THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume, Numer 4/0, pp. 9 95 UNSTEADY POISEUILLE FLOW OF SECOND GRADE FLUID IN A TUBE OF ELLIPTICAL CROSS SECTION S. ISLAM, Z. BANO, T. HAROON, A. M. SIDDIQUI COMSATS Institute of Information Technology, Department of Mathematics, Islamaad, Pakistan Pennsylvania State University, York Campus, Edecom 703, USA E-mail: xarqa@hotmail.com The unsteady flow of an incompressile second grade fluid in an infinitely long tue of elliptical cross-section is considered under constant pressure gradient. The resulting governing equation is a time dependent PDE which is solved for exact solution using separation of variales method. Key words: Second-grade fluid, Elliptic cross-section, Exact solution, Velocity profile, Separation of variales.. INTRODUCTION Many industrial materials including clay coatings, drilling muds, suspensions, certain oils and greases, polymer melts, elastomers and many emulsions have een categoried as non-newtonian fluids. Newtonian fluids can e descried y a single model ut it is difficult to suggest a single model which exhiits all properties of non-newtonian fluids. The classification of non-newtonian fluids has remained much confused. However, non-newtonian fluids may e classified as: (i) fluids for which the shear stress depends on the shear rate (ii) fluids for which the relation etween the shear stress and shear rate depends on time; (iii) fluids which possess oth elastic and viscous properties called visco-elastic fluids or elastico-viscous fluids[]. To recommend a single constitutive equation for use in the cases descried in (i), (ii) and (iii) it does not seem possile ecause of great diversity in the physical structure of non-newtonian fluids. Therefore, many constitutive equations for non-newtonian fluids have een proposed. Most of them are empirical or semi-empirical. Although many constitutive equations have een suggested, many questions are still unsolved. Some of the continuum models do not give satisfactory results in accordance with the availale experimental data. Therefore, in many practical applications, empirical or semi-empirical equations have een used. The constitutive equation of a second grade fluid is a linear relation etween the stress and the first Rivlin- Ericksen tensor, the square of the first Rivlin-Ericksen tensor and the second Rivlin-Ericksen tensor [5]. The constitutive equation has three coefficients. There are some restrictions on these coefficients due to the Clausius-Duhem inequality and due to assumption that the Helmholt free energy is minimum in equilirium. A comprehensive discussion on the restrictions for these coefficients has een given y Dunn and Fosdick [6], and Dunn and Rajagopal [7]. The equation of motion of incompressile second grade fluids is of higher order than the Navier-Stokes equation. The Navier-Stokes equation is a second order partial differential equation, ut the equation of motion of a second grade fluid is a third-order partial differential equation. A marked difference etween the case of the Navier-Stokes theory and that for fluids of second grade is that ignoring the non-linearity in Navier-Stokes does not lower the order of the equation, however, ignoring the higher order non-linearities in the case of the second grade fluids, reduces the order of the equation. Exact solutions are very important for many reasons. They provide a standard for checking the accuracies of many approximate methods such as numerical and empirical. Although computer techniques make the complete numerical integration of the non-linear equations feasile, the accuracy of the results can e estalished y a comparison with an exact solution. Many attempts to collect the exact solution of the non-linear equations for unsteady flow of second grade fluid have een done y different researcher for
9 S. Islam, Z. Bano, T. Haroon, A. M. Siddiqui different geometries. The only comprehensive review is that due to [8,9]. However, there are many new exact solutions which have een pulished in journals or in review articles. Two recent reviews, one for unsteady flows [0-] and the other for steady flows of non-newtonian fluids have een pulished y [3]. An exact solution is defined as a solution of the non-linear governing equations and the continuity equation. An exact solution may e in a closed form, in a series form or in a form expressed y a numerical method. Most of the exact solutions for unsteady flows are in series forms. They may e slowly convergent or rapidly convergent. However,it is possile to replace a rapidly convergent series with a slowly convergent one. Then this provides very important facilities. Steady or unsteady flows, one can e otained as a result of several effects. This can e some kind of motion of the oundaries, application of a ody force, wall that applies a tangential stress on the fluid or application of a pressure gradient. One or two of these effects can e applied together to the fluid. If the fluid is initially at rest, the motion of the fluid may eventually ecome steady or remain unsteady. This fact depends on the oundary condition and the kind of effects exerted on the fluid to set it in motion. In this paper, the flow is otained as a result of sudden application of a pressure gradient, therefore this is time-dependent prolem. The examination of this flows is not only done to characterie it ut also to follow its development in time.. GOVERNING EQUATIONS The asic equations governing the flow of an incompressile second grade fluid in the asence of ody forces and thermal effects are div V = 0, () ρv = p + div τ, () ρ is the constant density, V is the velocity vector, p is the pressure, τ is the stress tensor, V denotes the material derivative. Assuming that the flow is unsteady and two dimensional, we seek the velocity profile of the form V =( u( y,, t),0,0). (3) The stress tensor τ defining a second grade fluid is given y [7 9] τ= µ A +α A +α A, µ is the coefficient of viscosity and α, and α are material constants associated with the non-linear terms. The Rivlin-Erickson tensor, A n, are defined as: A = I -the identity tensor, and 0 DA A A V V A n Dt n t n = + n ( ) + ( ) n,. For unsteady two dimensional flow of a second grade fluid, equation () in components form yield: x-component: y-component: u u u u u ρ = +µ ( + ) +α ( + ); t x y t y (4) -component: u u u 0= + ( α +α ) + ( ); y y y y (5)
3 Unsteady Poiseuille flow of second grade fluid in a tue of elliptical cross section 93 u u u y u (6) 0= + ( α +α ) ( ) + ( ) =0. The Clausius-Duhem inequality and the condition that the Helmholt free energy is minimum in equilirium provide the following restrictions [, ]: µ 0, α 0 and ( α +α) = 0. (7) A comprehensive discussion on the restrictions for µ, α and α can e found in the work y Dunn and Rajagopal []. The sign of the material moduli α and α is the suject of much controversy [ 4]. Making use of equation (7) in equations (5) and (6) we get =0 and =0 y showing that p = p( x), Therefore equations(4), (5) and (6) reduces to single equation, i.e., u dp u u u u = +ν ( + ) +β ( + ). t ρ dx y t y (8) β = α ρ. 3. PROBLEM FORMULATION Consider the flow of an incompressile, isothermal second grade fluid in an infinitely long tue, under constant pressure gradient and negligile gravity. The tue has an elliptical cross-section with semi-axes x a and (Fig. ). The flow is considered to e unsteady, and two dimensional. Accordingly the flow velocity u has one non-vanishing component u x, which depends on the coordinates y and given in equation(8). Boundary conditions require that the flow velocity vanishes at the wall of the tue, i.e. on the ellipse y a + = and that the gradient of the velocity vanishes at the centre of the tue, y = = 0. Fig. Two-dimensional Poiseuille flow in a tue of elliptical cross-section. 4. SOLUTION OF THE PROBLEM Erdogan has presented the unsteady flows of an incompressile viscous fluid in rectangular and circular cross-sections. In this paper we have solved unsteady two dimensional flow prolem exactly using separation of variales [5]. We have converted the unsteady prolem given in quation (8) into steady and transient prolems using following transformation
94 S. Islam, Z. Bano, T. Haroon, A. M. Siddiqui 4 Steady prolem is given y u( y,, t)= f ( y, ) + g( y,, t). f f dp =. µ (0) y + dx we solve the steady prolem y assuming f ( y, ) of the following form [4] (9) using equation () in equation (0), we find that y f( y, )= k a () dp a y f( y, )=. µ dxa + a () The unsteady part is given y g g g g g = ν + +β + t y t y (3) suject to following oundary and initial conditions g g g( a,, t)=0, g( y,, t)=0, g( y,,0)= f( y, ), (0,, t)=0, ( y,0, t)=0 y (4) we will solve the aove IBVP using separation of variales method and assuming following in equations (3) and (4) g( y,, t)= Y( y) Z( ) T( t). (3) The resulting system of differential equations is Y + J Y = 0, Y (0) = Y( a) = 0, Z + L Z = 0, Z (0) = Z( ) = 0, (6) k T + ( ) T =0, +βk T(0)= f( y, ). The solutions otained for differential equations in (6) are The solution of the unsteady prolem is given y (m+ ) πy Ym = Bm cos, a m= 0,,,..., (n+ ) π Zn = Dn cos, n= 0,,,..., (m+ ) π (n+ ) π ( ) + ( ) =exp a Tmn t. (m+ ) π (n+ ) π +β (( ) + ( ) ) (m+ ) π (n+ ) π ( ) + ( ) (m+ ) π y (n+ ) π (,, ) = cos cos exp a g yt Dmn t, ( ) =0 =0 m (n ) m n a + π + π (8) +β (( ) + ( ) ) (7)
5 Unsteady Poiseuille flow of second grade fluid in a tue of elliptical cross section 95 or 4 a (m+ ) π y (n+ ) π Dmn = f( y, )cos cos dyd a 0 0 a (9) m+ n 8 d p a ( ) 8 8 Dmn = + + µ d xa + (m+ )(n+ ) π (m+ ) π (n+ ) π and the complete velocity distriution is given y a Dmn µ d a + a m=0 n=0 d p y (m+ ) π y (n+ ) π u( y,, t)= ( ) + cos cos x a (m+ ) π (n+ ) π ( ) + ( ) exp a t, (m+ ) π (n+ ) π +β (( ) + ( ) ) a D mn is given y equation 0. (0) () 5. CONCLUSION In this paper, a prolem is studied in order to show the effect of the applied pressure gradient in a channel of elliptical cross-section on unsteady flow of a fluid of second grade. Exact solution is otained using separation of variales. REFERENCES. K.R. RAJAGOPAL, P.N. KALONI, Continuum Mechanics and its Applications, Hemisphere Press, Washington, DC, 989.. H. GIESEKUS, Several comments on the paper some remarks on useful theorems for the second-grade fluid y P.N. Kaloni, J. Non Newtonian Fluid Mech., 33, pp. 343 348, 989. 3. N. AKSEL, A rief note from the editor on the second-order fluid, Acta Mech., 57, pp. 35 36, 003. 4. R.I. TANNER, Plane creeping flows of incompressile second order fluids, Phys. Fluids, 9, pp. 46 47, 996. 5. K. WALTERS, Relation etween Coleman-Nall, Rivlin-Ericksen, Green-Rivlin, and Oldroyd fluids, ZAMP,, pp. 59 600, 970. 6. J.E. DUNN, R.L. FOSDICK, Thermodynamics, staility and oundedness of fluids of complexity and fluids of second grade, Arch. Ration. Mech. Anal., 56, pp. 9 5, 974. 7. J.E. DUNN, K.R. RAJAGOPAL, Fluids of differential type-critical review and thermodynamic analysis, Int. J. Eng. Sci., 33, pp. 689 79, 995. 8. K.R. RAJAGOPAL, Flow of viscoelastic fluids etween rotating discs, Theor. Compt. Fluid Dyn., 3, pp. 85 06, 99. 9. R.L. FOSDICK, K.R. RAJAGOPAL, Anomalous features in the model of second order fluids, Arch. Ration. Mech. Anal., 70, pp. 45 5, 979. 0. K.R. RAJAGOPAL, On the oundary conditions for fluids of the differential types, in: A. Sequierra (Ed.), Navier-Stokes Equation and Related Non-Linear Prolems, Plenum Press, New York, 995, pp. 73 78.. R. BANDELL, K.R. RAJAGOPAL, C.P. GALDI, On some unsteady motions of fluids of second grade, Arch. Mech., 47, pp. 66 676, 995.. M.E. ERDOGAN, On the flows produced sudden application of a constant pressure gradient or y impulsive motion of a oundary, Int. J. Non-Linear Mech., 38, pp. 78 797, 003. 3. R.I. TANNER, Plane creeping flows of incompressile secondorder fluids, Phys. Fluids, 9, pp. 46 47, 996. 4. TASOS C. PAPANASTASSIOU, GERGIOS C. GEORGIOU, ANDREAS N. ALEXANDROU, Viscous Fluid Flow, CRC press, 000. 5. M. EMIN ERDOGAN, C. ERDEM IMRAK, Effects of the side walls on the unsteady flow of a second-grade fluid in a duct of uniform cross-section, Int. Journal of Non-Linear Mechanics, 39, pp. 379 384, 004. Receivde April 6, 0