Poiseuille Flows

Abstract

We consider steady flow of an upper convected Maxwell fluid through a channel (or a pipe) with wavy walls. The vorticity of this flow will change type when the velocity in the center of the channel is larger than a critical value defined by the propagation of shear waves. There is then a central region of the channel in which the vorticity equation is hyperbolic and a low speed region near the walls where the vorticity equation is elliptic. We linearize the problem for small amplitude waviness and the linearized problem is solved in detail. The characteristic nets depend on the viscoelastic “Mach” number, which is the ratio (M=U/c) of the unperturbed maximum velocity U to the speed of shear waves c into the fluid at rest, and on the elasticity number E. There is a supercritical (hyperbolic) region around the center of the channel when M>1. When M»1, the thickness of this hyperbolic region is small when E is large, and large when E is small. Regions of positive and negative vorticity are swept out along forward facing characteristics in the hyperbolic region.