Chapter 3 — The Rayleigh–Plesset equation
A momentum balance on a spherical bubble.
The previous chapter explained why a bubble appears. This chapter explains what it does once it is there. The single bubble — spherical, immersed in an infinite liquid, with a known internal pressure history — is the elementary object of cavitation physics. Its dynamics are governed by a second-order nonlinear ordinary differential equation in the bubble radius, derived from Newton’s second law applied to the liquid that surrounds it. Lord Rayleigh wrote down the inviscid surface-tension-free version in 1917. Milton Plesset added surface tension, viscosity, and vapour content over the next three decades. The full form bears their names.
The equation is the workhorse of the rest of the book. Chapters 4 (growth), 5 (collapse), 6 (noise and luminescence), and 7 (oscillation) all live downstream of it: each adds one regime of behaviour, with the underlying ODE held constant.
Four lessons develop the equation in full:
- 3.1 Derivation from momentum balance — apply Newton’s second law to a thin spherical shell of liquid; integrate from the bubble wall to infinity; pick up the surface-tension and viscous boundary corrections.
- 3.2 Bubble contents — the pressure inside the bubble. Permanent gas vs vapour. The polytropic-exponent approximation and when it fails.
- 3.3 Static equilibrium and the Blake threshold — the radius at which all four pressure terms balance, the stability of the equilibrium against radial perturbation, and the critical tension at which no equilibrium exists.
- 3.4 Solving the Rayleigh–Plesset equation — analytical regimes (Rayleigh’s pure-inertial collapse, the linearised small-oscillation problem) and numerical integration of the full nonlinear equation under arbitrary .