Rayleigh-Plesset equation: There are a few cases where an exact solution to theNavier-Stokes…

Rayleigh-Plesset equation: There are a few cases where an exact solution to theNavier-Stokes equation is possible when the geometry is spherical. Here we investigatespherically symmetric fluid motions, as occurs for example when a cavitation bubblegrows or collapses, or when a small bubble periodically oscillates radially owing toperiodic changes in the far-field pressure as occurs in sonoluminescence, the namegiven to the phenomenon when brief pulses of light are emitted regularly from thebubble.It is natural at first to work this problem in dimensional form. Assume that thebubble remains spherical but that its radius R(t) changes in time. Indeed, as weshall see, knowledge of the time-dependent radius is the primary goal of the analysis.(a) Consistent with the above assumptions, take the velocity field to be purelyradial, u = ur(r, t)er , then integrate the continuity equation in spherical coordinatesand use the boundary conditions ur(r = R, t) = dR/dt to deduceur(r, t) = R2r2dRdt .