drag force scheme {#s:drag}
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The dust component is treated as a number of individual superparticles, the position ${{\boldsymbol x}}$ and velocity ${{\boldsymbol v}}$ of each evolved according to $$\begin{aligned}
\label{eq:eqmotpar}
  \frac{{\mathrm{d}}{{\boldsymbol x}}}{{\mathrm{d}}t} &=& {{\boldsymbol v}} - \frac{3}{2} \varOmega x
  {{\boldsymbol e}}_y \, , \\
  \frac{{\mathrm{d}}{{\boldsymbol v}}}{{\mathrm{d}}t} &=& 
    2 \varOmega v_y {{\boldsymbol e}}_x - \frac{1}{2} \varOmega v_x {{\boldsymbol e}}_y + {{{\boldsymbol \nabla}}}\varPhi - \frac{1}{\tau_{\rm f}} ({{\boldsymbol v}}-\overline{{{\boldsymbol u}}}) \, .
    \label{eq:dvpdt}\end{aligned}$$ The particles feel no pressure or Lorentz forces, but are subjected to the gravitational potential $\varPhi$ of their combined mass. Particle collisions are taken into account as well (see Sect. \[s:coll\] below).

Two-way drag forces between gas defined on a fixed grid and Lagrangian particles are calculated through a particle-mesh method [see @YoudinJohansen2007 for details]. First the gas velocity field is interpolated to the position of a particle, using second order spline interpolation. The drag force on the particle is then trivial to calculate. To ensure momentum conservation we then take the drag force and add it with the opposite sign among the 27 nearest grid points, using the Triangular Shaped Cloud scheme to ensure momentum conservation in the assignment [@HockneyEastwood1981].

Units
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All simulations are run with natural 