Numerical Experiments and Theoretical Analysis of the Flow of an Elastic Liquid of the Maxwell-Oldroyd Type in the Presence of Geometrical Singularities

摘要

The Maxwell-B model of an elastic liquid was chosen for finite element calculations with the penalty method of the plane flow over a slot. Increasing oscillations in the solution propagate in an upstream direction for increasing Deborah number (De) (proportional to the relaxation time of the liquid) and for decreasing mesh size until the algorithm fails to converge. A singularity exists near geometrical singularities. The complete set of equations is of a mixed hyperbolic-elliptic type and cannot be solved by global methods. The Jeffreys-Oldroyd-B models give a set of parabolic equations, resembling the problematic convection-diffusion equations. The equations governing the plane flow of a liquid of the second degree are of mixed hyperbolic-elliptic type: the particle trajectories are characteristics. Stable solution schemes for arbitrary values of De are illustrated.