Calculations of steady-state viscoelastic flow in an undulating tube

Abstract

The flow of a viscoelastic fluid in an undulating tube is of importance both for modeling the flow of polymeric fluids through porous media and for testing numerical techniques for the simulation of viscoelastic flows. This geometry represents one of the simplest pore models for the consideration of the converging-diverging nature of the flow field in applications ranging from enhanced oil recovery to manufacturing of advanced composites using resin transfer molding. For an upper convected Maxwell fluid, a mixed pseudospectral/finite difference method (PSFD) is developed, employing the coordinate system provided by the streamlines and the lines orthogonal to them. For an Oldroyd-B fluid, a modification of the PSFD method is used, implemented in a stretched cylindrical coordinate system (PCFD method). For both the Maxwell and the Oldroyd-B fluid, solutions converged with mesh refinement are obtained up to high Deborah numbers. The results obtained using the Maxwell fluid show no increase in the flow resistance with increasing flow elasticity, in reference to the purely viscous flow. The results for the Oldroyd-B fluid show a very small increase in the flow resistance. These findings are in disagreement with experimental data reported in the literature, which have been obtained with Boger fluids. For the Maxwell fluid, a perturbation analysis valid for small amplitudes of undulation shows that extremely steep boundary layers are developed with increasing elasticity in the flow. The boundary layers are developed both near the wall and the centerline of the tube and their resolution becomes a key element for any potentially successful numerical method. In addition, their presence explains, at least partially, the failure of previous attempts by other investigators to obtain converged solutions at high values of elasticity.