A Semi-Analytical Solution for a Porous Channel Flow of a Non-Newtonian Fluid

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1 Journal o Applied Fluid Mechanics, Vol. 9, No. 6, pp , 6. Available online at ISSN , EISSN A Semi-Analytical Solution or a Porous Channel Flow o a Non-Newtonian Fluid P. Vimala and P. Blessie Omega Department o Mathematics, Anna University, Chennai, India Corresponding Author vimalap@annauniv.edu (Received July, 5; accepted February 3, 6) ABSTRACT A theoretical study o steady laminar two-dimensional low o a non-newtonian luid in a parallel porous channel with variable permeable walls is carried out. Solution by Dierential Transorm Method (DTM) is obtained and the low behavior is studied. The non-newtonian luid considered or the study is couple stress luid. Thus, in addition to the eects o inertia and permeabilities on the low, the couple stress eects are also analyzed. Results are presented and comparisons are made between the behaviour o Newtonian and non- Newtonian luids. Keywords: Laminar low; Couple stress luid; Porous channel; Variable permeability; Dierential transorm method. NOMENCLATURE, g generic unctions h height o the channel l dimensionless couple stress parameter q velocity vector Reynolds number in case (i) R R Reynolds number in case (ii) * R entrance Reynolds number U () entrance velocity u V V v axial velocity permeability velocity at the lower plate permeability velocity at the upper plate normal velocity suction/injection parameter or case (i) suction/injection parameter or case (ii) dimensionless y coordinate luid density luid viscosity ratio o injection/suction Reynolds number to entrance Reynolds number-case ratio o injection/suction Reynolds number to entrance Reynolds number-case couple stress parameter. INTRODUCTION Laminar low through porous channels (or) ducts have gained considerable importance because o their applications in industries and biophysical laboratories, such as binary gas diusion, iltration, ablation cooling, transpiration cooling, paper manuacturing, oil production, blood dialysis in artiicial kidneys and blood low in capillaries. Several researchers have studied the wall porosity eects on the two-dimensional steady laminar low o an incompressible Newtonian luid through uniorm porous walls. Berman (953) has investigated the eect o wall porosity on the twodimensional steady laminar low o an incompressible Newtonian luid in a rectangular channel and obtained solution by perturbation. Yuan (956) has analyzed the two-dimensional steady laminar low o an incompressible Newtonian luid in channels with porous walls at moderate Reynolds numbers. Terrill (964) has solved the low in a two-dimensional channel with permeable walls by a numerical technique with

2 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. uniorm injection or suction. Terrill and Shrestha (965) have studied the low through parallel porous walls o dierent permeabilities or small Reynolds number and solved using perturbation technique and numerical method. Many studies on luid lows are conined to the use o Newtonian luids owing to its simple nature o the linear constitutive equation. However, in many practical applications, the luids used are non- Newtonian. In the classical continuum theory, various non-newtonian models are used to describe the non-linear relation between stress and rate o strain. In particular, power-law model, cubic equation model, Oldroyd B model, Rivlin-Ericksen model, Hershel-Bulkley model, Bingham plastic model and Maxwell model have received remarkable attention among luid dynamists. Further, analytical solution o luid low problems using Newtonian luid is mostly possible and available in literature whereas it is rare in low problems using non-newtonian luids. Thereore, it is reasonable to attempt at determining an analytical or a semi-analytical solution procedure or a non- Newtonian luid low problem. The theory o couple stress luids proposed by Stokes (966) shows all the important eatures and eects o couple stresses in luid medium. The basic equations are similar to Navier Stokes equations. The importance o couple stress eects in low between parallel porous plates have been analyzed by several researchers. Kabadi (987) has studied the low o couple stress luid between two parallel horizontal stationary plates with luid injection through the lower porous plate. Ariel () has provided an exact solution or low o second grade luid through two parallel porous lat walls. Kamisili (6) has analyzed the laminar low o a non-newtonian luid in channels with the upper plate stationary, while the lower plate is uniormly porous and moving in x-direction with constant velocity. Srinivacharya et al. (9) have analyzed the low and heat transer o couple stress luid in a porous channel with expanding and contracting walls. Srinivacharya et al. () have investigated the low o couple stress luid between two porous plates or suction at both plates with dierent permeabilities. A powerul semi-analytical technique namely Dierential Transorm Method (DTM) proposed by Zhou (986) has been used to solve both linear and non-linear initial value problems in electric circuit theory. Several researchers like Chen and Ho (999), Bert (), Kurnaz et al. (5), Erturk et al. (7), Rashidi and Sadri (), Rashidi et al. (), Rashidi and Mohimanian Pour (), Rashidi and Erani (), Vimala and Blessie (3) have used DTM to solve partial dierential equations, system o algebraic equations and luid low problems with highly non-linear terms. In this paper, a steady laminar two-dimensional low o couple stress luid in a porous channel with variable permeablities is considered. The eects o inertia, porosity and couple stresses on the low behavior are studied. At every stage o the ormulation and solution, the present problem in the Newtonian case is compared with existing literature and the results agree well, which validate the present results.. MATHEMATICAL FORMULATION The problem o two-dimensional steady laminar low o an incompressible viscous couple stress luid in a parallel porous channel is considered. Various types o low occur and they all under two major categories: (i) V V and (ii) V V. y=h y Fig.. Flow Geometry. Based on the Stokes microcontinuum theory, the governing equations o the low in the absence o body orces and body moments are the continuity equation and the momentum equations (Stokes 966) given by. q, () v y= u x V V q. q p( q) ( ( ( q))). The boundary conditions are ux (,), vx (,) V uxh (, ), vxh (, ) V (), (3) and the no-couple stress conditions at the boundary are u y at y, u y at y h. y Taking, the above equations become h. (4) u v, (5) x h u v u p u u u, x h x x h h 78

3 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , u u u x h h x v v v p v v u, x h h x x h v v v x h h x (6) (7) where p is the pressure and is the kinematic viscosity. The boundary conditions are given by ux (,), vx (,) V, (8) ux (,), vx (,) V. and u at, u at.. (9) Using a stream unction (Berman, 953) o the orm hu () V V ( x, ) x s( ) s() s() s() s(), () where U() u(, ) d is the entrance velocity, the partial dierential Eqs. (5)-(7) reduce to an ordinary dierential equation. Without loss o generality, it is assumed that V V in the irst case and V V in the second case.. Case(i) V V The stream unction o Eq.() in this case becomes hu () ( x, ) V x ( ), () where s( ) V. () ( ), s() V Further, Eq. (6) and Eq. (7) respectively become V U Vx h h 4 () p,(3) v x h h V V p V h h iv, (4) 3 Eq.(4) is a unction o only and thereore p x. (5) Using Eq.(5) in Eq.(3), a ith order dierential equation is obtained as l R ( ) A, (6) v where A is an arbitrary constant, l is the h dimensionless couple stress parameter and R Vh / is the Reynolds number. It may be noted that Eq.(6) reduces to its Newtonian counterpart when l is made to vanish. The boundary conditions rom Eq. (8) and Eq. (9) become (), (), (), (), (), (). (7) For the case o suction at the upper wall and injection at the lower wall, takes the range and R. For injection at the upper wall and suction at the lower wall, the range is and R. For suction at both walls, it is required that R,, and or injection at both walls, it is R,.. Case (ii) V V In this case, the stream unction becomes hu () ( x, ) V x g( ), (8) where V s( ) and g( ). V s() Further Eq. (6) and Eq. (7) in this case reduce to V gg g U Vx h h g 4 () p,(9) v x g h h V V iv p g V gg g h 3 h. () Eq.() is independent o x and thereore Eq.(5) holds good in this case also. Using Eq. (5) in Eq. (9) as in case (i), a ith order dierential equation in g( ) is obtained as l g g R ( g gg ) A, () v where A is an arbitrary constant and R V h/ is the Reynolds number in this case. Here again Eq.() reduces to the Newtonian case when l is made equal to zero. The boundary conditions rom Eq. (8) and Eq.(9) become 79

4 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. g(), g(), g(), g(), g (), g(). () For the case o suction at the upper wall and injection at the lower wall, takes the range and R. For injection at the upper wall and suction at the lower wall, the range o values is and R. For suction at both the walls, it is required that R, and or injection at both the walls, it is R,. 3. SOLUTION OF THE PROBLEM 3. Solution by DTM or Case (i) Using dierential transorm about, Eq. (6) can be transormed into l k k k k F k (k)( )( 3)( 4)( 5) ( 5) ( k )( k )( k 3) F( k 3) k ( k)( k k) F( k) F( k k ),(3) k R k ( k k)( k k) F( k) F( k k ) k A ( k) and the irst two boundary conditions and ourth condition rom Eq. (7) are F(), F(), F(3). (4) It is assumed that F() b, F(4) d, where b& d are undetermined constants. Using these and Eq. (4) in Eq. (3), the values o F(k) are obtained iteratively and the other three boundary conditions are written as N () or F( k) (5) k N () or ( k) F( k) (6) N 3 k () or ( k )(k )(k 3) F( k 3) (7) k For N=7, solving the three Eqs. (5), (6) and (7), the values o b, d and A are obtained. Using Eqs. (3) and (4), the general expression or pressure distribution is obtained as px (, ) p(,) V V ( ) ( ) () h,(8) V A U() V x ( ) x 3 h h h From this general expression the pressure distribution or x and directions are deduced. The non- dimensional pressure drop in the x and directions are respectively given by p x p(, ) p( x, ) U () / x * 8A R h, (9) px (,) px (, ) p U () / 4R 4 ( ) () * R ( ) * R. (3) R 4 Rl * ( ) () R where R x * R h is a non-dimensional variable * and R 4 hu()/ is the entrance Reynolds number. The non-dimensional stream unction is given by ( x, ) ( x, ) 4 ( ). hu () The skin riction is deined as c (3) wall ( u / ) wall, (3) U () / hu() where τ wall is the shear stress at the wall and the skin riction here becomes c 8h 4 [ ( )] Rx wall 3. Solution by DTM or Case (ii) In this case, Eq. () is transormed to l k k k G k. (3) ( )( )( 3)(k4)(k5) ( 5) ( k )( k )( k 3) G( k 3) k ( k )( k k ) G( k ) G( k k ),(33) k R k ( k k )( k k ) G( k) G( k k ) k A ( k) and the irst two boundary conditions and ourth condition rom Eq. () are G(), G(), G(3). (34) It is assumed that G() b, G(4) d, where b,d are undetermined constants. Using these and Eq. (34) in Eq. (33), the values o G(k) are obtained iteratively and the other three boundary conditions are written as 7

5 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. N g() or G( k) (35) k N g() or ( k ) G( k ) (36) N 3 k k g() or ( k )(k)(k3) G( k 3) (37) For N=7, solving the three equations (35),(36) and (37), the values o b,d and A are obtained. Using (9) and (), the general expression or pressure distribution in this case is obtained as px (, ) p(,) V V, (38) g( ) g ( ) g () h V A U() V x g( ) x 3 h h h From this general expression, the pressure distribution or x and directions are deduced. The non-dimensional pressure drop in the x and directions are respectively given by x px * 8A R h, (39) at upper wall and suction at lower wall correspond to negative values o R. In particular, V in case (i) reduces to the problem o Kabadi (987). Figures -8 correspond to the results o case (i). Fig. shows the eects o on normal velocity or the Reynolds number R, couple stress parameter l =. &.5 and or dierent values o the wall permeability parameterα. ( ) R =,l =.5 R =,l = = -. = -.6 = - = -.4 = Fig.. Eects o wall permeability on normal velocity (case (i)). 4R g ( ) ( ) p R * 4 4 * * R g( ) Rl g ( ) g() R R.(4).9.8 R =,, 3, 5 where R x * R h is a non-dimensional variable. In this case, the non-dimensional stream unction and the skin riction are given by ( x, ) ( x, ) 4 g( ) hu () c 8h 4 [ g ( )] Rx wall 4. RESULTS AND DISCUSSION & (4). (4) The problem o a steady, two-dimensional laminar low o couple stress luid in a porous channel has been solved using DTM under two cases (i) V V and (ii) V V. In case (i), R is taken to be positive and the values o wall permeability.,.6 correspond to suction at the upper wall and injection at the lower wall, that o correspond to a solid upper wall and injection at lower wall and those o.4,.8 correspond to injection at both the walls. Other cases o suction at both walls, injection ( ) = -.5, l = Fig. 3. Eects o inertia on normal velocity (case (i)). () R =5, = -.5 l =.8,.6,.4,., Fig. 4. Eects o couple stresses on normal velocity (case (i)). 7

6 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. ' () R =, l =.5 R =, l = = -. = -.6 = - = -.4 = Fig. 5. Eects o wall permeability on axial velocity (case(i)). ' ( ) R =,, 3, 5 = -.5, l = Fig. 6. Eects o inertia on axial velocity (case (i)).. -. normal velocity increases with increasing Reynolds number. Figure 4 shows the eects o couple stresses on the normal velocity. The normal velocity component decreases throughout the domain or increasing values o the couple stress parameter as seen in Fig.. Fig. 5 gives the eects o on axial velocity or R, l. &.5 and dierent values o. l. corresponds to the Newtonian case where the axial velocity proiles are skewed close to the upper wall. For l.5, increase in magnitude o increases the magnitude o axial velocity, with minimum magnitudes at both walls and maximum magnitudes near the middle o the wall. Thus couple stresses are seen to permit a smooth low. Fig. 6 shows the inertia eects on axial velocity or.5, l.5 and or dierent values o R. It is observed that the axial velocity decreases in magnitude as the value o R is increased up to a certain value o. Above this value, the velocity increases in magnitude due to increase in lower wall velocity V which is greater than upper wall velocity V. The eects o couple stresses on axial velocity are shown in Fig. 7. It can be seen that the axial velocity decreases or increasing couple stress parameter near the lower wall and the trend is reversed near the upper wall. Here, it is clearly seen that the low becomes smoother as the couple stress parameter increases rom l. to l.8. Fig. 8 shows the pressure drop in axial direction or various values o Reynolds number and or various values o couple stress parameter. The pressure drop decreases with decreasing Reynolds number and with decreasing couple stress parameter. -. ' () l =.8,.6,.4,., R =, l = R =5, = R =,l =.5 R =5,l =.5 R =5,l = Fig. 7. Eects o couple stresses on axial velocity (case (i)). P x 6 4 = -.5 Here l =. correspond to the Newtonian case. The normal velocity decreases with increase in magnitude o. Also, the eect o couple stresses l =.5on the low behavior is to urther decrease the normal velocity. Fig. 3 shows the inertia eects on normal velocity or.5, l.5 and dierent values o R. It is observed that the x/h Fig. 8. Axial pressure drop (case (i)). Similar graphs are plotted or case (ii) in Figs In case (ii), R is taken to be positive and the 7

7 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. values o.,.6 correspond to suction at the upper wall and injection at the lower wall, that o correspond to a solid lower wall and suction at upper wall and those o.4,.8 correspond to suction at both walls. Other cases o injection at both walls, injection at upper wall and suction at lower wall correspond to negative values o R. Fig. 9 shows the eect o on normal velocity or the Reynolds number R, couple stress parameter l. &.5 and or dierent values o the wall permeability parameter. The normal velocity decreases as the value o decreases or l.5. Also, the eect o couple stresses on the low behavior is to urther decrease the normal velocity. Fig. shows the inertia eects on normal velocity or.5, l.5 and dierent values o R. It is observed that the normal velocity decreases with increasing Reynolds number. Figure shows the eects o couple stresses on the normal velocity. The normal velocity decreases with decrease in couple stress parameter. Fig. gives the eects o on axial velocity or R, l.&.5 and dierent values o. l. corresponds to the Newtonian case where the axial velocity proiles are skewed close to the upper wall. For l.5, increase in increases the axial velocity, with minimum values at both walls and maximum values near the middle o the wall. Thus couple stresses are seen to admit a smooth low. Fig. 3 shows the eects o inertia on axial velocity proile or.5, l.5 and dierent values o R. It is observed that the axial velocity decreases as the value o R increases up to a certain value o. Above this value, the velocity increases due to increase in V which is greater than V. The eects o couple stresses on the axial velocity or ixed values o R, l.5 are shown in Fig. 4. The axial velocity is increasing with increasing couple stress parameter or certain values o and ater that it is decreasing. Here, it is clearly seen that the low becomes smoother as the couple stress parameter increases rom l. to l.8. In Figs. and 4, when the couple stress parameter is equal to zero, it represents the Newtonian luid case. Fig. 5 shows the pressure drop in axial direction or various values o Reynolds number and or various values o couple stress parameter. The pressure drop decreases with increasing Reynolds number and with increasing couple stress parameter. () =.8 =.4 = =.6 =. R =,l =.5 R =,l = Fig. 9. Eects o wall permeability on normal velocity (case (i)). () =.5, l =.5 R =,,3, Fig.. Eects o inertia on normal velocity (case (ii)). () R =5, =.5 l =.8,.6,.4,., Fig.. Eects o couple stresses on normal velocity (case (ii)). ' () R =, l =.5 R =, l = =.8 =.4 = =.6 = Fig.. Eects o wall permeability on axial velocity (case (ii)). 73

8 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. l Table Eects o couple stresses on skin riction by DTM case (i) R =, R =, R =, α =-.5 α = -. 5 α = - ( ) ( ) ( ) ( ) ( ) ( ) Table Eects o couple stresses on skin riction by DTM case (ii). R =, R =, R =, α =.5 α =. 5 α = l g ( ) g ( ) g ( ) g ( ) g ( ) g ( ) lower wall whereas it increases at the upper wall with an increase in Reynolds number or ixed values o α and l. ' ( ) R =,,3, =.5,l =.5 ' ().4 l =.8,.6,.4,., Fig. 3. Eects o inertia on axial velocity (case (ii))..3.. R =5, =.5 Tables and present comparisons between Newtonian ( l ) and non-newtonian results ( l ), revealing the eects o couple stresses on the skin riction in case (i) and case (ii) respectively. From Table o case (i), it is observed that the magnitude o skin riction increases at the lower wall and decreases at the upper wall with an increase in couple stress parameter or all values o α & R. Here, α =-.5 corresponds to the case o one wall injection and other wall suction, whereas α =- corresponds to the case o solid upper wall and injection at the lower wall. Further, the magnitude o the skin riction decreases at the Fig. 4. Eects o couple stresses on axial velocity (case (ii)). From Table o case (ii), it is observed that the magnitude o skin riction increases at the lower wall and decreases at the upper wall with an increase in couple stress parameter or all values o α & R as in case (i). In Table α =.5 corresponds to the case o one wall injection and other wall suction, whereas α = corresponds to the case o solid lower wall and suction at the upper wall. Also, the magnitude o the skin riction decreases at the lower wall whereas it 74

9 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. increases at the upper wall with an increase in Reynolds number or ixed values o α and l. Thus the eect o couple stresses on the magnitude o skin riction in both cases gets enhanced at one wall and diminished at the other wall. P x =.5 R =, l =.5 R =, l =.5 R =5, l =.5 R =5, l = x/h Fig. 5. Axial pressure drop (case(ii)). 5. CONCLUSION In this paper, dierential transorm method is used successully or inding the solution o two dimensional steady laminar low o an incompressible couple stress luid in a porous channel. It is seen that this method is simple and easy to use and solves the problem without any discretization and does not require exertion o any low parameter as in perturbation method. Although the non-newtonian luid low problem deals with more complicated governing equations than the Newtonian case, the method does not get disturbed and works well. Besides this, the reliability o the method and reduction in size o computational domain gives this method a wider applicability. Hence, the Dierential Transorm Method is an eective and reliable tool in inding the semi- analytical solution to many non-linear systems. This paper uses DTM or the low problem with the couple stress model o the non- Newtonian luid. However, there is no limitation or applying the DTM or such problems in similar geometry with any other non-newtonian luid model. REFERENCES Ariel, P. D. (). On exact solutions o low problems o a second grade luid through two parallel porous walls. Int. J. Eng. Sci. 4, Berman, A. S. (953). Laminar low in channels with porous walls. Journal o. Applied. Physics 4, Bert, C. W. (). Application o dierential transorm method to heat conduction in tapered ins. Journal o Heat Transer 4, 8-9. Chen, C. K. and S. H. Ho (999). Solving partial dierential equations by two dimensional dierential transorm method. Applied Mathematics and Computation 6, Erturk, V. S., S. Momani (7). Dierential transorm method or obtaining positive solutions or two-point nonlinear boundary value problems. International Journal: Mathematical Manuscripts (), Kabadi, A. S. (987). The inluence o couple stresses on the low o luid through a channel with injection. Wear 9, Kamsili, F. (6). Laminar low o a non- Newtonian luid in channels with wall suction or injection. Int. J. Eng. Sci. 44, Kurnaz, A., G. Oturanc and M. E. Kiris (5). n- dimensional dierential transorm method or PDE. International Journal o Computer Mathematics 8(3), Rashidi, M. M. and S. M. Sadri (). Solution o the laminar viscous low in a semi-porous channel in the presence o a uniorm magnetic ield by using the dierential transorm method. Int. J. Contemp. Math. Sciences 5(5), 7-7. Rashidi, M. M. and E. Erani (). The Modiied Dierential Transorm Method or Investigating Nano Boundary-Layers over Stretching Suraces. International Journal o Numerical Methods or Heat and Fluid Flow (7), Rashidi, M. M. and S. A. Mohimanian Pour (). A Novel Analytical Solution o Heat Transer o a Micropolar Fluid through a Porous Medium with Radiation by DTM-Padé. Heat Transer-Asian Research 39(8), Rashidi, M. M., S. A. Mohimanian Pour and N. Laraqi (). A semi-analytical solution o micro polar low in a porous channel with mass injection by using dierential transorm method. Nonlinear Analysis: Modeling and Control 5(3), Srinivasacharya, D, N. Srinivasacharyulu and O. Ojjela (9). Flow and heat transer o couple stress luid in a porous channel with expanding and contracting walls. International Communication in Heat and Mass Transer 36(), Srinivasacharya, D, N. Srinivasacharyulu and O. Ojjela (). Flow o Couple Stress Fluid Between Two Parallel Porous Plates. International Journal o Applied Mathematics 4(), -4. Stokes, V. K. (966). Couple Stresses in Fluids. Phys. Fluids 9, Terrill, R. M. (964). Laminar low in a uniormly porous channel. The Aeronautical Quarterly 5,

10 P. Vimala and P. Blessie Omega / JAFM, Vol. 9, No. 6, pp , 6. Terrill, R. M. and G. M. Shrestha (965). Laminar low through parallel and uniormly porous walls o dierent permeability. ZAMP 6, Vimala, P. and P. Blessie Omega (3). Laminar low o a Newtonian luid through parallel porous walls solution by dierential transorm method. In Proceedings o the Fortieth National Conerence on Fluid Mechanics and Fluid Power Yuan, S. W. (956). Further investigation o laminar low in channels with porous walls. Journal o Applied Physics 7, Zhou, J. K. (986). Dierential transormation and its application or electric circuit analysis (in Chinese), Huazhong University Press. 76