An Analytical Solution of the Navier-Stokes Equation for Flow over a Moving Plate Bounded by Two Side Walls M. Emin Erdogan - C. Erdem imrak* Istanbul Technical University, Faculty of Mechanical Engineering, Turkey An exact solution of the Naiver-Stokes equation for unsteady flow over a moving plate between two side walls is given. This solution solves the problem that arises calculating shear stress at the bottom wall when the expression of velocity presented in literature is used. The variation of the shear stress at the bottom wall with the distance between two side walls for various values of the non-dimensional time is illustrated and it is shown that when the value of non-dimensional time is equal to unity, the shear stress approaches the asymptotic value. Furthermore, the volume flux across a plane normal to the flow is calculated and it is found that when the value of the non-dimensional time is equal to unity, the volume flux approaches the asymptotic value. © 2009 Journal of Mechanical Engineering. All rights reserved. Keywords: Naiver-Stokes equation, analytical solution, sine transform, steady flow, unsteady flow 0 INTRODUCTION An exact solution of the Navier-Stokes equation for unsteady flow over a plane wall and between two side walls perpendicular to the plane is given. Obtaining the exact solutions of the Navier-Stokes equation is very important because they provide a standard for checking the accuracies of many approximate numerical or empirical methods. Furthermore, the accuracies of the boundary layer approximation and the slow flow approximations can be tested by using the exact solutions of the Navier-Stokes equation. Although computer techniques make the complete integration of the Navier-Stokes equation feasible, the accuracy of the results can be established by a comparison to an exact solution. The exact solution given in this paper is for an unsteady flow in a viscous fluid generated by a plane wall moved suddenly and by two side walls held stationary. In the absence of the side walls, the flow reduces to the flow caused by a plane wall moved suddenly, which is termed Stokes problem or Rayleigh-type flow. The steady flow over a plane wall moving at a constant speed between two side walls was considered in [1] to present a simple model of a paint-brush due to Taylor. The mathematical problem is of the familiar boundary value form. The solution which satisfies the boundary conditions at the bottom wall, at the side walls and at infinity is given in the form of a series [1]. This series which gives the velocity of the steady state flow is a convergent series, but term-by-term differentiation of this series, which is used to calculate the shear stress at wall, does not give a convergent series. This situation arises not only due to the flow extending to infinity but also due to the flow bounded either by a horizontal plane at the top or by a horizontal free surface at the top [2]. A similar problem occurs in the conduction of heat [3], since the governing equation is the same for both problems. Unsteady flows of both Newtonian and non-Newtonian fluids over a plane wall in the absence of the side walls have been investigated by many authors [4] to [11]. The aim of this paper is to investigate the unsteady flow of a viscous fluid generated by a plane wall between two side walls. By using the Laplace transform method, the velocity is given in a series form. When the distance between two side walls increases, the expression of the velocity reduces to that of the velocity for an unsteady flow of a fluid generated by a plane wall moved suddenly. When non-dimensional time moves towards infinity, the expression of velocity approaches to that of the velocity for the steady flow. The shear stress at the bottom wall cannot be calculated by the expression of the velocity obtained by using the Laplace transform method. For this reason, by using the sine transformation, another expression for velocity is obtained. At any time, shear stress at the bottom wall is at its minimum at the middle of the plane and increases towards the side walls. The volume flux across the plane normal to the flow is given in terms of a *Corr. Author's Address: ITU Makina Fakultesi, Inonu Caddesi no. 67, Gumussuyu 34437, Istanbul, Turkey, imrak@itu. edu.tr definite integral. This integral provides a new function which is defined by the integral [12]. 1 UNSTEADY FLOW The fluid is over a plane wall and between two side walls perpendicular to the plane. The side walls extend to infinity in the x- and z-direction as shown in Figure 1. The fluid is suddenly set in motion by moving the bottom wall at constant speed in the absence of an imposed pressure gradient. The governing equation is du ~dt ( ■ = v cfu dy d2u ^ 2 +" 2 2 dz (1) The boundary and the initial conditions are u(±b, z, t) = 0, for all t, u(y, 0, t) = U for all t > 0 and -b < y < b, u(y, 0. (2) Fig. 1. Flow geometry and coordinate system By using the Laplace transform method one finds [3] u = 1 y (-1)" U ~ n y 2" +1 +e "zerfc —k"Z r e " erfc 2V 2-V cos Ky — Kypi (3) where kn = (2n + 1)n / 2b and 2 " 2 erfc x = [ d% Vi x is the complementary error function [13]. When t goes to infinity the expression of the velocity given by Eq. (3) approaches the steady state which is given in the following form u=1 y (—1)" u ~ny 2"+1 ' !cos kny • When z/b goes to infinity u/U becomes zero. In the absence of the side walls, namely, when b goes to infinity Eq. (3) takes the following form limU = erfc—. U J 2-V The variation of u/U at y = 0 with z/b for various values of vt/b2 is illustrated in Fig. 2. For vt/b2 = 0.01, z/b, is about 0.5 and for vt/b2 = 1 the velocity approaches the asymptotic value. The volume flux across the plane normal to the flow can be found in the following form Q 32 Uty (2"- Ub n "=0 1)—3 erf n(2" +1) vt 2 W (4) When t goes to infinity Eq. (4) reduces to Q = 32 y 1 Ub2 n3 y (2" +1)3 ' Fig. 2. The variation of u/U at y = 0 with z/b for various values of vt/b2 The numerical value of the right hand side is about 1.0855. The flux corresponds to a mean thickness of the order 0.5427b. For vt/b2 = 1, the volume flux approaches the asymptotic value. The variation of the volume flux with vt/b2 is illustrated in Fig. 3. The shear stress at the bottom wall cannot be calculated by the expression of the velocity given by Eq. (3). For this reason, another expression for velocity by using the sine transformation is obtained. The solution of Eq. (1) can be written as u = u0(y, z) + uj(y, z, t), where u0 and u1 satisfy the following differential equations d2u0 d2u0 j du, —0 +—f = 0 and —- = V ■2 dz2 dt dy2 ( d2u1 d\ ^ ~dyr "dz2 u0 shows the steady-state part and u1 corresponds to the transient part. The boundary and initial conditions become u0(±b, z) = 0, uo(y, 0) = U, u0(y, x) = 0, u1(±b, z, t) = 0, u1(y, 0, t) = 0, u1(y, x, t) = 0, u0(y, z) + u1(y, z, t) = 0. 0.8 1.0 vt/b2 Fig. 3. The variation of Q/Ub2 with vt/b2 The sine transform of u0 is [14] x u0 = J u0 sin Xzdz 0 and the boundary condition becomes u0 (±b ) = 0. The problem is reduced to the solution of the following ordinary differential equation d 2u„ dy 20 -A\ = -XU. The solution is _ U ( cosh Xy u0 =—I 1-- X v cosh Xb The inverse of u0 is given by the relation [14] u0 = — J u0 sin XzdX; w ' and since J • A r sin Xz ,. n f—^d X = - 0 X u0 can be written as u0 2 J cosh Xy -0 = 1 - —[ U nf U n{X cosh Xb sinXzdX . The sine transform of u1 is u1 = J u1 sin Xzdz , 0 and the conditions are u1 (±b, t) = 0 , u ( y )+u; ( y,0) = 0. The problem is reduced to the solution of the following partial differential equation du1 d2u IT = V "dy2 The boundary condition u1 (±b, t ) = 0 suggests a solution in the following form = v- vX%. =x A e-v(kn2 + X cos kny, where kn = (2n + 1)n/2b and An can be obtained by the equation U f _ cosh^y^ An cos k„y = 0. X V cosh Xb J n=0 This equation gives 2(-1)2U X A =- bkn K +X2-The inverse of u1 is u1 = -4 v (-1)ne"vk"t cos kny J Xe-vXt sin Xz bk 0 K +X2 dX. «1 —t ^^ U n n=0 Therefore, the application of the sine transform method to Eq. (1), gives x u 2 r cosh Xy . = 1--I --— sin XzdX X U n J Xcosh Xb -4 v (-1)ne "vf coskny r Xe~X vt sinXz n^-i bkn J . () n=0 n 0 k2 +X2 It can be shown that Eq. (5) satisfies Eq. (1). When t goes to infinity Eq. (5) reduces to the expression of the velocity in the case of the steady state in the following form 2 r cosh Xy . U nJnX cosh Xb u = 1 -2 J. 77 irJ : sin XzdX, n=0 or by using the integral r coshXy . . ,, n + _! cos (ny/2b) I-— sin XzdX =--tan ---J- 0 X cosh Xb 2 sinh (nz /2b) one finds u 2 , cos(ny/2b) — = - tan-1-^-'- U n sinh (nz/2b ) When t = 0 Eq.(5) gives 1 - ^ j cosh A sin AzdA = n 0 A cosh Ab 4 (-1)" -n —Y e knZ coskny . n n=o (2n +1) The expression of the velocity given by Eq. (5) satisfies the following conditions: when z/b goes to infinity u/U vanishes and in the absence of the side walls u/U reduces to the expression that is given by the Stokes problem. Eq. (5) can be written in the following form u 2 -1 cos(n y /2b) — = — tan 1 U n sinh(nz / 2b) - 2_ y (-1)" cos kny j2 ^ n n=0 2n +1 e Kzerfc +ek"zerfc 2V - k"Vt 'k"Vt tan 2-V When t goes to zero, Eq. (6) gives 1 cos(ny/2b) = 2Ycoskny. (6) sinh(nz /2b) n=0 2n +1 The volume flux across a plane normal to the flow is = ff1 _ tanh Xb ^ X Q = n K Xb I b2 - 8U Y — f n Y k2 J -(A + k2)vt nb "=0 k2n 0 A2 + k2 -dA. or by using the integral r _(X2+kl)vt n I exr+kr dX = Hkerfc ), 0 Kn ^n one finds Q-n F «»> _ 4g bk, 'rfc ^^ where F(r) is given by an integral (7) F (f) = J tanh Ç d \ and the numerical value of e F(r) is 1.0855090289 correct to ten decimal places [12]. If one uses the equality F(r) = 4" £ (2n +1)_3 , n n=0 then, this expression of the volume flux reduces to Eq. (4). The shear stress at the bottom wall can be calculated by Eq. (5). Taking the derivative of Eq. (5) with respect to z and then putting z = 0 one finds K )z=0 = Hff 2pU f cosh Ay n { cosh Ab dA 4/uU y (-1)" coskny} X2e 2 - (A + )t n — bkn and by using the integral A2 + K -dA. J 2 -(A2 + k„)t A2 e 0 A2 + k2 -dA = n 1 =ierfc (knV) , 2 V where inerfcx = | in _lerfcgdg, i 0erfc x = erfcx , are the integrals of the complementary error functions [13], one obtains (axz )z=0 juU / b 1 (ny/2b ) cos +-V (-1) cos kny v +2Y—, ¡— ierfckn\lvt n=0 knj Vt (8) When t goes to infinity, since ierfc(r) = 0, Eq. (8) reduces to the expression for the steady state K) z 1 /U / b cos (ny/2b ) The variation of (axz )z/(/U /b) with y/b for various values of vt/b2 is illustrated in Figure 4. When vt/b2 = 1, the shear stress approaches the asymptotic value. 10.0 KK.o 9 0 flU/b 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -0.75 -0.5 -0.25 0.0 0.25 0.5 0.75 1.0 ylb Fig. 4. The variation of (axz)z=0/(pU/b) withy/b for various values of vt/b2 From Fig. 4 it can be seen that the shear stress at the bottom wall is at its minimum at the middle of the plate and increases towards the side walls and at the side walls, namely, y=b, since cos knb = 0 and since [15] x> JcosXzdX = nS(z), 0 where 8(z) is the Dirac delta function, becomes It is can be clearly seen that this result is independent of time and it shows the corner singularity of the solution. This situtation is remarked in the second of the boundary conditions given by Eq. (2). 2 CONCLUSIONS An analytical solution for an unsteady flow generated by a plane wall moved suddenly and by two side walls held stationary is given. In the absence of the side walls, the flow reduces to the flow caused by a flat plate moved suddenly, which is termed Stokes problem or Rayleigh-type flow. When non-dimensional time moves towards infinity the solution tends to that of the steady flow in the same geometry. The steady and the unsteady flows in the geometry considered are investigated to give a correct result for shear stress at the bottom wall. The shear stress at the bottom wall can be calculated from the velocity obtained by using the sine transformation. It is shown that the shear stress at the bottom wall is at its minimum at the middle of the plate and it increases towards the side walls. Furthermore, the volume flux across a plane normal to the flow is calculated and it is shown that when value of non-dimensional time is equal to unity, the volume flux approaches the asymptotic value. 4 REFERENCES [1] Batchelor, G.K. (1967) An introduction to fluid dynamics, Cambridge University Press, Cambridge, ISBN 0521 66396 2. [2] Erdogan, M.E. (1997) Effects of the side walls on the flow over a moving plane wall, Applied Mechanics and Engineering, vol. 2, no. 3, p. 415-432. [3] Carslaw, H.S., Jaeger, J.C. (1973) Conduction of heat in solid, 2nd Ed. Oxford University Press, Oxford. 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