Abstract

We present a conservative finite difference method designed to capture elastic wave propagation in viscoelastic fluids in two dimensions. We model the incompressible Navier–Stokes equations with an extra viscoelastic stress described by the Oldroyd-B constitutive equations. The equations are cast into a hybrid conservation form which is amenable to the use of a second-order Godunov method for the hyperbolic part of the equations, including a new exact Riemann solver. A numerical stress splitting technique provides a well-posed discretization for the entire range of Newtonian and elastic fluids. Incompressibility is enforced through a projection method and a partitioning of variables that suppresses compressive waves. Irregular geometry is treated with an embedded boundary/volume-of-fluid approach. The method is stable for time steps governed by the advective Courant–Friedrichs–Lewy (CFL) condition. We present second-order convergence results in L¹ for a range of Oldroyd-B fluids.

Keywords

viscoelasticity, Oldroyd-B fluid, Godunov method, Riemann solver, projection method, embedded boundaries

Mathematical Subject Classification

Primary: 65N06, 76D05

Milestones

Received: 7 August 2008
Revised: 22 May 2009
Accepted: 25 May 2009

Authors