Rayleigh–Plesset equation

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The Rayleigh–Plesset equation is often applied to the study of cavitation bubbles, shown here forming behind a propeller.

In fluid mechanics, the Rayleigh–Plesset equation is an ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of liquid.[1][2][3][4] Its general form is usually written as


is the pressure within the bubble, assumed to be uniform
is the external pressure infinitely far from the bubble
is the density of the surrounding liquid, assumed to be constant
is the radius of the bubble
is the kinematic viscosity of the surrounding liquid, assumed to be constant
is the surface tension of the bubble

Provided that is known and is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius .

The Rayleigh–Plesset equation is derived from the Navier–Stokes equations under the assumption of spherical symmetry.[4] Neglecting surface tension and viscosity, the equation was first derived by John Strutt, 3rd Baron Rayleigh in 1917. The equation was first applied to traveling cavitation bubbles by Milton S. Plesset in 1949.[5]


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 , where is time. Assume that the bubble contains a homogeneously distributed vapor/gas with a uniform temperature and pressure . Outside the bubble is an infinite domain of liquid with constant density and dynamic viscosity . Let the temperature and pressure far from the bubble be and . The temperature is assumed to be constant. At a radial distance from the center of the bubble, the varying liquid properties are pressure , temperature , and radially outward velocity . Note that these liquid properties are only defined outside the bubble, for .

Mass conservation[edit]

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).[5] Therefore, letting 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, the rate of mass increase inside the bubble is given by

with being the volume of the bubble. If is the velocity of the liquid relative to the bubble at , then the mass entering the bubble is given by

with being the surface area of the bubble. Now by conservation of mass , therefore . Hence


In many cases, the liquid density is much greater than the vapor density, , so that can be approximated by the original zero mass transfer form , so that[5]

Momentum conservation[edit]

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 and rearranging gives

whereby substituting from mass conservation yields

Note that the viscous terms cancel during substitution.[5] Separating variables and integrating from the bubble boundary to gives

Boundary conditions[edit]

Let 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 is the surface tension.[5] 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[5]

Using dot notation to represent derivatives with respect to time, the Rayleigh–Plesset equation can be more succinctly written as


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.

Recently, analytical closed-form solution were found for the Rayleigh–Plesset equation for both an empty and gas—filled bubble [6] and were generalized to the N-dimensional case.[7] The case when the surface tension is present due to the effects of capillarity were also studied.[7][8]

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

In the static case, the Rayleigh–Plesset equation simplifies, yielding to the Young-Laplace equation:

When only infinitesimal periodic variations in the bubble radius and pressure are considered, the RP equation also yields to the expression of the natural frequency of the bubble oscillation.


  1. ^ Rayleigh, Lord (1917). "On the pressure developed in a liquid during the collapse of a spherical cavity". Phil. Mag. 34: 94–98. 
  2. ^ Plesset, M.S. (1949). "The dynamics of cavitation bubbles". ASME J. Appl. Mech. 16: 228–231. 
  3. ^ a b Leighton, T. G. (17 April 2007). "Derivation of the Rayleigh–Plesset equation in terms of volume". Southampton, UK: Institute of Sound and Vibration Research. 
  4. ^ a b Lin, Hao; Brian D. Storey; Andrew J. Szeri (2002). "Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation". Journal of Fluid Mechanics. 452. doi:10.1017/S0022112001006693. ISSN 0022-1120. 
  5. ^ a b c d e f Brennen, Christopher E. (1995). Cavitation and Bubble Dynamics. Oxford University Press. ISBN 0-19-509409-3. 
  6. ^ Kudryashov, Nikolay A.; Sinelshchikov, Dnitry I. (18 September 2014). "Analytical solutions of the Rayleigh equation for empty and gas-filled bubble". arXiv:1409.6699v1free to read. Bibcode:2014JPhA...47N5202K. doi:10.1088/1751-8113/47/40/405202. 
  7. ^ a b Kudryashov, Nikolay A.; Sinelshchikov, Dnitry I. (31 December 2014). "Analytical solutions for problems of bubble dynamics". arXiv:1608.00811free to read. Bibcode:2016arXiv160800811K. doi:10.1016/j.physleta.2014.12.049. 
  8. ^ Mancas, Stefan C.; Rosu, Haret C. (7 August 2015). "Cavitation of spherical bubbles: closed-form, parametric, and numerical solutions". arXiv:1508.01157free to read. Bibcode:2016PhFl...28b2009M. doi:10.1063/1.4942237. 
  9. ^ Obreschkow, D.; Bruderer M.; Farhat, M. (5 June 2012). "Analytical approximations for the collapse of an empty spherical bubble". Physical Review E. 85. arXiv:1205.4202free to read. Bibcode:2012PhRvE..85f6303O. doi:10.1103/PhysRevE.85.066303.