Development of Boundary Layer of Highly Elastic Flow of the Upper Convected Maxwell Fluid over a Stretching Sheet

Document Type: Original Article

Authors

Department of Mechanical and Aerospace Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

Abstract

High Weissenberg boundary layer flow of viscoelastic fluids on a stretching surface has been studied. The flow is considered to be steady and two dimensional. Flows of viscoelastic liquids at high Weissenberg number exhibit stress boundary layers near walls. These boundary layers are caused by the memory of the fluid. Upon proper scaling and by means of an exact similarity transformation, the non-linear momentum and constitutive equations of each layer transform into the respective system of highly nonlinear and coupled ordinary differential equations. Effects of variation in pressure gradient and Weissenberg number on velocity profile and stress components are investigated. It is observed that the value of stress components decrease by Weissenberg number. Moreover, the results show that increasing the pressure gradient results in thicker velocity boundary layer. It is observed that unlike the Newtonian flows, in order to maintain a potential flow, normal stresses must inevitably develop in far fields.

Keywords


[1]     Crane, L. J., “Flow past a stretching sheet”, ZAMP, Vol. 21, 1970, pp. 645–647.

[2]     Prasad, K. V., Santhi, S. R., and Datti, P. S:, “Non-Newtonian power-law fluid flow and heat transfer over a non-linearly stretching surface”, Appl. Math., Vol. 3, No. 5, 2012, pp. 425-435.

[3]     Xu, H., and Liao, S. J:, “Laminar flow and heat transfer in the boundary-layer of non-Newtonian fluids over a stretching flat sheet”, Comput. Math. Appl., Vol. 57, 2009, pp. 1425-1431.

[4]     Abel, M. S., Datti, P. S., and Mahesha, N., “Flow and heat transfer in a power-law fluid over a stretching sheet with variable thermal conductivity and nonuniform heat source”, Int. J. Heat Mass Transfer, Vol. 52, 2009, pp. 2902–2913.

[5]     Wang, C., “Analytic solutions for a liquid film on an unsteady stretching surface”, Heat Mass Transfer, Vol. 42, 2006, pp.759–766.

[6]     Ashrafi, N., Mohamadali, M., “High Weissenberg Number Stress Boundary Layer for the Upper Convected Maxwell Fluid”, Proceedings of the ASME International Mechanical Engineering Congress & Exposition, Vol. 8B, Heat Transfer and Thermal Engineering, Montreal , 2014, pp. 201-207.

[7]     Hassanien, I. A., “Flow and heat transfer on a continuous flat surface in a parallel free stream of viscoelastic second-order fluid”, Appl. Sci. Res., Vol. 49, 1992, pp. 335-344.

[8]     Schichting., H., “Boundary Layer Theory”, sixth ed., McGraw Hill, New York, 1964, Chap 7.

[9]     Hayat, T., Fetecau, C., Abbas, Z., and Ali, N., “Flow of a viscoelastic fluid with fractional Maxwell model between two side walls due to suddenly moved plate”, Nonlinear Anal. Real World Appl., Vol. 9, 2008, pp. 2288-2295.

[10]  Shateyi, S., “A new numerical approach to MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction”, Bound. Val. Prob., Vol. 196, 2013.

[11]  Fetecau, C., Jamil, M., Fetecau, C., and Siddique, I., “A note on the second problem of Stokes for Maxwell fluids”, Int. J. Non-Linear Mech., Vol. 44, 2009, pp. 1085-1090.

[12]  Hayat, T., Shehzad, S. A., and Alsaedi, A., “Study on three-dimensional flow of Maxwell fluid over a stretching surface with convective boundary conditions”, Int. J. Phys. Sci., Vol. 7, No. 5, 2012, pp.761-768.

[13]  Awais, M., Hayat, T., Alsaedi, A., and Asghar, S., “Time-dependent three-dimensional boundary layer flow of a Maxwell fluid”, Computers & Fluids, Vol. 91, 2014, pp. 21–27.

[14]  Rajagopal, K. R., Boundary layers in non-linear fluids, in: M.D.P. Monteivo Marques, Trends in Applications of Mathematics to Mechanics, in: Pittman Monographs and Surveys in Pure and Applied Mathematics, Vol. 77, Longman, New York, 1995.

[15]  Renardy, M., and Wang, X., “Boundary layers for the upper convected Maxwell fluid”, J. Non-Newtonian Fluid Mech., Vol. 189, 2013, pp. 14–18.

[16]  Renardy, M., “High Weissenberg number boundary layers for the upper convected Maxwell fluid”, J. Non-Newtonian Fluid Mech., Vol. 68, 1997, pp. 125-132.

[17]  Hagen, T., and Renardy, M., “Boundary layer analysis of the Phan–Tien–Tanner and Giesekus model in high Weissenberg number flow”, J. Non-Newtonian Fluid Mech., Vol. 73, 1997, pp. 181–189.

[18]  Renardy, M., “Prandtl boundary layers for the Phan-Thien Tanner and Giesekus fluid”, Z. Angew. Math. Phys., Vol. 66, 2014, pp. 1061- 1070.

[19]  Renardy, M., “Wall Boundary Layers for Maxwell Liquids”, Arch. Rational Mech. Anal., Vol. 152, 2000, pp. 93–102.

[20]  Renardy, M. and Wang, X., “Well-posedness of boundary layer equations for time-Dependent flow of Non-Newtonian fluids”, J. Math. Fluid Mech., Vol. 16, 2014, pp. 179–191

[21]  Renardy, M., “The initial value problem for creeping flow of the upper convected Maxwell fluid at high Weissenberg number”, Math. Meth. Appl. Sci., Vol. 38, 2014, pp. 959–965.

[22]  Ogilvie, G. I., Proctor, M. R. E., “On the relation between viscoelastic and magneto-hydrodynamic flows and their instabilities”, J. Fluid Mech., Vol. 476, 2003, pp. 389-409.

[23]  Bird, R. B., Armstrong, R. C., “Dynamics of polymeric Liquids”, second ed., John Wiley & Sons, New York, 1987, Chap 5.

[24]  [24] Evans, J. D., “Re-entrant corner flows of the Upper Convected Maxwell fluid,” Proc. Roy. Soc. A, Vol.461, 2005, pp.117–142.

[25]  Renardy, M.,“A matched solution for corner flow of the upper convected Maxwell fluid”, J. Non-Newtonian Fluid Mech.,Vol. 58, 1995, pp. 83-89.

[26]  Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P., Numerical Recipes )in Fortran 77(, 2nd ed., Cambridge University Press, New York, 2007.Chap12.