An algorithm for the grade-two rheological model

We develop an algorithm for solving the general grade-two model of non-Newtonian fluids which for the first time includes inflow boundary conditions. The algorithm also allows for both of the rheological parameters to be chosen independently. The proposed algorithm couples a Stokes equation for the velocity with a transport equation for an auxiliary vector-valued function. We prove that this model is well posed using the algorithm that we show converges geometrically in suitable Sobolev spaces for sufficiently small data. We demonstrate computationally that this algorithm can be successfully discretized and that it can converge to solutions for the model parameters of order one. We include in the appendix a description of appropriate boundary conditions for the auxiliary variable in standard geometries.