Gas in hydrostatic balance has zero vertical velocity. A dust grain subject to linear drag with <aerodynamic stopping time> $\tau$ therefore obeys
$$
\boxed{\ddot z+\frac1\tau\dot z+\Omega_z^2z=0}.
$$
This is a <damped harmonic oscillator>. For $\Omega_z\tau>1/2$, the motion is underdamped, with angular frequency
$$
\sqrt{\Omega_z^2-\frac1{4\tau^2}}
$$
and envelope $e^{-t/(2\tau)}$. <Critical damping> occurs at $\Omega_z\tau=1/2$. For $\Omega_z\tau<1/2$, the motion is overdamped. In the strong-drag limit $\Omega_z\tau\ll1$, a rapid transient on timescale $\tau$ leaves slow <dust settling in an astrophysical disk> at rate $\Omega_z^2\tau$; in the weak-drag limit the grain makes many damped vertical oscillations.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 321 / 3 / d / Solution

Dust settling is the vertical drift of solid particles toward a disk's midplane under gravity and <drag force>. In the strong-drag limit, a particle with <aerodynamic stopping time> $\tau$ settles through a linear vertical gravitational field at rate $\Omega_z^2\tau$.

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 Incoming links: Dust settling in an astrophysical disk