Upwind discretization of a time-dependent two-dimensional grade-two fluid model

In this paper, we propose a finite-element scheme for solving numerically the equations of a transient two dimensional grade-two fluid non-Newtonian Rivlin-Ericksen fluid model. This system of equations is considered an appropriate model for the motion of a water solution of polymers. As expected, the difficulties of this problem arise from the transport equation. As one of our aims is to derive unconditional a priori estimates from the discrete analogue of the transport equation, we stabilize our scheme by adding a consistent stabilizing term. We use the IP2 − IP1 Taylor-Hood finite-element scheme for the velocity v and the pressure p, and the discontinuous IP1 finite element for an auxiliary variable z. The error is of the order of h + k, considering that the discretization of the transport equation loses inevitably a factor h.