Nonlinear superposition of unidirectional and plane flows of viscoelastic Lodge fluids

Abstract

The paper deals with steady fully developed flows through straight channels of constant cross-section under the influence of an axial pressure gradient and under the action of a moving wall. Those flow fields are characterized by Cartesian velocity components of the type u(x,y), v(x,y) and w(x,y) and thus may be considered to be the superposition of a longitudinal and a transverse plane part. They possess some striking properties, not only in case of a Newtonian fluid, but also for quasi-linear viscoelastic fluids with single-integral constitutive equations. We find a one-sided coupling between the axial component w and the transverse components u and v. Accordingly, the transverse flow does not depend on the axial boundary conditions and can be treated first as a two-dimensional problem. Afterwards, a linear integro-differential boundary value problem concerning w(x,y) results, from which some superposition theorems follow, especially with respect to integral quantities like the axial volume flux and the power input. This has some general implications regarding the pumping characteristics of the combined pressure-drag flow, and that at arbitrary values of the Reynolds number and of the Deborah number.