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Concept
Rayleigh–Plesset equation
Résumé
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.
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.
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