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[[File:Cavitacion.jpg|thumb|right|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 (mechanics)|dynamics]] of a spherical [[liquid bubble|bubble]] in an infinite body of [[liquid]].<ref name="Leighton">{{cite journal|last=Leighton|first=T. G.|date=17 April 2007|title=Derivation of the Rayleigh–Plesset equation in terms of volume|publisher=Institute of Sound and Vibration Research|location=[[Southampton]], UK|url=http://eprints.soton.ac.uk/45698/}}</ref><ref name="Lin2002">{{cite journal|last=Lin|first=Hao|coauthors=Brian D. Storey, Andrew J. Szeri|year=2002|title=Inertially driven inhomogeneities in violently collapsing bubbles: the validity of the Rayleigh–Plesset equation|journal=Journal of Fluid Mechanics|volume=452|issn=0022-1120|doi=10.1017/S0022112001006693|url=http://digitalcommons.olin.edu/cgi/viewcontent.cgi?article=1002&context=facpub_2002&sei-redir=1&referer=http%3A%2F%2Fscholar.google.com%2Fscholar%3Fq%3DRayleigh–Plesset%2Bequation%26hl%3Den%26as_sdt%3D0%26as_vis%3D1%26oi%3Dscholart%26sa%3DX%26ei%3Du_2rT8q8GsqLgwe8zsnfAQ%26ved%3D0CBgQgQMwAA#search=%22Rayleigh–Plesset%20equation%22}}</ref>  Its general form is usually written as


:<math>\frac{P_B(t)-P_\infty(t)}{\rho_L} = R\frac{d^2R}{dt^2} + \frac{3}{2}\left(\frac{dR}{dt}\right)^2 + \frac{4\nu_L}{R}\frac{dR}{dt} + \frac{2S}{\rho_LR}</math>
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where
:<math>P_B(t) </math> is the [[pressure]] within the bubble, assumed to be uniform
:<math>P_\infty(t)</math> is the external pressure infinitely far from the bubble
:<math>\rho_L </math> is the [[density]] of the surrounding liquid, assumed to be constant
:<math>R(t)</math> is the radius of the bubble
:<math>\nu_L</math> is the [[kinematic viscosity]] of the surrounding liquid, assumed to be constant
:<math>S</math> is the [[surface tension]] of the bubble
 
Provided that <math>P_B(t) </math> is known and <math>P_\infty(t)</math> is given, the Rayleigh–Plesset equation can be used to solve for the time-varying bubble radius <math>R(t)</math>.
 
The Rayleigh–Plesset equation is derived from the [[Navier–Stokes equations]] under the assumption of [[spherical symmetry]].<ref name="Lin2002" />  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.<ref name="Brennen">{{cite book|last=Brennen|first=Christopher E.|title=Cavitation and Bubble Dynamics|year=1995|publisher=[[Oxford University Press]]|isbn=0-19-509409-3}}</ref>
 
== Derivation ==
 
The Rayleigh–Plesset equation can be derived entirely from [[first principle]]s using the bubble radius as the dynamic parameter.<ref name="Leighton" /> Consider a [[sphere|spherical]] bubble with time-dependent radius <math> R(t) </math>, where <math> t </math> is time. Assume that the bubble contains a homogeneously distributed vapor/gas with a uniform temperate <math> T_B(t) </math> and pressure <math> P_B(t) </math>. Outside the bubble is an infinite domain of liquid with constant density <math> \rho_L </math> and [[dynamic viscosity]] <math> \mu_L </math>. Let the temperature and pressure far from the bubble be <math> T_\infty </math> and <math> P_\infty(t) </math>.  The temperature <math> T_\infty </math> is assumed to be constant.  At a radial distance <math> r </math> from the center of the bubble, the varying liquid properties are pressure <math> P(r,t) </math>, temperature <math> T(r,t) </math>, and radially outward velocity <math> u(r,t) </math>. Note that these liquid properties are only defined outside the bubble, for <math> r \ge R(t) </math>.
 
=== Mass conservation ===
 
By [[conservation of mass]], the [[inverse-square law]] requires that the radially outward velocity <math> u(r,t) </math> must be inversely proportional to the square of the distance from the origin (the center of the bubble).<ref name="Brennen" />  Therefore, letting <math> F(t) </math> be some function of time,
 
:<math> u(r,t) = \frac{F(t)}{r^2} </math>
 
In the case of zero mass transport across the bubble surface, the velocity at the interface must be
 
:<math> u(R,t) = \frac{dR}{dt} = \frac{F(t)}{R^2} </math>
 
which gives that
 
:<math> F(t) = R^2dR/dt </math>
 
In the case where mass transport occurs, the rate of mass increase inside the bubble is given by
 
:<math> \frac{dm_V}{dt} = \rho_V\frac{dV}{dt} = \rho_V\frac{d(4\pi R^3/3)}{dt} = 4\pi\rho_VR^2\frac{dR}{dt} </math>
 
with <math> V </math> being the volume of the bubble. If <math> u_L </math> is the velocity of the liquid relative to the bubble at <math> r = R </math>, then the mass entering the bubble is given by
 
:<math> \frac{dm_L}{dt} = \rho_LAu_L = \rho_L(4\pi R^2)u_L </math>
 
with <math> A </math> being the surface area of the bubble. Now by conservation of mass <math> dm_v/dt = dm_L/dt </math>, therefore <math> u_L = (\rho_V/\rho_L)dR/dt </math>. Hence
 
:<math> u(R,t) = \frac{dR}{dt} - u_L = \frac{dR}{dt} - \frac{\rho_V}{\rho_L}\frac{dR}{dt} = \left(1-\frac{\rho_V}{\rho_L}\right)\frac{dR}{dt}</math>
 
Therefore
 
:<math> F(t) = \left(1-\frac{\rho_V}{\rho_L}\right)R^2\frac{dR}{dt} </math>
 
In many cases, the liquid density is much greater than the vapor density, <math> \rho_L \gg \rho_V </math>, so that <math> F(t) </math> can be approximated by the original zero mass transfer form <math> F(t) = R^2dR/dt </math>, so that<ref name="Brennen" />
 
:<math> u(r,t) = \frac{F(t)}{r^2} = \frac{R^2}{r^2}\frac{dR}{dt} </math>
 
=== Momentum conservation ===
 
Assuming that the liquid is a [[Newtonian fluid]], the [[Navier–Stokes equation]] in [[spherical coordinate]]s for motion in the radial direction gives
 
:<math> \rho_L\left(\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial r}\right) = -\frac{\partial P}{\partial r} + \mu_L \left[ \frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial u}{\partial r}\right) - \frac{2u}{r^2}\right]</math>
 
Substituting [[kinematic viscosity]] <math> \nu_L = \mu_L/\rho_L </math> and rearranging gives
 
:<math> -\frac{1}{\rho_L}\frac{\partial P}{\partial r} = \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial r} - \nu_L \left[ \frac{1}{r^2}\frac{\partial}{\partial r}\left( r^2\frac{\partial u}{\partial r}\right) - \frac{2u}{r^2}\right] </math>
 
whereby substituting <math> u(r,t) </math> from mass conservation yields
 
:<math> -\frac{1}{\rho_L}\frac{\partial P}{\partial r} = \frac{2R}{r^2}\left(\frac{dR}{dt}\right)^2 + \frac{R^2}{r^2}\frac{d^2R}{dt^2} - \frac{2R^4}{r^5}\left(\frac{dR}{dt}\right)^2 = \frac{1}{r^2}\left(2R\left(\frac{dR}{dt}\right)^2 + R^2\frac{d^2R}{dt^2}\right) - \frac{2R^4}{r^5}\left(\frac{dR}{dt}\right)^2</math>
 
Note that the viscous terms cancel during substitution.<ref name="Brennen" /> [[Separation of variables|Separating variables]] and integrating from the bubble boundary <math> r = R</math> to <math> r \rightarrow \infty </math> gives
 
:<math> -\frac{1}{\rho_L}\int_{P(R)}^{P_\infty} dP = \int_R^\infty \left[ \frac{1}{r^2}\left(2R\left(\frac{dR}{dt}\right)^2 + R^2\frac{d^2R}{dt^2}\right) - \frac{2R^4}{r^5}\left(\frac{dR}{dt}\right)^2 \right] dr</math>
 
:<math>{ \frac{P(R) - P_\infty}{\rho_L} = \left[- \frac{1}{r}\left(2R\left(\frac{dR}{dt}\right)^2 + R^2\frac{d^2R}{dt^2}\right) + \frac{R^4}{2r^4}\left(\frac{dR}{dt}\right)^2 \right]_R^\infty = R\frac{d^2R}{dt^2} + \frac{3}{2}\left(\frac{dR}{dt}\right)^2 }</math>
 
=== Boundary conditions ===
 
Let <math> \sigma_{rr} </math> 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,
 
:<math> \sigma_{rr} = -P +2\mu_L\frac{\partial u}{\partial r} </math>
 
Therefore at some small portion of the bubble surface, the net force per unit area acting on the lamina is
 
:<math>\begin{align}
\sigma_{rr}(R) + P_B - \frac{2S}{R} & = -P(R) + \left.2\mu_L\frac{\partial u}{\partial r}\right|_{r=R} + P_B - \frac{2S}{R} \\
& = -P(R) + 2\mu_L\frac{\partial}{\partial r}\left( \frac{R^2}{r^2}\frac{dR}{dt} \right)_{r=R} + P_B - \frac{2S}{R} \\
& = -P(R) - \frac{4\mu_L}{R}\frac{dR}{dt} + P_B - \frac{2S}{R} \\
\end{align}</math>
 
where <math> S </math> is the [[surface tension]].<ref name="Brennen" />  If there is no mass transfer across the boundary, then this force per unit area must be zero, therefore
 
<math> P(R) = P_B - \frac{4\mu_L}{R}\frac{dR}{dt} - \frac{2S}{R} </math>
 
and so the result from momentum conservation becomes
 
:<math>\frac{P(R) - P_\infty}{\rho_L} = \frac{P_B - P_\infty}{\rho_L} - \frac{4\mu_L}{\rho_LR}\frac{dR}{dt} - \frac{2S}{\rho_LR} = R\frac{d^2R}{dt^2} + \frac{3}{2}\left(\frac{dR}{dt}\right)^2</math>
 
wherby rearranging and letting <math> \nu_L = \mu_L/\rho_L </math> gives the Rayleigh–Plesset equation<ref name="Brennen" />
 
:<math> \frac{P_B(t) - P_\infty(t)}{\rho_L} = R\frac{d^2R}{dt^2} + \frac{3}{2}\left(\frac{dR}{dt}\right)^2 + \frac{4\nu_L}{R}\frac{dR}{dt} + \frac{2S}{\rho_LR} </math>
 
Using [[Newton's notation|dot notation]] to represent derivatives with respect to time, the Rayleigh–Plesset equation can be more succinctly written as
 
::<math> \frac{P_B(t) - P_\infty(t)}{\rho_L} = R\ddot{R} + \frac{3}{2}(\dot{R})^2 + \frac{4\nu_L\dot{R}}{R} + \frac{2S}{\rho_LR} </math>
 
== Solutions ==
[[File:Rayleigh-Plesset numerical solution for bubble in sine-wave pressure 01.jpg|thumb|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.]]
 
[[File:Rayleigh-Plesset numerical solution for bubble in step-like pressure.jpg|thumb|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.]]
 
No [[closed-form expression|analytical closed-form solution]] is known for the Rayleigh–Plesset equation. However, [[numerical analysis|numerical solutions]] to any accuracy can be easily obtained. In the special case, where surface tension and viscosity are neglected, high-order analytical approximations are known.<ref name="Obreschkow">{{cite journal|last=Obreschkow|first=D.|coauthors=Bruderer M., Farhat, M.|date=5 June 2012|title=Analytical approximations for the collapse of an empty spherical bubble|journal=Physical Review E|url=http://arxiv.org/abs/1205.4202}}</ref>
 
In the static case, the Rayleigh-Plesset equation simplifies, yielding to the [[Young-Laplace equation]]:
:<math>P_B-P_\infty = \frac{2S}{R}</math>
 
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 [[Liquid bubble#Pulsation|bubble oscillation]].
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Rayleigh-Plesset equation}}
[[Category:Equations of fluid dynamics]]
[[Category:Ordinary differential equations]]

Latest revision as of 18:51, 28 October 2014


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