This paper is primarily concerned with the stability of the non Newtonian fluid between two porous cylinders in the case of wide gap. The problem is discussed for Mhu&gt;0,Mhu=0 and Mhu,0. The results show that the Taylor number depends on the gap size in the case of non Newtonian fluids and the presence of suction stabilizes the flow whereas the injection destabilizes the flow. Its is found the stability of the fluid decreases when the gap increases. The non Newtonian fluid is less stable when compared to the Newtonian fluid in the case of wide gap. It is also found in the case of wide gap(for non Newtonian fluid and the cylinders are counter rotating equally) the application of injection at the outer cylinder disturbs the radial velocity will not effect any appreciable change or disturbance in the vortex cell pattern at the onset of instability

Reddy, A.G.S.

Reddy, P. G.

Reddy, Y. B.

English

Defence Science Journal

THE STABILITY OF NON-NEWTONIAN FLUlD BETWEEN THE TWO ROTATING POROUS CYJ'INDERS (WIDE GAP CASE) Indian Institute of Technology, New Delhi (Received 20 April 1977) This paper is primarily concerned with 'the stability of the non-Newtonian fluid between two porous cylin- ders in the case of wide gap. The problem is discussed for p > 0, p = 0 and p < 0. The results show that the Taylor number depends on the gap sizein the case of non-Newtonian fluids and the presenae of duction stabilizes the flow whereas the injection destabilizes the flow. It is found that the stability of the fluid de- creases when the gapinoreases. The non-Newtonian fluidis less atable when compared to the Newtonian fluid in the case of wide gap. It is also found is the case of wide gap (for non-Newtonian fluid and the eylind.ers are oounter ratating equally) the appliaation of injection a t  the outer aylinder disturbs the radial velocity and no change when suction is applied. I t  is. also conaluded that the presenoe of suotion or in- jection will not effect any appreciable ahange ar disturbance in the vortex cell patter4 a t  the onset of instability. The stability of the Newtonian h i d  between two concentric cylinders has been extensively stu- died by many authors like Taylor4 Jeffreyp2, Meksyns , etc. in the case of narrow gap approximation. Bhaskarao4, Iyenger696~7 respective1~- studied the stability of the second order fluid and non-Newtonian fluid with the narrow gap approximation. Recently Chan Man Fongs studied the stability of the visco- elastic fluid in the case of wide gap approximation and found that the stability of visco elastic fluid depends on the size of the gap between the two cylinders. Also Westbrookg, in his research paper No: 58 studied the stability of the convective flow in a porous medium. ReddyloJl gtudied the problem of porous cylinders for narrow gap case. This problem is the extension of Reddy's problem with wide gap. The r-esults are flexible for slightly wide gap and the fluid is more stable between the cylin- d ders in the present case. In the case of wide gap parameter -- is zero, the results are coinciding with Reddy's results. RO / The present paper mainly concerns the stability of the non-Newtonian fluid between two po- rous cylinders (in the presence of suction ol injection) when the gap is wide. The stability of fluid is discussed for different values of p, i.e., when the cylinders are co-rotating (p  > O), the outer cylinder is at rest ( p  = o), the cylinders are ~ounter otating ( p  < 0). In all cases the presence of ~uction or injection disturbs the non-Newtonian fluid appreciably when the cylinders are counter rotating (for the wide gap approximation). Also it is noted that the application of suction does not disturb the radial velocity. When the cylinders are counter rotating equally ( p  = -1.0) whereas the presence of injec- tion disturbs the radial velocity. Also we can find little disturbance in radial velocity for the application of suction or injection when the cylinders are co-rotating or the outer cylinder is at rest. Also it is found that the effect of suction or injection in wide gap case when the non-Newtonian fluid is between the two cylinders is considerably l e s  on the~vortex cells at  the outset of instability. EQUATION O F  MOTION The equations of motion and continuity in the cylindrical coordinates (r, 8, x) with axial sym- metry are (Navier-Stokes equation) DEF. Sex. J.,  VOL. 28, JDLP 1978 and the strew tensor for an in~omprewitle non-Newtonian flui& is given by . [ , v9.g 7t.S [em erg e, 1 Tor , 0 1 0  o 1 + 7 ;  l e v .  ego egz + - 7zr  728 7z.z J L O  0 14 ezt e ~ g  ezz J (9 where p is the pressure, p the density, u, v and w are the velacity components in the direction of r, e and z respectively. T, and p-, are the coefficients of viscosity and crasa vkicosity. - au 2 w e W 2  - ilu er, = eF=e;l. = - + - ar ifr a~ a~ , 1 e g g = 2 ( g +  :) e g z = % = -  -- 1 a2 r ae (6) 3v e,, = 2 - 1 a~ + a. v -- - --- - . ar e r ~ = e l ? r =  r a o  The Navier-Stokes equations (1)-(4) admit the steady state solution of the form R1u1 1 ='u(T) = - = - r  9' v = v ( r )  = Ar W - 1  $- - r w = o  J R1ul the suction parameter u, the radial velocity of the fluid and a the EO-efficient where X = -a linematic viscosity and R,, R2, SZ, and-Q2 are the radii and angular velocities of the inner and outer a a cylinde~b and R2 - R, = d .  Also it is assumed R, = R, - - = R, + - and d << (R,  + Rz)/2. 2 2 The arbitrary constant A and B can be determined by lip condition and are given by A = - 01q2 (1 - plq2) , B =  R,2Q, (1 P V X  ) R2h ( 1  - qA + 2) (1 - V',A + 2) (8) - where - -  - Ri " and 1 = - P =  -*I Rz By considering the s~mmetric perturbation of-solution (7) by a- periodic .djsturban?q_h the direction parallel to the axir, of rotation, the equations - go-veining - the ma@&l stability'are gi<en By - r . - ,. 2 2u + -- ' V  p2 0% -- $4; D D , ~ )  -s 2 -- V-- + - (DD* -7 P - T- . - - r -  - a -, 1 ,  2 ") ... 2:- - 6 ) I & t - A - - - -  . +  _ -  - P % ~ _ - ~ + - r - p  -I a - - - f-2 4B + k v - : )  ( A ( A - ~ ) A ~ A - + ~ ) - ]  - - (9) REDDY, el : Stability of Non-Newtonian Wuid 1 A (DD, - p2) 2) = - u D,r + - a 9' d 2Aa \ 2B - - P ~ ~ + ( A A ~ ~ - - ) ( D D , - $ ) ~ ]  r - r 2 (10) where 1 d a  DD, = D,D - - P = &(&+ t) d D =  -, D* = a - l  T1 'Pa d; +J' " =  - , y = -  (11) dr P J with the boundary, conditions u = D*u = v = 0 at r = 14 and r = R, - (12) d2 Here the wide gap approximation is considered and neglected the terms of - and higher powers. \ a Ro2 Since we are considering the terms upto - only, we cannot find the difference between D and D,. Ro Using &his approximation in the equations (11) and (12) can be written as by using the transformation 2a2d2 U - f  - S2,u a d [ (  - 2 - - D(D2 - Ro A-3  d + ((1 - P )  6 - 5 ' ) ~  -i- C ( P  - 1 )  ( 1 + ( A  - 2 )  ( 5  - 2) $) (D2 -a2)]v  (13) . [ D 2 - a s  - A  D l v  = - T a 2  ( P  - 1)  - & I , +  1) d i l -  R, R,, ( P - 1 )  7 + 2 t ( 1  - p) (5A2 - 4 A -I- 8 )  - A ( l  + 3/45 12 - 2 A2 - A(A - 1) (1 -- p) 1 2 (1 - P )  5 -t 2 Ro ( P - 1 )  7 + 2 1 I u  (14) . where J 2Q12 a 4  T =-  -S = 7% 2ds Dm. M. i-, 8%. %, 3 m  19!78 with the boundary conditions -- - - - U = . D U = V = O  at g & i  (15) The equations (13) - (15) deter&ne the eigenvalue pob1em f6r T as a function of the parameters and & 2 P, a, A . - RO M E T R O D  O F  S O L U T I O N  The eigenvalue problem givea in equations (13) and (14) with the boundary conditions (15) oan be, solved b y  using the Galerkin's methodmhicb hav-been employed in several studies by Dipr-la and KwzeglS. This method consists of expanding $he solution of fhe problem in series of a complete set of orthonormal functions e6eh of which s&s$ the boundary conditions imposed on the solution. The trial functions s a t i s h  the boundary conditions (15) are given by u = a1 (1 - 4t2J2 + a25 (1 - 452)2 1 j. v = b1 (1 - 4c2) + bk5 (1 - 4g2) (16) where a,, a,, b, and b, are constants. Applying the orthogonal conditionls j [ ., ] U ,  v l5 = D (17) -t where L (u, v) is theLenor matrix obtained on subatituting (15) in (16) and (17) we get the se&lar equa- tion as .- . I 011 612 ~ 1 3 -  G14 @l 622 623 624 , I 631 a32 633 - 1 =. (18) O L 6 4 1  G42 0 G44 1 where G l l  = 16a4 + 384a2 + 8064 622 = 8a4 + 704a2 f 31680 633 = 7a2 $ 10 + A - ( Ro 1 ( ) r -644 = 3a2 + 42 + h - Ro d 612 = &(a2 + 36) A - 3  ' d -  G13 = 18(p, $ l) -  - - 68 ( p  - IJ (3u2.+ 28) 2 - ( A  - 2) - * 2  Ro- - ( 17 d: d 614 - ( p  - 1)  f 16 (A - 3) ( p  - 1 )  ~6 -- 8 ( P  - 1 )  ( A  - 2) (a2 f 36) -- Ro * d G21 = - 176A(a2 1 3q - - R, -- d d G23 = 22(p - 1)  - 11 ( A  - 3) ( p  - 1 )  - * 22s(p - 1)  ( A  - 2 )  (a2 12) - 8, &v 148 - , .  REDDY, et a1 : Stsbility of Non-Newtoaia~~ Fluid 3 a ' ~ 2 4  = 11 ( p  +- 1) - - ( A  - 3) - - 11 (a2 4- 36) ( p  - 1) ( (2  - ( A  - 2)  4 KO (131 = a2 [- 3( 1  - 2 -1- K 2  -1 K 3  \ i -. d , 3  d K3 K + a K5 E)a2 and ( P  - l ) A  K ,  = Ro ( P - 1 )  7 + 2  1 i K 3  - ( ( p  - 1 )  (5A2 -- 4A + 8)  - 12A K ,  = A(A--1) ( p - 1 )  / 2  { ( p - 1 )  % + 2 )  6.6 With respect to the gap size and irrespective ' of the fluid the Taylor number does not differ - - - A = I S  when the cylinders are co-rotating equally, when - A = O  suction or injection is made at the outer cylinder - - - - - A  =- I  5 5. s d For constant values of d and - p, S; the Ro variation of the critical Taylor number when the suction or injection is made a t  the outer cylin- 4 s der shows that the fluid ia stable in the case of 7' suction (The result has been concluded\ after , 0 - X calculation of Taylor number for different values i cU of suction and injection in the cage of narrow 3 s gap and wide gap). It is also noteworthy that Taylor number depends on the gap size both for Newtonian and non-Newtonian fluids, which satisfy the results of Chan Man Fongs in the abgence of suction or injection. In the case of wide gap, we can fbd from Rg. 1 and Pig. 2 that the presence of suction or injection, effects more in the case of non-New- tonian fluids than in the case- of Newtonian fluids. 1.s]- r 0 1 0 -0 5 The Fig. 3 is drawn from the critical wave number P a ,  again& p, In the Pig. 3, the curves for guction or injection in different c a w  of 8 are not shown Fig. I-The behaviour of criticd Taylor number with ,because the change in wave number is little d p &-I. = 0.08, (negligibly small). The curves are highly damped Re even in wide gap is noted in the case of Newtonian . 148 a DEF. SOT. J., VOL. 28, JULY 1978 fluids for suction and injection ( ~ i ~ .  2), which differs for non-Newtoriian fluids (Fig. 2 and 3). The results obtained here from Fig. 2 and 3 are satisfying the results of Harris and Reid14 (the curve in the Galerkin methodisused where Reid use some of the Chandrasekhar's results). Fig. 2-The beha~iour of critical Taylor number with p . Fig. 3-The behaviour of critical wave number with p. The disturbance in radial velocity is calcvlated for p>O, p=O and p <O and found that the distur- bance in the radial velocity is negligibly small when they are co-rotating or the outer cylinder is at rest. The author has calculated for many values of p when p > 0 and p<O but it is given here in Table 1 only when p 2: -- 1.0. The resdts are coinciding with Reddy'sloJl results in the absence of wide gap case. The disturbance in the case of wide gap approximation for non-Newtonian fluid is more when com- pared to the narrow gap approximation. When the cylinders .are counter rotating equally (i.e. p = -. 1-0) the presence of injection disturbs the radial velocity whereas with suction there will not be any change in the radial velocity (in the case of narrow gap approximation or wide gap approximation and non-Newtonian fluid). But in the case of Newtonian fluid there is equal disturbance in the case of wide gap approximation. This can be observed clearly from Table 1. The results are coinciding with Bhaskar Rao's4 results in the absence of wide gap and suction parameber. For non-Newtonian fluids in the case of wide gap approximation when the cylinders are co-ro- tating, for applictbtion of suction or injection the variation in vortex cells 4s little (negligibly small i.e. we cannot show the difference on the graph clearly). When the other cylinder is at rest (i.e. p = 0) the effect of suction OF injection have still less disturbance on the vortex cell patter& - - 1w THE BADIAL VELOCITY uISTRIBUTLOI~ FOB V= 1 '0 DEF. SCI. d., VOL. 28, JUI,Y 1978 ' For non-Newtonian fluids (8=0.01) the presence of suction (Fig. 4) does not disturb the vortex cells much (i.e. disturbance 's negligible) for all values of in the case of wide gap but in the case of injection we can find slight disturbance (Big. 5). Also when the non-Newtonian f lu3  is between the porous walls the effect of suction or injection makes little disturbance either in the case of narrow gap approximation or in the case of wide gap approximation. Fig. 4 and Fig. 5 respectively show the case of wide gap approximation in the presence of ~uction, injection at the outer cylinder respectively. We do not h d  much disturbance either on Newtonian- or on non-Newtonian fluids in the presence of suction or injection. Here in these figures (4 and 5) i t  is n6t shown because the disturbance is very less. Fig. 4-The Vortex cells at the onset of instability p = -1.0 Fig. 5-The Vortex cella at the onset of instability p - - 1 .0  and h = 1.5. and h -- - 1.6. Here i t  can be concluded that when the non-Newtonian fluid is placed between two cylinders is a t  rest the application of suction or injection disturbs the vortex cells little (both in narrow gap ap- p~oximation and wide gap app~oximation). - C O N C L U S I O N  (i) The non-Newtonian fluid is more unstable in the wide gap case when compared to the narrow gap case. (i ) The fluid is more unstable in the wide gap approximation in the presence of injection, but suction stabilizes the flow. (iii) When the cylinders are co-rotating equally (p=l .O) the Taylor number will not be changed for any fluid when suction or injection is applied at  he outer cylinder.. (iv) Irres- pective of the gap size when the non-Newtonian fluid is kept between the two counter rotating cylinders equally ( p  = -1.0) the presence of suction will not 'disturb the radial velocity. (v) In the case of non- Newtonian fluid between the two CO-rotating or counter rotating cylinders (when $he wide gap is considered) the presence of suction or injection makes little disturbance on the vortex cells a t  the onset of instability. R E F E R E N C E S  TAYLOR, G.I., Phil. Tram. A,  (1923), 289. JEW~I$YS,  H., Phil. flag., a (1926), 833. MEKSYN., Proo. Hoy. Soc. A,  181 (1946), 480. RAO, BRASKAR., App. Sci. Res., l4(A), (1964), 199. IYENGAR, S.R.K., A r d .  Mech. Stos., 4 (1964), 7 .  IYENGAR, S.R.K., J. Sc. Engg. Res., VIII-(1964), 31. IYENGAI~, S.R.K., AppZ. Sci. Res,, 14(A) (1964), 91. CHAN MAN BONG, C.P.J., Appl. Math .  and Phys. (ZAMP)  21, Fasc, 6 (1970), 972. W E S ~ R O O H ,  D.R., Research Paper No. 58, October, 1968. %EDDY, Y.B.,'Stability of Vicn-Elaatic Fluid between two rotating porous cylinders' REDDY, P.G. & REDDY, Y.B., Def. Sci. J., 26 (1976), 47-54. DIPLOBLA, R.C., J. Fluid Xech., 9 (1960), 621. KuBzEa, U.H., 'Technical Report', (Prinaeton University) 11-29, N o ~ r .  1858 (1961), BIRRIS, D.L. & REID, W.H., J. Fluid flech., 20 (1964), 95. ' (Tensor, 20. N.S.), 

The Stability of Non Newtonian Fluid Between the Two Rotating Porous Cylinders (Wide Gap Case)

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The Stability of Non Newtonian Fluid Between the Two Rotating Porous Cylinders (Wide Gap Case)

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