniform The equation of state is dT 0 1 T T0 , (6) adverse temperature gradient is Where the suffix zero refer to the values at the dz maintained. The fluid is acted upon by a uniform reference level z = 0. Here g 0,0, g is acceleration due to gravity and is the coefficient vertical magnetic field H 0,0, H . This fluid layer of thermal expansion. In writing the equation (1), is flowing through an isotropic and homogeneous we made use of the Boussinesq approximation, porous medium of porosity and of medium which states that the density variations are ignored permeability k 1 . in all terms in the equation of motion except the external force term. The kinematic viscosity , Let , p, T, , e and q u , v, w denote respectively the fluid density, pressure, couple-stress viscosity , magnetic permeability temperature, resistivity, magnetic permeability and e , thermal diffusivity , and electrical resistivity filter velocity of the fluid, respectively Then the , and the coefficient of thermal expansion are momentum balance, mass balance, and energy balance equation of couple-stress fluid and all assumed to be constants. Maxwell’s equations through porous medium, The basic motionless solution is governing the flow of couple-stress fluid in the q 0,0,0 , 0 (1 z ) , p=p(z), presence of uniform vertical magnetic field (Stokes 8 ; Joseph 2 ; Chandrasekhar 1 ) are T z T0 , (7) given by Here we use the linearized stability theory and p 1 the normal mode analysis method. Assume small1 q 1 perturbations around the basic solution, and let , q . q g 1 2 q k t o 0 1 0 p , , q u, v, w and h hx , h y , hz denote respectively the perturbations in density , pressure e ( H ) H , (1) 4 o p, temperature T, velocity q (0,0,0) and the . q 0 , magnetic field H 0,0, H . The change in (2) density , caused mainly by the perturbation in dT E ( q .)T 2T , (3) temperature, is given by dt 1267 | P a g e
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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp. 0 1 T T0 0 , i.e. Where we have introduced new coordinates 0 . (8) x , y , z = (x/d, y/d, z/d) in new units of length d and D d / dz . For convenience, the dashes are Then the linearized perturbation equations dropped hereafter. Also we have substituted of the couple-sress fluid reduces to nd 2 a kd, , p1 , is the thermal1 q 1 1 p g 2 q e h H t 0 k1 0 4 0 Prandtl number; p 2 , is the magnetic Prandtl, (9) k1 number; Pl . q 0 , (10) d2 is the dimensionless medium /( 0 d 2 ) E w 2 , (11) permeability, F , is the t dimensionless couple-stress viscosity parameter; . h 0 , g d 4 (12) R , is the thermal Rayleigh number and h e H 2 d 2 H . q 2 h . (13) Q , is the Chandrasekhar number. t 4 0 Also we have Substituted W W , 3. NORMAL MODE ANALYSIS Analyzing the disturbances into two- d 2 Hd dimensional waves, and considering disturbances , K K , characterized by a particular wave number, we assume that the Perturbation quantities are of the and D dD , and dropped for convenience. form Now consider the case for any combination of w, , h W z , z , K ( z ) exp the horizontal boundaries as, rigid-rigid or rigid-free z, or free-rigid or free-free at z=0 and z=1, as the case ik x ik y nt , x y (14) may be, and are perfectly conducting. The boundaries are maintained at constant temperature, Where k x , k y are the wave numbers along thus the perturbations in the temperature are zero at the boundaries. The appropriate boundary the x- and y-directions, respectively, conditions with respect to which equations (15)- , is the resultant wave number, n 1 (17), must possess a solution are k kx k y 2 2 2 W = 0= , on both the horizontal boundaries, (18) is the growth rate which is, in general, a complex DW=0, on a rigid boundary, (19) constant and, W ( z ), ( z ) and K (z ) are the D 2W 0 ,on a dynamically free boundary, 20) functions of z only. K = 0,on both the boundaries as the regions Using (14), equations (9)-(13), Within the outside the fluid are perfectly conducting, (21) framework of Boussinesq approximations, in the non-dimensional form transform to Equations (15)-(17) and appropriately adequate boundary conditions from (18)-(21), pose 1 an eigenvalue problem for and we wish toD 2 a 2 F 2 D a2 P W Ra QDD 2 2 a2 K Characterize i , when r 0 . Pl l , (15) D 2 2 a p 2 K DW , (16) 4. MATHEMATICAL ANALYSIS We prove the following theorems: and Theorem 1: If R 0 , F 0, Q 0, r 0 D and 2 a 2 Ep1 W , (17) i 0 then the necessary condition for the existence of non-trivial solution W , , K of equations (15) - (17) and the boundary conditions (18), (21) and any combination of (19) and (20) is that 1268 | P a g e
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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp. And equating the real and imaginary parts R Pl p 2 on both sides of equation (28), and cancelling . i ( 0) throughout from imaginary part, we get Ep1 p 2 (1 4 F ) Pl 2 2 r 1 1 1 1 DW a 2 W dz D 2W 2a 2 DW a 4 W dz Ra 2 D a 2 dz 2 2 F 2 2 2 2 Proof: Multiplying equation (15) by W (the P complex conjugate of W) throughout and integrating l 0 Pl 0 0 the resulting equation over the vertical range of z, we get 1 1 1 Q D 2 K 2a 2 DK a 4 K dz r Ra 2 Ep1 dz Qp2 DK a 2 K dz 2 2 2 2 2 2 1 W D 2 a 2 Wdz W D 2 a 2 Wdz 1 1 F 2 P 0 0 0 l 0 Pl 0 (29) and Ra 2 W dz Q W DD 2 a 2 Kdz , 1 1 (22) 1 1 1 1 DW a W dz Ra Ep1 dz Qp2 DK a K dz 2 2 2 2 2 2 2 2 0 0 0 Taking complex conjugate on both sides of equation 0 0 (17), we get , (30) D 2 a 2 Ep1 W , (23) Equation (30) implies that, dz , (31) 1 1 Ra 2 Ep1 dz Qp 2 DK a 2 K 2 2 2 Therefore, using (23), we get W dz D a Ep1 dz , (24) 1 1 0 0 2 2 is negative definite and also, dz ap 1 2 1 0 0 Q DK a 2 K W 2 2 2 Also taking complex conjugate on both sides of dz , (32) 0 2 0 equation (16), we get D 2 a 2 p 2 K DW , (25) We first note that since W , and K satisfy Therefore, using (25) and using boundary condition W (0) 0 W (1) , (0) 0 (1) and (18), we get K (0) 0 K (1) in addition to satisfying toW DD a 2 Kdz DW D 2 a 2 Kdz K D 2 a 2 D 2 a 2 p2 K dz ,1 1 1 governing equations and hence we have from the 2 Rayleigh-Ritz inequality 240 0 0 1 1 DW dz 2 W dz ; (26) 2 2 Substituting (24) and (26) in the right hand side of 0 0 equation (22), we get 1 1 1 W D 2 a 2 Wdz W D 2 a 2 Wdz 1 1 DK dz 2 K dz , F 2 2 2 P l 0 Pl 0 0 0 Ra 2 D 2 a 2 Ep1 dz Q K D 2 a 2 D 2 a 2 p 2 K dz 1 1 and Banerjee et al. 25 have proved that, 1 1 DW dz 2 DW dz 0 0 2 2 2, (27) 0 0 Integrating the terms on both sides of 1 1 equation (27) for an appropriate number of times by D K dz 2 DK dz 2 2 2 making use of the appropriate boundary conditions (33) (18) - (21), we get 0 0 1 1 1 Further, multiplying equation (17) and its DW a 2 W dz D 2W 2a 2 DW a 4 W dz F 2 2 2 2 2 P complex conjugate (23), and integrating by parts l 0 Pl 0 each term on right hand side of the resulting equation for an appropriate number of times and making use of boundary conditions on namely (0) 0 (1) along with (22), we get 1 Ra 2 D a 2 Ep1 dz 2 2 2 D a dz 2Ep1 r D a dz E p1 1 1 1 1 dz W dz 0 2 2 2 2 2 2 2 2 2 2 2 dz 1 1 Q D 2 K 2a 2 DK a 4 K dz Qp 2 DK a 2 K 2 2 2 2 2 0 0 0 0 0 0 , (34). (28) since r 0 , i 0 therefore the equation (34) gives, 1269 | P a g e
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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp. 2 a2 1 1 2 2 D a dz W dz , 2 is 4 at 2 2 2 (35) Since the minimum value of 0 0 a2 And a 2 2 0 . 1 1 1 Hence, if dz W dz , 2 2 (36) r 0 and i 0, E 2 p1 2 2 0 0 Pl p 2 It is easily seen upon using the boundary conditions then R . Ep1 p 2 (1 4 F ) Pl 2 2 (18) that D 1 And this completes the proof of the theorem. a 2 dz Real 2 2 part of 0 D 1 1 2 5. CONCLUSIONS D a dz a 2 dz , 2 2 0 0 The inequality (40) for r 0 and i 0 , can be written as 1 D 2 a 2 dz 2 R Pl p 2 r i 2 2 , Ep1 p 2 (1 4 F ) Pl 0 2 2 D 2 a 2 dz , 1 The essential content of the theorem, from the point of view of linear stability theory is that for 0 the configuration of couple-stress fluid of infinite horizontal extension heated form below, having top D 2 a 2 dz 1 and bottom bounding surfaces are of infinite horizontal extension, at the top and bottom of the 0 fluid and are perfectly conducting with any arbitrary 1 1 combination of dynamically free and rigid 1 2 2 1 2 dz D 2 a 2 dz , (37) 2 boundaries, in the presence of uniform vertical magnetic field parallel to the force field of gravity, 0 0 the complex growth rate of an arbitrary oscillatory motions of growing amplitude, lies inside a semi- (Utilizing Cauchy-Schwartz-inequality) circle in the right half of the r i - plane whose Upon utilizing the inequality (35) and (36), Centre is at the origin and radius is equal inequality (37) gives Pl p 2 D to R 1 1 1 , Where R is the a 2 dz W 2 2 2 dz , (38) Ep1 p 2 (1 4 F ) Pl 2 2 0 Ep1 0 thermal Rayleigh number, F is the couple-stress Now R 0, Pl 0 , 0 , F 0 and r 0 , thus parameter of the fluid, Pl is the medium upon utilizing (31) and the inequalities (32),( (33) and (38), the equation (29) gives, permeability, is the porosity of the porous F 1 2 R 1 medium, p1 is the thermal Prandtl number and p 2 is a2 W 2 dz 0 , (39) Ep1 I 1 a 2 ( 2 a 2 ) 2 P p the magnetic Prandtl number. The result is Pl l 2 0 important since the exact solutions of the problem Where investigated in closed form, are not obtainable, for 1 a2 1 2 any arbitrary combinations of perfectly conducting 1 1 I 1 r DW dz r W dz Qa 2 DK a 2 K dz 2 2 2 P 0 dynamically free and rigid boundaries and it l 0 0 provides an important improvement to Banyal and Khanna 26 ., is positive definite. Therefore , we must have REFRENCES 2 1 ( 2 a 2 ) 2 F , 1 . R P Chandrasekhar, S., Hydrodynamic and Ep1 p 2 Pl a2 l Hydromagnetic Stability, Dover which implies that Publications, New York 1981. R 4 2 F 2 1 (40) 2 . Joseph, D.D., Stability of fluid motions, Ep1 Pl p 2 Pl vol.II, Springer-Verlag, berlin, 1976. 1270 | P a g e
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Ajaib S. Banyal, Daleep K. Sharma / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue 5, September- October 2012, pp.3 .Stommel, H., and Fedorov, K.N., Small 17 . Kumar, V. and Kumar, S., On a couple- scale structure in temperature and salinity stress fluid heated from below in near Timor and Mindano, Tellus 1967, vol. hydromagnetics, Appl. Appl. Math.2011, 19, pp. 306-325 Vol. 05(10), pp. 1529-15424 . Linden, P.F.,Salt fingers in a steady shear 18 . Sunil, Devi, R. and Mahajan, A., Global flow, Geophys. Fluid Dynamics, 1974, stability for thermal convection in a couple vol.6,pp.1-27. stress-fluid,Int. comm.. Heat and Mass5 . Nield, D.A., and Junqueira, S.L.M. and Transfer 2011, 38,pp. 938-942. Lage,J.L., Fprced convection in a fluid 19 . Pellow, A., and Southwell, R.V., On the saturated porous medium ahannel with maintained convective motion in a fluid isothermal or isoflux boundaries, J. fluid heated from below. Proc. Roy. Soc. Mech. 1996, vol.322, pp. 201-214. London A, 1940, 176, 312-43.6 . Nield, D.A., and Bejan, A., Convection in 20 . Banerjee, M.B., Katoch, D.C., Dube,G.S. porous medium, Springer and Verlag, and Banerjee, K., Bounds for growth rate Newyark, 1999. of perturbation in thermohaline7 . Brinkman, H.C., problems of fluid flow convection. Proc. R. Soc. A, 1981,378, 301-04 21 . through swarms of particles and through macromolecules in solution, Banerjee, M. B., and Banerjee, B., A research(London) 1949,Vol. 2,p.190. characterization of non-oscillatory motions8 . Stokes, V.K., Couple-stress in fluids, Phys. in magnetohydronamics. Ind. J. Pure & Appl Maths., 1984, 15(4): 377-382 22 . Fluids, 1966, Vol. 9, pp.1709-15.9 . Lai, W.M., Kuei,S.C., and Mow,V.C., Gupta, J.R., Sood, S.K., and Bhardwaj, U.D., On the characterization of Rheological equtions for synovial fluids, J. nonoscillatory motions in rotatory of Biomemechanical eng.1978, vol.100, hydromagnetic thermohaline convection, pp. 169-186. Indian J. pure appl.Math. 1986,17(1) pp10 . Walicka, A., Micropolar flow in a slot 100-107. between rotating surfaces of revolution, Zielona Gora, TU Press, 1994. 23 . Banyal, A.S, The necessary condition for the onset of stationary convection in11 . Sinha, P., Singh, C., and Prasad, K.R., couple-stress fluid, Int. J. of Fluid Mech. Couple-stresses in journal bearing Research, Vol. 38, No.5, 2011, pp. 450- lubricantsand the effect of convection, 457. Wear, vol.67, pp. 15-24, 1981. 24 . Schultz, M.H., Spline Analysis, Prentice 12 . Bujurke, N.M., and Jayaraman, G., The Hall, Englewood Cliffs, New Jersy, 1973. influence of couple-stresses in squeeze films Int. J. Mech. Sci. 1982, Vol. 24, 25 . Banerjee, M.B., Gupta, J.R. and Prakash, J. On thermohaline convection of Veronis pp.369-376. type, J. Math. Anal. Appl., Vol.179, No. 2,13 . Lin, J.R., Couple-stress effect on the 1992, pp. 327-334. squeeze film characteristics hemispherical bearing with reference of 26 . Banyal, A. S. and Khanna, M., Bounds for the Complex Growth Rate of a Perturbation tosynovial joints, Appl. Mech.engg.1996, in a Couple-Stress Fluid in the Presence of vol. 1, pp.317-332. Magnetic Field in a Porous Medium, Int. J.14 . Walicki, E. and Walicka, A., Inertial effect Advances in Engg. And Technology, Vol. in the squeeze film of couple-stress fluids 4, Issue 1, 2012, pp. 483-491. in biological bearings, Int. J. Appl. Mech. Engg. 1999, Vol. 4, pp. 363-373. 15 . Sharma, R.C. and Thakur, K. D., Couple stress-fluids heated from below in hydromagnetics, Czech. J. Phys. 2000, Vol. 50, pp. 753-58 16 . Sharma, R.C. and Sharma S., On couple- stress fluid heated from below in porous medium, Indian J. Phys 2001, Vol. 75B, pp.59-61. 1271 | P a g e
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