A Mathematical Theorem on the Onset of Stationary Convection in Couple-Stress Fluid

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1 Journl of Applied Fluid Mechnics, Vol. 6, No., pp. 9-96,. Avilble online t ISSN 75-57, EISSN A Mthemticl Theorem on the Onset of Sttionry Convection in Couple-Stress Fluid A. S. Bnyl eprtment of Mthemtics, NSCBM, Govt. College Hmirpur, (HP,) 775, Indi Emil: jibbnyl@rediffmil.com (Received April, ; ccepted August 6, ) ABSTRACT The therml instbility of couple-stress fluid heted from below is investigted. Following the linerized stbility theory nd norml mode nlysis, the pper mthemticlly estblishes tht the onset of instbility t mrginl stte, cnnot mnifest itself s sttionry convection, if the therml Ryleigh number R nd the couple-stress prmeter F, stisfy the inequlity F F F R 7F F, nd when the couple-stress prmeter F is 7 infinitesimlly smll, R F, the result which lso clerly mthemticlly estblished the stbilizing chrcter of the couple-stress. Keywords: Therml convection; Couple-stress fluid; Ryleigh number. F g dimensionless wve number couple-stress prmeter ccelertion due to grvity, m / s k wve number, /m kx, ky wve numbers in x nd y-directions, /m n growth rte, /s Ryleigh number R T temperture, K qu,, v w components of velocity fter perturbtion NOMENCLATURE coefficient of therml expnsion, /K uniform temperture grdient, K / m perturbtion in temperture, K therml diffusivity, m / s kinemtic viscosity, m / s kinemtic viscoelsticity, m / s,, el opertor, Curly opertor nd erivtive with respect to z (=d/dz). INTROUCTION Right from the conceptuliztions of turbulence, instbility of fluid flows is being regrded t its root. The therml instbility of fluid lyer with mintined dverse temperture grdient by heting the underside plys n importnt role in Geophysics, interiors of the Erth, Ocenogrphy nd Atmospheric Physics etc. A detiled ccount of the theoreticl nd experimentl study of the onset of Bénrd Convection in Newtonin fluids, under vrying ssumptions of hydrodynmics nd hydromgnetics, hs been given by Chndrsekhr (98). The use of Boussinesq pproximtion hs been mde throughout, which sttes tht the density chnges re disregrded in ll other terms in the eqution of motion except the externl force term. Shrm (976) hs considered the effect of suspended prticles on the onset of Bénrd convection in hydromgnetics. The fluid hs been considered to be Newtonin in ll bove studies. Chndr (98) observed tht in n ir lyer, convection occurred t much lower grdients thn predicted if the lyer depth ws less thn 7mm nd clled this motion columnr instbility. However for

2 A. S. Bnyl et l. / JAFM, Vol. 6, No., pp. 9-96,. lyer deeper thn mm, Bénrd type cellulr convection ws observed. Thus there is contrdiction between the theory nd experiment. Scnlon nd Segel (97) hve considered the effect of suspended prticles on the onset of Bénrd convection nd found tht the criticl Ryleigh number ws reduced solely becuse the het cpcity of the pure fluid ws supplemented by tht of the prticles. With the growing importnce of non-newtonin fluids in modern technology nd industries, the investigtions on such fluids re desirble. Stokes (966) proposed nd postulted the theory of couple-stress fluid. One of the pplictions of couple-stress fluid is its use to the study of the mechnism of lubriction of synovil joints, which hs become the object of scientific reserch. A humn joint is dynmiclly loded bering which hs rticulr crtilge s the bering nd synovil fluid s lubricnt. When fluid film is generted, squeeze film ction is cpble of providing considerble protection to the crtilge surfce. The shoulder, knee, hip nd nkle joints re the lodedbering synovil joints of humn body nd these joints hve low-friction coefficient nd negligible wer. Norml synovil fluid is cler or yellowish nd is viscous, non-newtonin fluid. According to the theory of Stokes (966), couple-stresses re found to pper in noticeble mgnitude in fluids very lrge molecules. Since the long chin hyluronic cid molecules re found s dditives in synovil fluid. Wlicki nd Wlick (999) modeled synovil fluid s couple-stress fluid in humn joints. Shrm nd Shrm () hve studied the couple-stress fluid heted from below in porous medium. The use of mgnetic field is being mde for the clinicl purposes in detection nd cure of certin diseses with the help of mgnetic field devices. Shrm nd Thkur () hve studied the therml convection in couple-stress fluid in porous medium in hydromgnetics. Shrm nd Shrm () hve studied the effect of suspended prticles on couple-stress fluid heted from below in the presence of rottion nd mgnetic field nd found tht rottion hs stbilizing effect while dust prticles hve destbilizing effect on the system. Sunil et l. () hve studied the effect of suspended prticles on couple-stress fluid heted nd soluted from below in porous medium nd found tht suspended prticles hve stbilizing effect on the system. Kumr nd Kumr () hve studied the combined effect of dust prticles, mgnetic field nd rottion on couplestress fluid heted from below nd for the cse of sttionry convection, found tht dust prticles hve destbilizing effect on the system, where s the rottion is found to hve stbilizing effect on the system, however couple-stress nd mgnetic field re found to hve both stbilizing nd destbilizing effects under certin conditions. Shivkumr et l. () hve studied the effect of non-uniform temperture grdients on the onset of convection in couple-stress fluid sturted porous medium. Keeping in mind the importnce of non-newtonin fluids, the present pper is n ttempt to chrcterize the onset of instbility t mrginl stte s sttionry convection nlyticlly, in lyer of incompressible couple-stress fluid heted from below in the presence of suspended prticles.. FORMULATION OF THE PROBLEM AN PERTURBATION EQUATIONS Considered n infinite, horizontl, incompressible couple-stress fluid lyer, of thickness d, heted from below so tht, the temperture nd density t the bottom surfce z = re, respectively nd t the lower surfce z = d ret, nd tht uniform dverse temperture grdient T nd q u, v, w d d dt dz is mintined. Let, p, denote respectively the density, pressure, temperture nd velocity of the fluid respectively. Then the momentum blnce, mss blnce equtions of the couple-stress fluid (Stokes 966, Chndrsekhr 98 nd Scnlon nd Segel 97) re q q. q p g q t. q The eqution of stte is T T Where the suffix zero refer to the vlues t the g,, g is ccelertion reference level z =. Here _. due to grvity nd x x, y, z Let c v denote the het cpcity of the fluid t constnt volume, ssuming tht the fluid is in therml equilibrium, the eqution of het conduction gives. c v t q T q T or T q. T T t The kinemtic viscosity, couple-stress viscosity, therml diffusivity q/ cv, nd coefficient of therml expnsion re ll ssumed to be constnts. The initil stte of the system is tken to be quiescent lyer (no settling). The bsic motionless solution is given by q,,, T T z, z (5) Assume smll perturbtions round the bsic solution nd let, p q u, v, w denote respectively, nd () () () () 9

3 A. S. Bnyl et l. / JAFM, Vol. 6, No., pp. 9-96,. the perturbtions in density, pressure p, temperture T nd couple-stress fluid velocity (,,). The chnge in density cused minly by the perturbtion in temperture is given by: (6) Then the linerized perturbtion equtions of the couple stress fluid become q p g q t. q w t. NORMAL MOE ANALYSIS (7) (8) (9) Anlyzing the disturbnces into norml modes, we ssume tht the Perturbtion quntities re of the form x y w, W z, z Exp ik x ik y nt () Where k x, k y re the wve numbers long the x nd y- directions respectively k k x k y, is the resultnt wve number nd n is the growth rte which is, in generl, complex constnt. Using Eq. (), Eq. (7) nd Eq. (9), on using Eq. (8) nd, in non-dimensionl form, become F W gd () d p W () nd d where kd,, p, F, d dz nd dropping for convenience. Here p, is the therml prndtl number nd F is the couple-stress prmeter. d Substituting W W nd in Eq. () nd Eq. () nd dropping for convenience, in non-dimensionl form becomes F W R () p W () gd where R, is the therml Ryleigh number. Since both the boundries re mintined t constnt temperture, the perturbtions in the temperture re zero t the boundries. The cse of two free boundries is little rtificil but it is most pproprite for Stellr tmospheres nd enbles us to find nlyticl solutions nd to mke some qulittive nd quntittive conclusions. The pproprite boundry conditions with respect to which Eq. () nd Eq. () must be solved re W =, t z = nd z = (5) nd the constitutive equtions of the couple-stress fluid re v v j ij e ij nd e i ij x j x i The conditions on free surfce re u w xz z x (6) v w yz z y From the eqution of continuity Eq. (8) differentiting with respect to z, we conclude tht w x y z z which implies tht (7) w w,, t z = nd z = d (8) z z Using Eq. (), the boundry conditions Eq. (8) in non-dimensionl form trnsform to W W t z = nd z = (9) We prove the following theorem. Theorem: If R, F nd then the necessry condition for the existence of non-trivil solution W, of Eqs. () nd () together with boundry conditions Eqs. (5) nd (9) is tht F F F R 7F F further, when the couple-stress prmeter F is 7 infinitesimlly smll, then R F 9

4 A. S. Bnyl et l. / JAFM, Vol. 6, No., pp. 9-96,. Proof: When the instbility sets in sttionry convection nd the `exchnge principle` is vlid, the neutrl or mrginl stte will be chrcterized by nd hence the relevnt governing Eqs. () nd () reduces to F W R () W () together with the relevnt boundry conditions, W W W () on both the horizontl boundries t z = nd z = Multiplying Eq. () by W (the complex conjugte of W) throughout nd integrting the resulting eqution over the verticl rnge of z, we get F W Wdz W Wdz () R W dz Tking complex conjugte on both sides of Eq. (), we get W () Substituting for W in the right hnd side of Eq. (), we get F W Wdz W Wdz R dz (5) Integrting the terms on both sides of Eq. (5) for n pproprite number of times by mking use of the pproprite boundry conditions Eq. (), we get 6 F W W W W dz W W W dz R dz (6) We first note tht since W nd stisfy W() W() nd () () in ddition to stisfying to governing equtions nd hence we hve from the Ryleigh-Ritz (97) inequlity dz dz (8) Further, for W () W (), Bnerjee et l. (99) hve shown tht W dz W dz (9) Further, multiplying Eq. () by (the complex conjugte of ), integrting by prts ech term of the resulting eqution on the right hnd side for n pproprite number of times nd mking use of boundry condition on nmely () (), it follows tht dz Rel prt dz Rel prt of Wdz Wdz W dz W dz W dz dz W dz Utilizing Cuchy Schwrtz inequlity () So tht using inequlity Eq. (8), we obtined from bove inequlity dz W dz And hence inequlity Eqs. () nd (), gives dz W dz () () Now, if R, utilizing the inequlities Eqs. (7), (9), nd Eq. (), the Eq. (6) gives R I F W dz () W dz W dz (7) nd 9

5 A. S. Bnyl et l. / JAFM, Vol. 6, No., pp. 9-96,. Where non-existence of sttionry convection t mrginl stte I W W W dz, is F F F positive definite. nd therefore, we must hve is tht R, 7F F R () F for the configurtion under considertion. nd thus we necessrily hve (b) When the couple-stress prmeter F is infinitesimlly smll, then the necessry condition for F F F the onset of instbility s sttionry convection t R (5) mrginl stte for configurtion under considertion is 7F tht the inequlity (7) is stisfied. Thus the sufficient F condition for the non-existence of sttionry convection 7 t mrginl stte is tht R F, for the Since the minimum vlue of F, is configurtion under considertion. F F F 7F F for F F (6) Further, when the couple-stress prmeter F is infinitesimlly smll, then in the expnsion of right hnd side of Eq. (6), the higher powers of F cn be ignored, nd then we hve, Eq. () gives 7 R F (7) The sme result Eq. (7) lso follows for very-very smll vlues of F nd ignoring the higher powers of F in the expnsion of right hnd side of Eq. (5) nd this completes the proof of the theorem. The theorem implies from the physicl point of view tht the onset of instbility t mrginl stte in couplestress fluid heted from below, cnnot mnifest itself s sttionry convection, if the therml Ryleigh number R nd the couple-stress prmeter F, stisfy the 7 inequlity R F, when couple-stress prmeter is infinitesimlly smll nd the boundries re dynmiclly free. (c) In the inequlities (5) nd (7), the therml Ryleigh number R, is directly proportionl to the couple-stress prmeter F, which clerly mthemticlly estblished the stbilizing chrcter of the couple-stress, for the configurtion under considertion s derived by Shrm nd Shrm (). ACKNOWLEGMENT The uthor is highly thnkful to the referees for their very constructive, vluble suggestions nd useful technicl comments, which led to significnt improvement of the pper. REFERENCES Bnerjee, M.B., Gupt, J.R. nd Prksh, J. (99). On thermohline convection of Veronis type, J. Mth Anl. Appl., Vol.79, 7-. Chndrsekhr, S. (98). Hydrodynmic nd Hydromgnetic Stbility, over Publiction, NewYork Chndr, K., (98). Stbility of fluids heted from below, Proc. Roy. Soc. London, vol. A6, Kumr, V. nd Kumr, S. (). On couple-stress fluid heted from below in hydromgnetics, Appl. Appl. Mth., Vol. 5(), 59-5 Scnlon, J.W. nd Segel, L.A. (97). Some effects of suspended prticles on the onset of Bénrd convection, Phys. Fluids. Vol. 6, CONCLUSIONS In this pper, the onset of instbility s sttionry convection of couple-stress fluid heted from below is considered nd the immedite nlytic conclusions of the theorem proved bove, re s follows: () The necessry condition for the onset of instbility s sttionry convection t mrginl stte for configurtion under considertion, is tht the inequlity (5) is stisfied. Thus the sufficient condition for the Schultz, M.H. (97). Spline Anlysis, Prentice Hll, Englewood Cliffs, New Jersy. Shrm, R.C., Prksh, K. nd ube, S.N. (976).Effect of suspended prticles on the onset of Bénrd convection in hydromgnetics, J. Mth. Anl. Appl., USA, Vol. 6, 7-5. Shrm, R.C. nd Thkur, K.. (). Couple stressfluids heted from below in hydromgnetics, Czech. J. Phys., Vol. 5,

6 A. S. Bnyl et l. / JAFM, Vol. 6, No., pp. 9-96,. Shrm, R.C. nd Shrm, M. (). Effect of suspended prticles on couple-stress fluid heted from below in the presence of rottion nd mgnetic field, Indin J. pure. Appl. Mth., 5(8), Shrm, R.C. nd Shrm S. (). On couple-stress fluid heted from below in porous medium, Indin J. Phys, Vol. 75B, Shivkumr, I.S., Kumr, S.S. ndevrju, N. (), Effect of non- uniform temperture grdients on the onset of convection in couple-stress fluid sturtd porous medium, J. of Appld Fluid Mechnics, (Accepted for publictions). Stokes, V.K.(966). Couple-stress in fluids, Phys. Fluids, Vol. 9, Sunil, Shrm, R.C. nd Chndel, R.S. (). Effect of suspended prticles on couple-stress fluid heted nd soluted from below in porous medium, J. of Porous Medi, 7() 9-8. Wlicki, E.nd Wlick, A. (999). Inertil effect in the squeeze film of couple-stress fluids in biologicl berings, Int. J. Appl. Mech. Engg., Vol.,