Rayleigh–Plesset equation

Remarks

Incompressibility forces the radial velocity of the liquid to be $u(r, t) = R^2 \dot R / r^2$. The equation follows by integrating the radial momentum equation from $r = R$ to $r = \infty$. The normal stress condition at the bubble wall then adds the Young–Laplace term $2 \sigma / R$ and the viscous term $4 \mu_L \dot R / R$, with $\mu_L = \rho_L \nu_L$.

Example: an empty cavity ($p_B = 0$) with $\nu_L = 0$, $\sigma = 0$ and constant $p_\infty > 0$ is Rayleigh's collapse problem. For a cavity at rest at radius $R_0$, the energy integral gives $\dot R^2 = \frac{2 p_\infty}{3 \rho_L} \big( R_0^3 / R^3 - 1 \big)$. The collapse time is about $0.915 \, R_0 \sqrt{\rho_L / p_\infty}$.

Non-example: a bubble near a wall does not stay spherical, and its radius alone does not determine the flow, so the equation does not apply.

For a gas bubble one often takes $p_B = p_v + p_{g0} (R_0 / R)^{3 \kappa}$, with the vapour pressure $p_v$ and a polytropic exponent $\kappa$.