Exact solutions for unsteady unidirectional flows of second grade fluid

Abstract

Second grade fluid or second grade fluid is one of the most popular used model to describe non-Newtonian fluids. In study, the problems involving unsteady unidirectional flows of second grade fluid, flows of second grade fluid through a porous medium and flows of heated second grade fluid are discussed. The unidirectional flows of second grade fluid considered in this study are Couette flow, Stokes’ first flow problem and plane Poiseuille flow. The exact solutions for Couette flow and Poiseuille flow are obtained using the Laplace transform while the Fourier transform method is used to obtain the exact solution of the Stokes’ first flow problem. The momentum equation for the flows of second grade fluid through a porous medium is derived based on the modified Darcy’s law. The momentum equation for flows of second grade fluid through a porous medium has an extra term, which is the viscous damping force caused by the micro-pore structures of the porous medium. It is found that the velocity of the flows decrease when the porosity, j and permeability, K parameter are increased. For flows of heated second grade fluids, the velocity and temperature distribution are determined using the simple and double Fourier sine transforms.