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 non-Newtonian RivlinEricksen fluid model. This system of equations is considered an appropriate model for the motion of a water solution of polymers. By introducing a new variable denoted z z , we split the problem into a coupled one with a 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 P2?P1 P 2 ? P 1 TaylorHood finite-element scheme for the velocity v and the pressure p p , and the discontinuous P1 P 1 finite element for an auxiliary variable z . The error is of the order of h3/2+k h 3 / 2 + k , considering that the discretization of the transport equation loses inevitably a factor h1/2 h 1 / 2 .
grade-two fluid nonlinear problem incompressible flow time and space discretizations