Finding steady Poiseuille-type flows for incompressible polymeric fluids by the relaxation method

Stabilization of flows of an incompressible viscoelastic polymeric fluid in a channel with a rectangular cross section under the action of a constant pressure drop is analyzed numerically. The flows are described within the Pokrovskii–Vinogradov rheological mesoscopic model. An algorithm for solving initial-boundary value problems for nonstationary equations of the model is developed. It uses spatial interpolations with Chebyshev nodes and implicit time integration scheme. It is shown analytically that, in the steady state, the model admits three highly smooth solutions. The question of which of these solutions is realized in practice is investigated by calculating the limit of the solutions of nonstationary equations. It is found that this limit coincides, with high accuracy, with one of the three solutions of the steady-state problem, and the values of parameters at which the switching from one of these solutions to another occurs are calculated.