In fluid dynamics, the Rayleigh-Plesset equation
describes how the radius of a spherical bubble evolves in time
inside an incompressible liquid.
Notably, it leads to cavitation.
Consider the main
for the velocity field :
We make the ansatz ,
where is the basis vector;
in other words, we demand that the only spatial variation of the flow is in .
The above equation then becomes:
Meanwhile, the incompressibility condition
in spherical coordinates yields:
This is only satisfied if is constant with respect to ,
leading us to a solution given by:
Where is an unknown function that does not depend on .
We then insert this result in the main Navier-Stokes equation,
and isolate it for , yielding:
Integrating this with respect to yields the following expression for ,
where is the (possibly time-dependent) pressure at :
From the definition of viscosity,
we know that the normal stress
in the liquid is given by:
We now consider a spherical bubble
with radius and interior pressure along its surface.
Since we know the liquid pressure ,
we can find from .
Furthermore, to include the effects of surface tension, we simply add
the Young-Laplace law to :
We isolate this for , and equate it to
our expression for
at the surface :
Isolating for ,
and inserting the fact that ,
such that ,
Rearranging this and defining
leads to the Rayleigh-Plesset equation:
- B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition,