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 aswhere 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-liquid interface , in which, is the pressure within the bubble, assumed to be uniform and is the external pressure infinitely far from 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.
History
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). 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. The kinetic energy of the inrushing fluid is where is the time-dependent radius of the void, and the radial velocity of the fluid there. The work done by the fluid pressing in at infinity is , and equating these two energies gives a relation between and . Then, noting that , separation of variables gives Besant's result. Rayleigh went further than Besant, in evaluating the integral (Euler's beta function) in terms of gamma functions. 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 in the fluid at a radius is given by
.
When the void is at least one quarter of its initial volume, then the pressure decreases monotonically from at infinity to zero at . 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.
Derivation
The Rayleigh–Plesset equation can be derived entirely from first principles using the bubble radius as the dynamic parameter. 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
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). 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
Therefore
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
Momentum conservation
Assuming that the liquid is a Newtonian fluid, the incompressible Navier–Stokes equation in spherical coordinates for motion in the radial direction gives