Dynamics of dusty vortices - I. Extensions and limitations of the terminal velocity approximation

Motivated by the stability of dust laden vortices, in this paper we study the terminal velocity approximation equations for a gas coupled to a pressureless dust fluid and present a numerical solver for the equations embedded in the FARGO3D hydrodynamics code. We show that for protoplanetary discs it is possible to use the barycentre velocity in the viscous stress tensor, making it trivial to simulate viscous dusty protoplanetary discs with this model. We also show that the terminal velocity model breaks down around shocks, becoming incompatible with the two-fluid model it is derived from. Finally we produce a set of test cases for numerical schemes and demonstrate the performance of our code on these tests. Our implementation embedded in FARGO3D using an unconditionally stable explicit integrator is fast, and exhibits the desired second-order spatial convergence for smooth problems.