A Marker-and-Cell Scheme for Viscoelastic Flows on Non Uniform Grids

In this paper, we develop a numerical scheme for the solution of the coupled Stokes and Navier-Stokes equations with constitutive equations describing the flow of viscoelastic fluids. The space discretization is based on the so-called Marker-And-Cell (MAC) scheme. The time discretization uses a fractional-step algorithm where the solution of the Navier-Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite-volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy reasons, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable from \({\mathbf{c}}\) to \(\log ({\mathbf{c}})\), and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the Stokes flow of an Oldroyd-B fluid in the lid-driven cavity at We = 1 confirm the scheme efficiency.