Some unsteady flows of a second grade fluid over a plane wall

The effects of the side walls on unsteady flow of a second grade fluid over a plane wall are considered. The following three problems have been studied: (1) flow due to oscillation of a plate; (2) flow imposed by a plate that applies constant tangential stress to the fluid and flow induced by a constantly accelerating plate. Exact solutions of some unsteady flows of a second grade fluid over a plane wall are obtained by the sine and cosine transforms. These solutions are expressed in terms of definite integrals. It is shown that the starting solution for the flow of the oscillating plate can be presented as the sum of the steady state and the transient solutions. For large times they tend to the steady state solutions which are independent of the initial conditions and are periodic in time. This gives a result for the shear stress at the bottom wall. The shear stress at the bottom wall is maximum or minimum at the middle of the plate and it varies more rapidly when it is closer to the side walls. (C) 2010 Elsevier Ltd. All rights reserved.