FLOW OF A WILLIAMSON FLUID OVER A STRETCHING SHEET

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1 Brazilian Journal of Cheical Engineering ISSN 4-66 Printed in Brazil Vol., No., pp , July - Septeber, FLOW OF A WILLIAMSON FLUID OVER A STRETCHING SHEET S. Nadee *, S. T. Hussain and Changhoon Lee Departent of Matheatics, Quaid-I-Aza University 45, Phone: , Islaabad 44, Pakistan. E-ail: snqau@hotail.co Departent of Coputational Science and Engineering, Yonsei University, Seoul, Korea. (Subitted: January 6, ; Revised: May 9, ; Accepted: June, ) Abstract - In the present article, we have exained the two diensional flow of Williason fluid odel over a stretching sheet. The governing equations of pseudoplastic Williason fluid are odelled and then siplified by using siilarity transforations and boundary layer approach. The reduced equations are then solved analytically with the help of hootopy analysis ethod. The physical features of the odel are presented and discussed through graphs. Keywords: Williason fluid; Stretching sheet; Hootopy analysis ethod. INTRODUCTION In non-newtonian fluids, the ost coonly encountered fluids are pseudoplastic fluids. The study of the boundary layer flow of pseudoplastic fluids is of great interest due to its wide range of application in industry such as extrusion of polyer sheets, eulsion coated sheets like photographic fils, solutions and elts of high olecular weight polyers, etc. The Navier Stokes equations alone are insufficient to explain the rheological properties of fluids. Therefore, rheological odels have been proposed to overcoe this deficiency. To explain the behaviour of pseudoplastic fluids any odels have been proposed like the power law odel, Carreaus odel, Cross odel and Ellis odel, but little attention has been paid to the Williason fluid odel. Williason (99) discussed the flow of pseudoplastic aterials and proposed a odel equation to describe the flow of pseudoplastic fluids and experientally verified the results. Lyubiov and Perinov () discussed the flow of a thin layer of a Williason fluid over an inclined surface in the presence of a gravitational field. Dapra and Scarpi (7) developed the perturbation solution for a Williason fluid injected into a rock fracture. Peristaltic flow of a Williason fluid has been discussed by Nadee et al. (). Vasudev et al. () studied the peristaltic puping of a Williason fluid through a porous ediu considering heat transfer. Craer et al. (968) showed that this odel fits the experiental data of polyer solutions and particle suspensions better than other odels. For pseudoplastic fluids the power law odel predicts that the apparent/effective viscosity should decrease indefinitely with increasing shear rate, which eans infinite viscosity at rest and zero viscosity as the shear rate approaches infinity. A real fluid has both iniu and axiu effective viscosities depending upon the olecular structure of the fluid. In the Williason fluid odel, both the iniu (μ ) and axiu viscosities (μ ) are considered. So, for pseudoplastic fluids (for which the apparent viscosity does not go to zero at infinity), it will give better results. To the best of the author's knowledge no one has *To who correspondence should be addressed

2 6 S. Nadee, S. T. Hussain and Changhoon Lee considered the boundary layer flow of a Williason fluid over a stretching sheet. Stretching sheet flows are of great iportance in any engineering applications like extrusion of a polyer sheet fro the die, the boundary layer in liquid fil condensation processes, eulsion coating on photographic fils, etc. Sakiadis (96) initiated the study of boundary layer flows over a continuous surface and forulated the two diensional boundary layer equations. Tsou et al. (967) extended the work of Sakiadis and considered the heat transfer in the boundary layer flow over a continuous surface and experientally verified Sakiadis results. Erickson et al. (966) included the heat and ass transfer on a stretching surface with suction or injection. Many researchers later investigated boundary layer flow over a stretching surface, such as Gupta and Gupta (977), Ishak (8), and Nadee (). The objective of the present paper is to present the odelling of a two-diensional Williason fluid for a stretching sheet. The highly non-linear partial differential equations are siplified by using suitable siilarity transforations and the resulting reduced equations are then solved analytically with the help of hootopy analysis ethod (HAM) (see Liao,, 4). The results are discussed through graphs for various physical paraeters of interest. The coefficient of skin friction, shear stress and apparent viscosity are also plotted. FLUID MODEL For an incopressible Williason fluid, the continuity and oentu equations are given as: div V =, () dv ρ = div S+ ρb, () dt where ρ is the density, V is the velocity vector, S is the Cauchy stress tensor, b represents the specific body force vector and d/dt represents the aterial tie derivative. The constitutive equations of the Williason fluid odel are given as: S= p I+ τ, () ( μ μ ) τ= [ μ + ] A, (4) Γγ in which p is the pressure, I is the identity vector, τ is the extra stress tensor, μ and μ are the liiting viscosities at zero and at infinite shear rate, Γ > is the tie constant, A is the first Rivlin- Erickson tensor and γ is defined as follows γ= π, π = trace( A ), (5) / γ= [( ) + ( + ) + ( ) ], (6) x x where π is the second invariant strain tensor. Here we have only considered the case for which μ = and Γγ <. Thus, the extra stress tensor takes the for: μ τ= [ ] A, (7) Γγ or, by using binoial expansion, we get: τ= μ [ +Γγ ] A, (8) The coponents of the extra stress tensor are: [ ], x τ xy = τ yx =μ [ +Γγ ]( + ), x τ xx = μ +Γγ [ ], y τ yy = μ +Γγ τ = τ = τ = τ =τ =. xz yz zx zy zz Matheatical Forulation (9) Let us consider a steady, two-diensional flow of an incopressible Williason fluid over a wall coinciding with the plane y =, the flow being confined to y >. Two equal and opposite forces are applied along the x -axis to produce stretching, keeping the origin fixed. The flow is generated due to the linear stretching. In the absence of a body force, the governing equations in coponent for can be defined as: + =, x () Brazilian Journal of Cheical Engineering

3 Flow of a Williason Fluid Over a Stretching Sheet 6 where u(x,y) and v(x,y) are the velocity coponents along the flow direction and noral to the flow direction, respectively: p τ τ xx xy ρ (u + v ) = + +, x x x p τyx τyy ρ (u + v ) = + +, x x () () at the wall and B> is the stretching paraeter. Introducing the following transforations: B u = Bxf η,v = Bνf η, η= y. (7) ν ( ) ( ) Making use of transforations (7), Eq. (5) takes the for: f f + ff +λ f f =, (8) where τxx, τxy, τ yy are the coponents of the extra stress tensor Eq. (7). Applying the boundary layer approxiations to the coponents of the tensor, where the order of x and u is while the order of Γ, y and v is δ. with the corresponding boundary conditions: f =,f = at η=, f as η, (9) / δ [ δ { δ } ] δ +Γ ( ) τ x x x xx μ =, ρ ρ / ( ) ( )} x { () / δ [ δ { δ } ] ( δ ) δ δ τ xy u u v v / = μ [ +Γ {( ) + ( + ) + ( )} ] ρ ρ x x (4) ( + ). x Each ter of Eq. () and Eq. () ust be of order. So we retained those ters of Eq. () and Eq. (4) that have order and δ, respectively. In the absence of a pressure gradient, we finally arrive at: u u u + v = ν + νγ. x The corresponding boundary conditions are: u = U = Bx;v = at y=, w u as y, where ν is the kineatic viscosity, (5) (6) U w is the velocity where λ, the diensionless Williason paraeter. is defined as: B λ=γx. ν () For λ =, Eq. (8) reduces to the classical boundary layer equation for a viscous flow. Physical quantities of interest are the wall shear stress τ w and the coefficient of skin friction c f. After using boundary layer approxiations, τ w is given by: Γ τ w =μ [ + ( ) ], y= The coefficient of skin friction is defined as: w w () τ c f =. () ρ U In diensionless for it is defined as: λ Rec f = (f + f ), where Re= η= Bx / ν is the Reynolds nuber. Hootopy Analysis Solutions We choose the set of base functions: () { η k exp( n η), k, n }, (4) Brazilian Journal of Cheical Engineering Vol., No., pp , July - Septeber,

4 6 S. Nadee, S. T. Hussain and Changhoon Lee and can write: k k,,n n= k= f( η ) = a + a η exp( n η), (5) k where a,n is the coefficient. With the help of boundary conditions (9), the initial approxiation and auxiliary linear operator are chosen as: f() η = exp( η ), (6) df df L(f) f =, d (7) η d η L f[c+ Cexp( η ) + Cexp( η )] =, (8) and C ( i ) i = are the arbitrary constants. Zeroth Order Deforation Equation If q [,] is the ebedding paraeter and the non-zero auxiliary paraeter is ħ, then the proble at the zeroth order deforation is: ( q)l [f ( η;q) f ( η )] = q N [f ( η;q)], (9) ( ) η= f f f( η;q) f( η ;q) =, =, η f( η;q) η η= η =, where the nonlinear operator is defined as: f( η;q) f( η;q) N[f(;q)] f η = η η f( η;q) + f( η;q) η ( f( η;q)) ( f( η;q)) +λ, ( η ) ( η ) When q = and q =, then: () () f ( η ;) = f ( η), f ( η ;) = f ( η ). () With the help of a Taylor series expansion: () = f( η ;q) = f ( η ) + f ( η)q, f( η;q) f ( η ) =,! q q= (4) the auxiliary paraeter is chosen in such a way that Eq. () converges at q = ; thus f( η ) = f ( η ) + f ( η), (5) = th Order Deforation Equations The th order approxiation is defined as: f f L[f ( η) χ f ( η )] = R ( η), (6) f () = f () = f ( ) =, (7) where: f k k k k k= R ( η ) = f + [f f f f ] + χ λ k k l k=,, =., > f f, The general solution for Eq. (6) is defined as: (8) (9) f ( η ) = f ( η ) + C + C exp( η ) + C exp( η ), (4) where f ( η ) is a special function. We can deterine constants using conditions (7) : f ( η) = = = η η= C, C,C C f (). (4) We can calculate any order of approxiation for =,,... with the help of Matheatica. Brazilian Journal of Cheical Engineering

5 Flow of a Williason Fluid Over a Stretching Sheet 6 RESULTS AND DISCUSSION We know that the convergence of the solution of Eq. () depends upon the non-zero auxiliary paraeter ħ. To obtain an adissible value of ħ, the th ħ-curve is plotted for the order of approxiation. In Fig. (), it can be observed that the adissible range of ħ is.5.5. In order to choose the best value of ħ, the residual error in f is plotted for different values of ħ (see Fig. ()). It is observed that we can choose any value between.4.9. It can be seen fro Table that convergence of the HAM th solution is obtained at the 5 order of approxiation. paraeter λ. It is observed that, with the increase in λ, the velocity field and boundary layer thickness decrease. Fig. 4 shows that f decreases with an increase in λ. When we increase the value of Γ or B (keeping the other paraeters fixed), it results in an increase in λ. So the velocity profile as well as the boundary layer thickness decrease with an increase in Γ or B. Table shows the effects of λ on the skin friction coefficient at ħ=-.. It also depicts that, with the increase of λ, the agnitude of the coefficient of skin friction decreases (see Fig. 5). In Figs. 6(a) and 6(b), the shear stress and apparent viscosity for the Williason fluid are plotted against the deforation rate. These figures show that the behaviour is quite siilar to pseudoplastic fluids. = f( η ).4. Figure : -curve of f''() η Figure : Variation of f' for various values of λ. =..8 f( η ).6.4. Figure : Residual error in f at η= Table : Convergence of the HAM solution for different order of approxiation when =. λ. Order of approxiation -f () Fig. () plots the non-diensional velocity f against η for various values of the Williason fluid η Figure 4: Variation of f for various values of λ Table : Values of the coefficient of skin friction for different values of λ. λ - Rec f Brazilian Journal of Cheical Engineering Vol., No., pp , July - Septeber,

6 64 S. Nadee, S. T. Hussain and Changhoon Lee nonlinear differential equation is solved analytically by using HAM. At the end we can conclude that:. f decreases with an increase in the Williason paraeter λ.. The coefficient of skin friction decreases parabolically with an increase in λ.. The Williason fluid odel describes the flow of pseudoplastic fluids. ACKNOWLEDGEMENT Figure 5: Variation of the coefficient of skin friction for different values of λ This research was supported by the WCU (World Class University) progra through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R REFERENCES (a) Figure 6(a): Shear stress against deforation rate when λ =.4 (b) Figure 6(b): Apparent viscosity against deforation rate when λ =.4 CONCLUDING REMARKS We have investigated the boundary layer flow of a Williason fluid over a stretching surface. The Abbasbandy, S., The application of hootopy analysis ethod to solve a generalized Hirota- Satsua coupled KdV equation. Physics Letters A, 6, No. 6, 478 (7). Abbasbandy, S., Hootopy analysis ethod for heat radiation equations. International Counications in Heat and Mass Transfer, 4, No., 8 (7). Abbasbandy, S., The application of hootopy analysis ethod to nonlinear equations arising in heat transfer. Physics Letters A, 6, No., 9 (6). Abbasbandy, S. and Shirzadi, A., A new application of the hootopy analysis ethod: Solving the Stur--Liouville probles. Counications in Nonlinear Science and Nuerical Siulation, 6, No., (). Craer, S. D. and Marchello, J. M., Nuerical evaluation of odels describing non-newtonian behavior. Aerican Institute of Cheical Engineers Journal, 4, 98 (968). Dapra and Scarpi, G., Perturbation solution for pulsatile flow of a non-newtonian Williason fluid in a rock fracture. International Journal of Rock Mechanics and Mining Sciences, 44, 7 (7). Erickson, L. E., Fan, L. T. and Fox, V. G., Heat and ass transfer in the lainar boundary layer flow of a oving flat surface with constant surface velocity and teperature focusing on the effects of suction/injection. Industrial & Engineering Cheistry Research, 5, 9 (966). Ellahi, R., Riaz, A., Analytical solutions for MHD Brazilian Journal of Cheical Engineering

7 Flow of a Williason Fluid Over a Stretching Sheet 65 flow in a third-grade fluid with variable viscosity. Matheatical and Coputer Modelling, 5, No. 9-, 78 (). Ellahi, R., Effects of the slip boundary condition on non-newtonian flows in a channel. Counications in Nonlinear Science and Nuerical Siulation, 4, No. 4, 77 (9). Ellahi, R. and Afzal, S., Effects of variable viscosity in a third grade fluid with porous ediu: An analytic solution. Counications in Nonlinear Science and Nuerical Siulation, 4, No. 5, 56 (9). Ellahi, R., Raza, M. and Vafai, K., series solutions of non-newtonian nanofluids with Reynolds odel and Vogels odel by eans of the hootopy analysis ethod, Matheatical and Coputer Modelling, 55, No. 7, 876 (). Gupta, P. S. and Gupta, A. S., Heat and ass transfer on a stretching sheet with suction or blowing. Canadian Journal of Cheical Engineering, 55, No. 6, 744 (977). Ishak, A., Nazar, R. and Pop, I., Heat Transfer over a stretching surface with variable heat flux in icropolar fluids. Physics Letters A, 7, No. 5, 559 (8). Lyubiov, D. V. and Perinov, A. V., Motion of a thin oblique layer of a pseudoplastic fluid. Journal of Engineering Physics and Therophysics, 75, No. 4, 9 (). Liao, S., Beyond Perturbation: Introduction to the Hootopy Analysis Method. Chapan & Hall/ CRC, Boca Raton (). Liao, S., On the analytic solution of agneto hydrodynaic flows of non-newtonian fluid over a stretching sheet. Fluid Mechanics, 488, 89 (). Liao, S., A new branch of solutions of unsteady boundary layer flow over an ipereable stretched plate. International Journal of Heat and Mass Transfer, 48, 59 (5). Liao, S., On the hootopy analysis ethod for nonlinear probles. Applied Matheatics and Coputation, 47, 499 (4). Nadee, S. and Akra, S., Influence of inclined agnetic field on peristaltic flow of a Williason fluid odel in an inclined syetric or asyetric channel. Matheatical and Coputer Modelling, 5, 7 (). Nadee, S. and Akbar, N. S., Nuerical solutions of peristaltic flow of Williason fluid with radially varying MHD in an endoscope. International Journal for Nuerical Methods in Fluids, 66, No., (). Nadee, S. and Akra, S., Peristaltic flow of a Williason fluid in an asyetric channel. Counications in Nonlinear Science and Nuerical Siulation, 5, 75 (). Nadee, S., Hussain, A. and Khan, M., HAM solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet. Counications in Nonlinear Science and Nuerical Siulation, 5, No., 475 (). Nadee, S., Hayat, T., Abbasbandy, S. and Ali, M., Effects of partial slip on a fourth-grade fluid with variable viscosity: An analytic solution. Nonlinear Analysis: Real World Applications,, No., 856 (). Nadee, S., Hussain, A. and Vajravelu, K., Effects of heat transfer on the stagnation flow of a third order fluid over a shrinking sheet. Zeitschrift für Naturforschung A, 65, 969 (). Nadee, S., Hussain, A., Malik, M. Y. and Hayat, T., Series solutions for the stagnation flow of a second-grade fluid over shrinking sheet. Applied Matheatics and Mechanics,, 55 (9). Nadee, S. and Hussain, A., MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with hootopy analysis ethod. Applied Matheatics and Mechanics,, No., 569 (9). Sakiadis, B. C., Boundary layer behavior on continuous solid flat surfaces. Aerican Institute of Cheical Engineers Journal, 7, 6 (96). Tsou, F. K., Sparrow, E. M. and Goldstein, R. J., Flow And heat transfer in the boundary layer on a continuous oving surface. International Journal of Heat and Mass Transfer,, 9 (967). Vasudev, C., Rao, U. R., Reddy, M. V. S. and Rao, G. P., Peristaltic puping of Williason fluid through a porous ediu in a horizontal channel with heat transfer. Aerican Journal of Scientific and Industrial Research,, No., 656 (). Williason, R. V., The flow of pseudoplastic aterials. Industrial & Engineering Cheistry Research,, No., 8 (99). Brazilian Journal of Cheical Engineering Vol., No., pp , July - Septeber,