Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / b / ii / Solution

Write $\Delta \mathbf{u}=(U,W,0)$. The background shear givesThe constant radial and azimuthal components of the dust equation are thereforeWith the Stokes number $\mathrm{St}=\tau {\Omega }_K$, their solution is

The gas is slower than a Keplerian particle, so the particle feels an aerodynamic headwind. Drag force removes its angular momentum, and it drifts radially inward because $U<0$. Its radial speed isDifferentiation with respect to $\mathrm{St}>0$ shows that the unique maximum occurs atwhere $|U|=|V|$.