# Dynamics of Dusty Vortices I: Extensions and limitations of the terminal velocity approximation

## TL;DR

This paper investigates the terminal velocity approximation in dusty vortices within protoplanetary discs, highlighting its limitations near shocks and providing a numerical solver implementation in FARGO3D.

## Contribution

It introduces a numerical solver for the terminal velocity equations in FARGO3D, demonstrating its application to viscous dusty protoplanetary discs and analyzing its limitations.

## Key findings

- The model works well for viscous discs using baricenter velocity.
- The approximation breaks down near shocks, conflicting with the two-fluid model.
- The implemented solver is fast, stable, and second-order accurate for smooth problems.

## Abstract

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 baricenter 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.