By considering polymeric solutions to be mixtures of two fluids and by applying standard continuum arguments, de Gennes, Doi, and Milner have developed equations to describe such solutions. In the present paper, the problem of the flow of a polymeric solution between two parallel plates under a pressure gradient is considered. A generalized Maxwell model constitutive equation is employed, and the boundary value problem is solved numerically. A number of simplifying assumptions are invoked, in order to solve the highly nonlinear coupled equations. In both steady- and transient-flow situations, migration of the polymer molecules from the wall toward the midplane is predicted.