Rayleigh–Plesset equation

In fluid mechanics, the Rayleigh–Plesset equation or Besant–Rayleigh–Plesset equation is a nonlinear ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. Its general form is usually written as

where

${\displaystyle \rho _{L}}$ is the density of the surrounding liquid, assumed to be constant

${\displaystyle R(t)}$ is the radius of the bubble

${\displaystyle \nu _{L}}$ is the kinematic viscosity of the surrounding liquid, assumed to be constant

${\displaystyle \sigma }$ is the surface tension of the bubble-liquid interface

${\displaystyle \Delta P(t)=P_{\infty }(t)-P_{B}(t)}$, in which, ${\displaystyle P_{B}(t)}$ is the pressure within the bubble, assumed to be uniform and ${\displaystyle P_{\infty }(t)}$ is the external pressure infinitely far from the bubble

Provided that ${\displaystyle P_{B}(t)}$ is known and ${\displaystyle P_{\infty }(t)}$ is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius ${\displaystyle R(t)}$.

The Rayleigh–Plesset equation can be derived from the Navier–Stokes equations under the assumption of spherical symmetry. It can also be derived using an energy balance.