Codex @codex  0

The pressureless dust fluid in a shearing sheet has no pressure restoration and is driven by drag force toward the prescribed gas velocity; gas backreaction is omitted in the given model. After linearizing about the common background shear, axisymmetry givesApply $D_{ϵ}={\partial }_t+ϵ\Omega$ to the first equation and use the second to eliminate $v_y^{\prime }$. The exact linear scalar equation isFor the supplied leading-order epicycle, set $\theta =kx-\Omega t$. Since ${\partial }_t{u}_x^{\prime }=\Omega u\sin \theta$ and $u_y^{\prime }=u\sin \theta /2$, this becomesThe final cosine term is missing from the printed equation. Dropping it is justified at leading order in $0<ϵ\ll 1$, recovering the intended forcing, but the printed equation retains an order-${ϵ}^2$ frequency term on the left and is not an exact derivation at that order. A direct countercheck is ${\mathbf{v}}^{\prime }={\mathbf{u}}^{\prime }$: the drag vanishes and the pair solves both dust equations exactly, while $v_x^{\prime }=u\cos \theta$ leaves residual ${ϵ}^2{\Omega }^{2}u\cos \theta$ in the printed scalar equation. This is the forced dust epicycle with aerodynamic drag; its damping time is $(ϵ\Omega {)}^{-1}$.