These equations were made using five major assumptions, Keller and Miksis obtained the following formula: : \dot{p} = \frac{3}{r} \left ( (\gamma - 1) K \frac {\partial T}{\partial r}\Bigg|_R - \gamma p \dot{R} \right ) where T is the temperature, K is the thermal conductivity of the gas, and r is the radial distance.
Flynn's formulation This formulation allows the study of the motions and the effects of heat conduction, shear viscosity, compressibility, and surface tension on small cavitation bubbles in liquids that are set into motion by an acoustic pressure field. The effect of
vapor pressure on the cavitation bubble can also be determined using the interfacial temperature. The formulation is specifically designed to describe the motion of a bubble that expands to a maximum radius and then violently collapses or contracts. This set of equations was solved using an improved
Euler method. :\left ( 1 - \frac{\dot{R}}{c} \right ) R \ddot{R} + \frac{3}{2} \dot{R^2} \left ( 1 - \frac{\dot{R}}{3c} \right ) = \left ( 1 + \frac{\dot{R}}{c} \right ) \frac{1}{\rho_l} \left [ p_B(R,t) - p_A(t) - P_\infty \right ] + \frac{R}{\rho_l c} \left ( 1 - \frac{\dot{R}}{c} \right ) \frac{dp_B(R,t)}{dt} where R is the radius of the bubble, the dots indicate first and second time derivatives, \rho_l is the density of the liquid, c is the speed of sound through the liquid, dp_B(R,t) is the pressure on the liquid side of the bubble's interface, t is time, and p_A(t) is the driving pressure.
Rayleigh–Plesset equation The theory of bubble dynamics was started in 1917 by
Lord Rayleigh during his work with the Royal Navy to investigate cavitation damage on ship propellers. Over several decades his work was refined and developed by
Milton Plesset,
Andrea Prosperetti, and others. The
Rayleigh–Plesset equation is: :R \ddot{R} + \frac{3}{2} \dot{R^2} = \frac{1}{\rho_l} \left ( p_g - P_0 - P \left ( t \right ) - 4\mu \frac{\dot{R}}{R} - \frac{2\gamma}{R} \right ) where R is the bubble radius, \ddot{R} is the second order derivative of the bubble radius with respect to time, \dot{R} is the first order derivative of the bubble radius with respect to time, \rho_l is the density of the liquid, p_g is the pressure in the gas (which is assumed to be uniform), P_0 is the background static pressure, P(t) is the sinusoidal driving pressure, \mu is the
viscosity of the liquid, and \gamma is the
surface tension of the gas-liquid interface. == Bubble surface ==