A New Boundary Condition for Rate-Type Non-Newtonian Diffusive Models and the Stable MAC Scheme

Abstract

We present a new Dirichlet boundary condition for the rate-type non-Newtonian diffusive constitutive models. The newly proposed boundary condition is compared with two such well-known and popularly used boundary conditions as the pure Neumann condition [1] and the Dirichlet condition by Sureshkumar and Beris [2]. Our condition is demonstrated to be more stable and robust in a number of numerical test cases. A new Dirichlet boundary condition is implemented in the framework of the finite difference Marker and Cell (MAC) method. In this paper, we also present an energy-stable finite difference MAC scheme that preserves the positivity for the conformation tensor and show how the addition of the diffusion helps the energy-stability in a finite difference MAC scheme-setting.