A projection scheme for Navier-Stokes with variable viscosity and natural boundary condition

Combiningthe Navier-Stokes systems with Neumann (or natural) boundary condition to characterize a fluid flow is frequent. The popular projection (or pressure correction) methods inspired by Chorin and Temam are not well adapted to such boundary condition, which translate in loss of accuracy. If some alternative projection methods have been proposed to reduce the accuracy loss due to the Neumann condition in case of Newtonian fluids, little has been proposed for generalized Newtonian fluids. In this work, we propose two methods derived from the incremental pressure correction projection that can be used for fluids with inhomogeneous or variable viscosity with natural boundary condition. Both time and space accuracy of the methods will be illustrated using a manufactured solution.