Spatial decay of steady perturbations of plane Poiseuille flow for the Oldroyd-B equation

Abstract

A numerical eigenfunction analysis of steady perturbations of plane Poiseuille flow for the Oldroyd-B equation is presented. From this the importance of the downstream boundary conditions used in entry-flow calculations may be determined. It is shown that the length scale over which the downstream boundary condition affects the upstream flow is much shorter than the length scale over which the polymer stress relaxes to that of fully developed flow. Consequently features such as vortex enhancement should not be significantly affected by the downstream boundary condition. A correction to the pressure drop is also derived, thereby allowing the Couette correction to be calculated even when the downstream section is too short for fully developed flow to be obtained at the downstream boundary of the computational domain. In addition we show that an instability criterion for time-dependent simulation of planar Couette flow derived in an earlier paper (R.A. Keiller, J. Non-Newtonian Fluid Mech., 43 (1992) 229–246) may also apply to steady-state calculations of Poiseuille flow.