Sharama, Shivani

Sharma, R C

IACS Institutional Repository

Indian J. Phys. 75B(2), 137-139 (2001)U P  B— an intematioinal joumalOn couple-stress fluid heated from below in porous mediumR C Sharma* and Shivant SharmaDepartment of Mathematics, Himachal Pradesh University, Summer Hill, Shimla-171 005, India E-mail : AniLMohil@yahoo.comReceived 14 June 2000, accepted 19 October 2000Abstrart : A layer of couple-stress fluid heated from below m porous medium is considered Using linearired stability theory and normal mode analysis, the dispersion relation is obtained For stationary convection, the couple-stress postpones the onset of convection whereas the medium permeability hastens the onset of convection The principle of exchange of stabilities is valid ibr the couple-stress fluid heated from below in porous medium.Kc‘>words : Couple-stress fluid, porous medium, principle of exchange of stabilities PACS Nos. : 47.20.Ma, 47.30 Td, 47.55.MhTlie Study of a 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 find the food process industry, chemical process industry, solidification and ccntnlugal casting of metals. The basic equations of a layer of fluid heated from below in porous medium, have been derived in a treatise by Joseph [ 1 ]. The use of tlie Boussinesq approximation has been made throughout which stales that the variations of density in the equations of motion can be Ignored cvcry'where except in its association with tlie external force. When the fluid permeates an isotropic and homogeneous porous medium, the gross effect is represented b\ the Darcy's law.With the growing importance of non-Newtonian fluids in modern technology and industries, the investigations on such fluids are desirable. Stokes [2] proposed and postulated the theory of couple-stress fluid. O r: of the applications of couple-stress fluid is its use to the study of the mechanisms oflubricalion of synovial joints, which has become the object of scientific research. A human joint is a dynamically loaded bearing which fias articular cartilage as the bearing iind synovial fluid as the lubricant. When a fluid film is generated, squeeze-film action is capable of providing considerable proteaion to the cartilage surihee. The shoulder, I'lp, knee and ankle joints are the loaded-bearing synovial^“•responding Authorjoints of file human body and tliese joints have a low friction coefficient and negligible wear.Normal synovial fluid is clear or yellowish and rs a viscous, non-Newtonian fluid. According to the theory of Stokes [2], 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 13] modeled synovial fluid as a couple-stress fluid in human joints.Keeping in mind the importance of non-Newtonian fluids, convection in fluid layer heated from below and porous medium, the present paper is devoted to study the couple-stress fluid heated from below in porous medium.Consider an infinite, horizontal, incompressible, couple- stress fluid layer of thickness d, heated from below, so that the temperatures and densities at the bottom surface r * 0 arc To, pd and at the upper surface z ^ d art Td, pd respectively, and that a uniform adverse temperature gradient )0 (=  I dTIdz I) is maintained. The gravity field g(0, 0, -g) pervades the system. This fluid layer is flowing through an isotropic and homogeneous porous medium of porosity e and medium permeabihty Ai.Let p, p, T and q(u, v\ W) denote respectively the fluid pressure, density, temperature and filter velocity. Then the momentum balance, mass balance and energy balance© 2001 I ACS138 R C Sharma and Shivani Sharmae q u a tio n s  o f  c o u p le -s tre s s  f lu id  th ro u g h  p o ro u s  m ed iu m  [ 1 ,2 ] a re^ 1 1, Po )  ’V . « - 0 ,E ^  + (q.S7)T = K^^T, T h e  e q u a tio n  o f  s ta te  isp  =  p o [ l - a ( r - 7 ’o ) ] .( 1)(2)(3)(4)w h ere  th e  su ffix  z e ro  re fe rs  to  v a lu es  a t th e  re fe re n c e  level z 0 . T h e  k in e m a tic  v isc o s ity  v, c o u p le -s tre s s  v isco sity  / / ' ,  th e rm a l difT iisiv ity  k  an d  c o e ff ic ie n t o f  th e rm a l e x p a n s io na  a re  a ll a ssu m e d  to  be  co n s ta n ts . ^  = e + ( l - -  e )  isPQCfc o n s ta n t w h e re  c, an d  po, cy stan d  fo r d e n s ity  an d  sp ec if ic  h ea t o f  so l id  (p o ro u s  m a trix )  m a te ria l an d  flu id  re sp ec tiv e ly .T h e  b as ic  m o tio n le ss  so lu tio n  is^ - ( 0 , 0 , 0 ) ,  r =  r o - / f e ,  p  =  A )(l ^  a/?z). (5 )H ere , w e  u se  lin e a riz e d  s ta b ility  th eo ry  an d  n o rm a l m o d e  a n a ly s is  m e th o d . A ssu m e  sm all p e r tu rb a tio n s  a ro u n d  th e  b a s ic  so lu tio n  a n d  le t Sp, 0  an d  q{u, v, cS) d e n o te  re s p e c tiv e ly  th e  p e r tu rb a tio n s  in p re s su re  p, d en sity  p , te m p e ra tu re  T a n d  v e lo c ity  (0 , 0 , 0 ). T h e  ch a n g e  in d en sity  Sp, c a u se d  m a in ly  b y  th e  p e r tu rb a tio n  0 in  te m p e ra tu re , is g iv en  byS p  »  - a p ^ e  . (6 )T h en  th e  l in e a r iz e d  p e r tu rb a tio n  e q u a tio n s  o f  th e  c o u p le -  s tre ss  flu id  b e c o m e1 dq€ dtV .^  =  0 ,—  ^ 5 p - g a 6 - — V--------^  ^  it. p^ ’P o3t ^' V72 i f---- — V^<B +  —G ^  k\\^ Po )  E q. (9 )  ca n  b e  re -w ritte n  asA n a ly s in g  th e  d is tu rb a n c e s  in to  n o rm a l m o d e s , w e  assum e th a t th e  p e r tu rb a tio n  q u a n titie s  a re  o f  th e  fo rm[o),0] = [W{z),e{z)]ex!p(ik,,x + ikyy + nt) , ( 1 2 )w h e re  it,, ky a re  w a v e  n u m b e rs  a lo n g  th e  x - a n d  y- d irec tio n s  re sp e c tiv e ly , ^ ( =  ) is th e  re s u lta n t w a v e  n u m b eran d  n is, in g e n e ra l, a  c o m p le x  c o n s ta n t.E x p re ss in g  th e  c o o rd in a te s  x , y , z in  th e  n e w  u n it o flen g th  d  an d  le ttin g  a = kd, a  = , P i =  — , ^/ =V K  ’F  = and ; Eqs. (10) and (11), using (12),yield(7)( 8) (9)W ritin g  th e  s c a la r  c o m p o n e n ts  o f  eq . (7 )  a n d  e lim in a tin g  u, V, Sp b e tw e e n  th e m  b y  u s in g  ( 8 ), w e  o b ta in{ D^ - a ^) [ ’ + f  )jlf '( £ ) 2 - f l 2 - £ p | c r ) 0 = -  W(13)(14)S in c e  b o th  b o u n d a r i e s  a r e  m a i n ta i n e d  a t  c o n s ta n t  te m p e ra tu re s , th e  p e r tu rb a tio n s  in te m p e ra tu re  a re  z e ro  a t the b o u n d a rie s . C o n s id e r  th e  c a se  o f  tw o  free  b o u n d a rie s . The ca se  o f  tw o  free  b o u n d a rie s  is a  little  a r tif ic ia l b u t it enab les us to  fin d  a n a ly tic a l so lu tio n s  an d  to  m a k e  so m e  q u a lita tiv e  co n c lu s io n s . T h e  a p p ro p r ia te  b o u n d a ry  c o n d itio n s , with re sp e c t to  w h ich  eq s. (1 3 )  an d  (1 4 )  m u s t be  so lv e d , areW = i9 w =Q, 0 ^ 0  a t 2  =  0 a n d l .  (15)U sin g  th e  b o u n d a ry  c o n d itio n s  (1 5 ) , it c an  b e  sh o w n  th a t all th e  ev en  o rd e r  d e r iv a tiv e s  o f  W m u s t v a n ish  fo r  z ^ 0  and 1 an d  h en ce  th e  p ro p e r  so lu tio n  o f  W c h a ra c te r iz in g  the lo w est m o d e  isW Wq sin  nz, (16)w h e re  ffo is a c o n s tan t.E lim in a tin g  © b e tw e e n  eq s . (1 3 )  an d  (1 4 )  an d  substitu tin g  th e  p ro p e r  so lu tio n  (1 6 )  in  th e  re su lta n t e q u a tio n , w e  obtain th e  d isp e rs io n  re la tio nn  1-t-Jc xP (l-»‘ X + i£ /7 ic r , ) [ ^ l+ ^ < T | -f F ;r2  1 +  x ] ^  (17)w h e re  R\ = ^  , x  = ia\ =  a n d  P '■ n^ Pi7T^ VKTC^W h en  th e  in s ta b ili ty  se ts  in as  s ta tio n a ry  c o n v e c tio n , the m arg in a l s ta te  w ill b e  c h a ra c te r iz e d  b y  <7 =  0 . P u tt in g  o * ' 0, th e  d isp e rs io n  re la tio n  (1 7 )  re d u c e s  toRi =  (* 1 +  Jt) . ( 18)(10)( 11)To study the effects of couple-stress parameter and mediumdR, dRip e rm e a b ility , w e  e x a m in e  th e  n a tu re s  o f  a n d  a n a ly tic a lly . E q . (1 8 )  y ie ld sdRi f f 2 ( l  +  x )5dF xP(19)On couple-stress fluid heated from below in porous medium 139^  =  +  +  ( 2 0 )T h u s  f o r  s t a t i o n a r y  c o n v e c t i o n ,  t h e  c o u p l e - s t r e s s  postpones th e  o n se t o f  c o n v e c tio n  w h e reas  th e  m ed iu m  p e rm e a b ility  h a s te n s  th e  o n s e t  o f  c o n v e c t io n , on  th e  c o u p le - s t r e s s  f l u id  h e a t e d  f r o m  b e lo w  in  p o r o u s  m edium .M u ltip ly in g  eq . (1 3 )  b y  th e  c o m p le x  c o n ju g a te  o f  W, and u sing  (1 4 )  to g e th e r  w ith  th e  b o u n d a ry  c o n d itio n s  (1 5 ), we o b ta inFli+\^\ + P , ^ ' j h = ^ ^ ^ P i { h  + Eptff l ,)^(2\)  w here f= ^ J ^ D ^ W '(+ 2 a'^ \D W ^+a*\w f^±,f  = ^J\D€\^+a^-\e\^)dz, (2 2 )i , = l \ e \ ^ d z ,and are all positive definite.The real and imaginary parts of cq. (21) must vanish separately; and the vanising of the imaginary part gives(2J)But the quantity inside the brackets is positive definite. Hence,i m ( a )  - 0. (24)This establishes that cr is real and that the principle of exchange of stabilities is valid for the couple-stress lluid heated from below in porous medium.Rercrence.s(I I D O  Joseph Stahihty of hluui Mot!on\ Vols. I, II (tJcrbn Sprmger-Verlag) (1976)|2| V K Stokes Phys Fluids 9 1709 (1966)[31 Walicki and A Walickrt Appl Mech 4 3(>3 (1999)

On Couple-Stress Fluid Healed from Below in Porous Medium

Sharma, Shivani

On Couple-stress Fluid Heated from Below in Porous Medium

http://arxiv.iacs.res.in:8080/jspui/bitstream/10821/6251/1/On%20Couple-Stress%20Fluid%20Heated%20from%20Below%20in%20Porous%20Medium_R%20C%20Sharma%202.pdf

On Couple-Stress Fluid Healed from Below in Porous Medium

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