In a liquid, cavitation is the spontaneous appearance of bubbles, occurring when the pressure in a part of the liquid drops below its vapour pressure, e.g. due to the fast movements. When such a bubble is subjected to a higher pressure by the surrounding liquid, it quickly implodes.
To model this case, we use the simple form of the Rayleigh-Plesset equation for an inviscid liquid without surface tension. Note that the RP equation assumes incompressibility.
We assume that the whole liquid is at a constant pressure , and the bubble is empty, such that the interface pressure , meaning . At first, the radius is stationary , and given by a constant . The simple Rayleigh-Plesset equation is then:
To solve it, we multiply both sides by and rewrite it in the following way:
It is then straightforward to integrate both sides with respect to time , from to :
Rearranging this equation yields the following expression for the derivative :
This equation is nasty to integrate. The trick is to invert into , and, because we are only interested in collapse, we just need to consider the case . The time of a given radius is then as follows, where we are using slightly sloppy notation:
The minus comes from the constraint that , but . We insert the expression for :
This integral needs to be looked up, and involves the hypergeometric function . However, we only care about collapse, which is when . The time at which this occurs is:
With our assumptions, a bubble will always collapse. However, unsurprisingly, reality turns out to be more complicated: as , the interface velocity . By looking at the derivation of the Rayleigh-Plesset equation, it can be shown that the pressure just outside the bubble diverges due to . This drastically changes the liquid’s properties, and breaks our assumptions.
- B. Lautrup, Physics of continuous matter: exotic and everyday phenomena in the macroscopic world, 2nd edition, CRC Press.