Rayleigh–Plesset equation

is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius

[5] Neglecting surface tension and viscosity, the equation was first derived by W. H. Besant in his 1859 book with the problem statement stated as An infinite mass of homogeneous incompressible fluid acted upon by no forces is at rest, and a spherical portion of the fluid is suddenly annihilated; it is required to find the instantaneous alteration of pressure at any point of the mass, and the time in which the cavity will be filled up, the pressure at an infinite distance being supposed to remain constant (in fact, Besant attributes the problem to Cambridge Senate-House problems of 1847).

[6] Besant predicted the time required to fill an empty cavity of initial radius

to be Lord Rayleigh found a simpler derivation of the same result, based on conservation of energy.

Rayleigh adapted this approach to the case of a cavity filled with an ideal gas (a bubble) by including a term for the work done compressing the gas.

For the case of the perfectly empty void, Rayleigh determined that the pressure

is given by: When the void is at least one quarter of its initial volume, then the pressure decreases monotonically from

As the void shrinks further a pressure maximum, greater than

appears at very rapidly growing and converging on the void.

The equation was first applied to traveling cavitation bubbles by Milton S. Plesset in 1949 by including effects of surface tension.

[7] The Rayleigh–Plesset equation can be derived entirely from first principles using the bubble radius as the dynamic parameter.

[3] Consider a spherical bubble with time-dependent radius

Assume that the bubble contains a homogeneously distributed vapor/gas with a uniform temperature

Outside the bubble is an infinite domain of liquid with constant density

from the center of the bubble, the varying liquid properties are pressure

By conservation of mass, the inverse-square law requires that the radially outward velocity

must be inversely proportional to the square of the distance from the origin (the center of the bubble).

be some function of time, In the case of zero mass transport across the bubble surface, the velocity at the interface must be which gives that In the case where mass transport occurs and assuming the bubble contents are at constant density, the rate of mass increase inside the bubble is given by with

can be approximated by the original zero mass transfer form

, so that[7] Assuming that the liquid is a Newtonian fluid, the incompressible Navier–Stokes equation in spherical coordinates for motion in the radial direction gives Substituting kinematic viscosity

from mass conservation yields Note that the viscous terms cancel during substitution.

[7] Separating variables and integrating from the bubble boundary

be the normal stress in the liquid that points radially outward from the center of the bubble.

In spherical coordinates, for a fluid with constant density and constant viscosity, Therefore at some small portion of the bubble surface, the net force per unit area acting on the lamina is where

[7] If there is no mass transfer across the boundary, then this force per unit area must be zero, therefore

and so the result from momentum conservation becomes whereby rearranging and letting

gives the Rayleigh–Plesset equation[7] Using dot notation to represent derivatives with respect to time, the Rayleigh–Plesset equation can be more succinctly written as More recently, analytical closed-form solutions were found for the Rayleigh–Plesset equation for both an empty and gas-filled bubble [8] and were generalized to the N-dimensional case.

[9] The case when the surface tension is present due to the effects of capillarity were also studied.

[9][10] Also, for the special case where surface tension and viscosity are neglected, high-order analytical approximations are also known.

[11] In the static case, the Rayleigh–Plesset equation simplifies, yielding the Young–Laplace equation: When only infinitesimal periodic variations in the bubble radius and pressure are considered, the RP equation also yields the expression of the natural frequency of the bubble oscillation.

The Rayleigh–Plesset equation is often applied to the study of cavitation bubbles, shown here forming behind a propeller.
Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to oscillatory pressure at its natural frequency undergoes expansion and then collapses.
Numerical integration of RP eq. including surface tension and viscosity terms. Initially at rest in atmospheric pressure with R0=50 um, the bubble subjected to pressure-drop undergoes expansion and then collapses.