Kragujevac J. Sci. 3 () 7-4. UDC 53.5:536. 4 THE EFFECT OF SUCTION AND INJECTION ON UNSTEADY COUETTE FLOW WITH VARIABLE PROPERTIES Hazem A. Aia Dep. of Mahemaics, College of Science,King Saud Universiy (Al-Qasseem Branch), P.O. Box 37, Buraidah 8999, KSA e-mail: ah3@yahoo.com (Received May 9, 6) ABSTRACT. The unseady Couee flow of an incompressible viscous fluid beween wo parallel porous plaes is sudied wih hea ransfer in he presence of a uniform sucion and injecion considering variable properies. The viscosiy and hermal conduciviy of he fluid are assumed o vary wih emperaure. The fluid is subjeced o a consan pressure gradien and a uniform sucion and injecion hrough he plaes which are kep a differen bu consan emperaures. The effec of he sucion and injecion velociy and he variable viscosiy and hermal conduciviy on boh he velociy and emperaure fields is sudied. INTRODUCTION The flow wih hea ransfer of a viscous incompressible fluid beween wo parallel plaes has imporan applicaions in many devices such as aerodynamics heaing, elecrosaic precipiaion, polymer echnology, peroleum indusry. Many researchers have considered his problem under differen physical effecs [-5]. Mos of hese sudies are based on consan physical properies, alhough some physical properies are varying wih emperaure and assuming consan properies is a good approximaion as long as small differences in emperaure are involved [6]. More accurae predicion for he flow and hea ransfer can be achieved by considering he variaion of hese physical properies wih emperaure. The effec of emperaure dependen viscosiy on he flow in a channel has been sudied in he hydrodynamic case [7] and he hydromagneic case [8,9]. In he presen work, he unseady Couee flow of a viscous incompressible fluid beween wo parallel porous plaes is sudied wih hea ransfer in he presence of uniform sucion and injecion hrough he plaes wih variable physical properies. The upper plae is moving wih a consan speed and he lower plae is kep saionary. The viscosiy and hermal conduciviy of he fluid are assumed o vary wih emperaure and he wo plaes are kep a wo consan bu differen emperaures. The fluid is aced upon by a consan pressure gradien. The coupled se of he nonlinear equaions of moion and he energy equaion including he viscous dissipaion erm is solved numerically using finie differences o obain he velociy and emperaure disribuions a any insan of ime.
8 FORMULATION OF THE PROBLEM The fluid is assumed o be flowing beween wo infinie horizonal plaes locaed a he y=±h planes. The fluid beween he wo plaes is subjeced o a uniform sucion from above and injecion from below wih velociy V o j. The moion is produced by a consan pressure gradien dp/dx in he x-direcion. The wo plaes are kep a wo consan emperaures T for he lower plae and T for he upper plae wih T >T. The viscosiy of he fluid is assumed o vary exponenially wih emperaure while he hermal conduciviy is assumed o depend linearly on emperaure. The viscous dissipaion is aken ino consideraion. The flow of he fluid is governed by he Navier-Sokes equaion which has he form [,5] u u dp u u ρ( + Vo ) = + µ + µ () dx where ρ is he densiy of he fluid, µ is he viscosiy of he fluid, and u=u(y,) is he velociy componen of he fluid in he x-direcion. I is assumed ha he pressure gradien is applied a = and he fluid sars is moion from res and for >, he no-slip condiion a he plaes implies ha =: u=, >: u=, y=-h& u=u o, y=h () The energy equaion describing he emperaure disribuion for he fluid is given by [,] T T T u ρ c p ( + Vo ) = ( k ) + µ ( ) (3) where T is he emperaure of he fluid, c p is he specific hea capaciy of he fluid a consan volume, and k is he hermal conduciviy of he fluid. The las erm in he lef-hand side of Eq. (3) represens he viscous dissipaion. The emperaure of he fluid mus saisfy he boundary condiions, =: T=T >: T=T, y=-h, T=T, y=h (4a) (4b) The viscosiy of he fluid is assumed o vary wih emperaure and is defined as, µ=µ o f (T). By assuming he viscosiy o vary exponenially wih emperaure, he funcion f (T) akes he form [7], f (T)=exp(-a (T-T )). In some cases a may be negaive, i.e. he coefficien of viscosiy increases wih emperaure [8,9]. Also, he hermal conduciviy of he fluid is assumed o vary wih emperaure as k=k o f (T). We assume linear dependence for he hermal conduciviy upon emperaure in he form k=k o [+b (T-T )] [], where he parameer b may be posiive or negaive []. The problem is simplified by wriing he equaions in he non-dimensional form. To achieve his, we define he following non-dimensional quaniies, ( x, y, z) µ h P hu T T dpˆ x y z ˆ o Pˆ ρ ρ ( ˆ, ˆ, ˆ) =, =, =, uˆ =, =, G =, h h ρ µ µ o T T dxˆ o ˆf () = exp(-a (T -T )) = exp(-a), a is he viscosiy exponen, ˆf () = +b (T -T ) = +b, b is he hermal conduciviy parameer, R=ρU o h/µ o, is he Reynolds number, Pr = µ o c p /k o is he Prandl number,
9 Ec = µ o /ρ h c p (T -T ) is he Ecker number, S=V o ρh/µ o, is he sucion parameer, τ L = ( û / ŷ ) ŷ =- is he axial skin fricion coefficien a he lower plae, τ U = ( û / ŷ ) ŷ = is he axial skin fricion coefficien a he upper plae, Nu L = ( / ŷ ) ŷ =- is he Nussel number a he lower plae, Nu U = ( / ŷ ) ŷ = is he Nussel number a he upper plae, In erms of he above non-dimensional quaniies Eqs. () o (4) read (he has are dropped for convenience) u u u f u + S = G + f + ) ( ( ) (5) =: u=, >: u=, y=- & u=, y= (6) + S = R Pr f f ( ) R Pr Ec R u ( ) + + f ( )( ) (7) =: = >: =, y=-, =, y= Equaions (5) and (7) represen a sysem of coupled non-linear parial differenial equaions which can be solved numerically under he iniial and boundary condiions (6) and (8) using he finie difference approximaions. The Crank-Nicolson implici mehod is used []. Finie difference equaions relaing he variables are obained by wriing he equaions a he mid poin of he compuaional cell and hen replacing he differen erms by heir second order cenral difference approximaions in he y-direcion. The diffusion erms are replaced by he average of he cenral differences a wo successive ime levels. The nonlinear erms are firs linearized and hen an ieraive scheme is used a every ime sep o solve he linearized sysem of difference equaions. All calculaions have been carried ou for G=5, R=, Pr=, and Ec=.. (8a) (8b) RESULTS AND DISCUSSION Figures a and b presen he ime developmen of he velociy componen u a he cener of he channel (y=), for various values of he parameers a and S and for b=. The figures show ha increasing he parameer a increases u for all values of S as a resul of decreasing he viscosiy. I is also shown ha he seady sae ime of u increases wih increasing a for all S. Comparing Figs. a and b indicaes ha increasing S decreases u for all values of a. Figures a and b presen he ime developmen of he emperaure a he cener of he channel (y=), for various values of he parameers a and S and for b=. The figures show ha increasing a increases for all values of S as a resul of increasing he viscous dissipaion. I is also shown ha he seady sae ime of increases wih increasing a for all values of S. The comparison beween Figs. a and b shows ha increasing S decreases for all values of a. Also, i can be seen from Fig. a ha may exceed he value which is he emperaure of he ho plae and his is due o he viscous dissipaion. Figures 3a and 3b presen he ime developmen of he emperaure a he cener of he channel (y=), for various values of he parameers b and S and for a=. Figure 3a shows ha, in he case of zero sucion, he variaion of he emperaure wih he parameer b depends on where a crossover in - chars occurs. For small, increases wih
increasing b, however, for large, increasing b decreases. This occurs because, a low imes, he cener of he channel acquires hea by conducion from he ho plae, bu afer large ime, when u is large, he viscous dissipaion is large a he cener and cener looses hea by conducion. I is noiced ha he parameer b has no significan effec on u in spie of he coupling beween he momenum and energy equaions. I is also shown in he figures ha increasing he parameer b decreases he seady sae ime of. Figure 3b indicaes ha, in he presence of sucion, increasing b increases for all ime and leads o he suppression of he crossover in - chars. Comparing Figs. 3a and b shows ha increasing S decreases for all values of b. Tables a and b presen he variaion of he seady sae axial and ransverse skin fricion coefficiens a boh walls for various values of a and for S= and, respecively. I is clear ha increasing a increases he magniude of τ L and τ U for he case S=, as depiced in Table a. Table b shows ha, in he presence of sucion, increasing a increases he magniude of τ U for all values of a. Increasing a increases τ L for small and moderae values of a, however, increasing a more decreases τ L. Increasing S decreases τ L bu increases he magniude of τ U. Tables and 3 presen he variaion of he seady sae emperaure a y=, he Nussel number a he lower and upper plaes for various values of he parameers a and b and, respecively, for S= and. I is clear from Table ha increasing a increases, Nu L and he magniude of Nu U for all values of b. In he sucion case, as shown in Table 3, increasing a increases, bu decreases Nu L and Nu U for all values of b. Table indicaes ha increasing b decreases and he magniude of N uu for all values of a. On he oher hand, increasing b increases Nu L for all values of a excep for negaive values of a and b where increasing he magniude of b decreases Nu L. Table 3 shows ha increasing b decreases Nu L, and he magniude of Nu U bu increases for all values of a. Table. - The Seady Sae Axial and Transverse Skin Fricion Coefficiens (a) S=, (b) S= (a) S= a=-.5 a=-. a= a=. a=.5 τ L 5.38 5.4494 5.4976 5.5396 5.583 τ U -.9647-4.36-4.4978-5.9-6.948 (b) S= a=-.5 a=-. a= a=. a=.5 τ L.994.3484.357.3546.348 τ U -3.56-5.59-5.647-6.4-9.36 Table. - Variaion of he Seady Sae Temperaure and he Nussel Number a Boh Walls of he Channel wih he Parameers a and b and for S= a=-.5 a=-. a= a=. a=.5 b=-.5.8594.38.888.547.4789 b=-..878.99.9546.4.997 b=.87.99.9453.98.58 b=..844.8989.987.965.3 b=.5.783.8564.8789.936.44 Nu L a=-.5 a=-. a= a=. a=.5 b=-.5.637.3867.4888.5995 3.493 b=-..66.3969.53.653 3.463 b=.678.43.573.63 3.583 b=..74.4.545.683 3.69 b=.5.49.4436.5474.667 3.74
Nu U a=-.5 a=-. a= a=. a=.5 b=-.5 -.97 -.69 -.897 -.36 -.8695 b=-. -.59 -.938 -.48 -.589 -.89 b= -.475 -.866 -.956 -.464 -.6473 b=. -.439 -.7464 -.8454 -.9549 -.53 b=.5 -.3358 -.5673 -.639 -.78 -.44 CONCLUSION The unseady Couee flow of a viscous incompressible fluid beween wo parallel plaes has been sudied wih emperaure dependen viscosiy and hermal conduciviy in he presence of uniform sucion and injecion. I was found ha increasing he viscosiy exponen a increases he velociy u and he emperaure for all values of he sucion parameer S. Increasing S decreases u and for all values of he parameer a. In he case of zero sucion, he effec of b on depends on he ime and leads o he appearance of crossover in - chars. On he oher hand, in he presence of sucion, increasing b increases for all ime and leads o he suppression of he crossover in - chars. I was observed ha he effec of sucion on he velociy u depends grealy on he viscosiy parameer. The parameer b has a marked effec on he emperaure field while is effec on he velociy field can be enirely negleced. Table 3. - Variaion of he Seady Sae Temperaure and he Nussel Number a Boh Walls of he Channel wih he Parameers a and b and for $= a=-.5 a=-. a= A=. a=.5 b=-.5.84.99.38.36.363 b=-..37.336.3355.339.3486 b=.33.345.3454.3489.358 b=..33.354.3553.3589.3679 b=.5.37.3894.393.3966.444 Nu L a=-.5 a=-. a= A=. a=.5 b=-.5.539.53.574.544.55 b=-..448.4.487.458.3996 b=.467.437.44.3985.387 b=..393.3889.3866.3837.366 b=.5.3464.3459.3436.344.35 Nu U a=-.5 a=-. a= A=. a=.5 b=-.5.7446.3378.87.6.6346 b=-..94.6985.6345.569.95 b=.834.69.55.499.449 b=..739.5349.488.478.9 b=.5.4878.3335.938.536.886 References: [] CRAMER, K.R. and PAI, S.-I.: Magneofluid dynamics for engineers and applied physiciss. McGraw-Hill Book Co. (973). [] TANI, I.: J. of Aerospace Sci. 9, 87 (96).
[3] SOUNDALGEKAR, V.M., VIGHNESAM, N.V., and TAKHAR, H.S.: IEEE Trans. Plasma Sci. PS-7(3), 78 (979). [4] SOUNDALGEKAR, V.M. and UPLEKAR, A.G.: IEEE Trans. Plasma Sci. PS-4(5), 579 (986). [5] ATTIA, H.A.: Can. J. Phys. 76(9), 739 (998). [6] HERWIG, H. and WICKEN, G.: Warme-und Soffuberragung, 47 (986). [7] KLEMP, K., HERWIG, H. and SELMANN, M.: Enrance flow in channel wih emperaure dependen viscosiy including viscous dissipaion effecs. Proc. Third In. Cong. Fluid Mech., Cairo, Egyp 3, 57 (99). [8] ATTIA, H.A. and KOTB, N.A.: Aca Mechanica 7, 5 (996). [9] ATTIA, H.A.: Mech. Res. Comm. 6(), 5 (999). [] WHITE, M.F.: Viscous fluid flow. McGraw-Hill (99). [] AMES, W.F.: Numerical soluions of parial differenial equaions, nd ed. Academic Press, New York (977).
3 u 5 4 3 3 4 a=-.5 a= a=.5 (a) u 4 3 3 4 a=-.5 a= a=.5 (b) Fig. Time developmen of u a y= for various values of a and S (a) S=; (b) S=. (b=).5.5 3 4 a=-.5 a= a=.5 (a).8.6.4. 3 4 a=-.5 a= a=.5 (b) Fig. Time developmen of a y= for various values of a and S (a) S=; (b) S=. (b=)
4.5.5 3 4 b=-.5 b= b=.5 (a).8.6.4. 3 4 b=-.5 b= b=.5 (b) Fig. 3 Time developmen of a y= for various values of b and S (a) S=; (b) S=. (a=)