For me, I think that would be any variation of the Rayleigh-Plesset equation, which describes the volume oscillations of a bubble in a liquid under the effect of a pressure field.
Below, see an example of the Keller-Miksis equation (similar to Rayleigh-Plesset, but accounting for mild liquid compressibility) simulating the radial oscillation of an inertially collpasing bubble.
clear,clc
T0 = 298 ; % Ambient temperature, K
freq = 26500; % Driving frequency, Hz
r0 = 4.5 ; % Initial bubble radius, microns
p0 = 1 ; % Ambient pressure, atm
pA = 1.2 ; % Acoustic pressure, atm
rho0 = 997 ; % liquid density, kg m-3
sigmaL = 0.072 ; % surface tension, N m-1
muL = 1e-3 ; % liquid viscosity, kg m-1 s-1
c = 1485 ; % speed of sound, m s-1
gamma = 1.4 ; % air specific heats ratio, -
tspan = [0 1/freq] ;
IC = [r0*1E-6,0,p0*101325+2*sigmaL/(r0*1E-6),T0] ;
odefun = @(t,w)keller_miksis(t,w,r0*1E-6,T0,rho0,freq,p0*101325,pA*101325,sigmaL,muL,c,gamma) ;
[time,SOL] = ode15s(odefun,tspan,IC) ;
t = time*freq ;
R = SOL(:,1)*1E+6 ;
S = SOL(:,2) ;
p = SOL(:,3)/101325 ;
T = SOL(:,4) ;
set(0,'defaulttextinterpreter','latex')
set(0,'defaultAxesTickLabelInterpreter','latex')
set(0,'defaultLegendInterpreter','latex')
figure(1)
subplot(2,2,1)
plot(t,R)
title('Bubble radius')
xlabel('$\omega t$')
ylabel('$R$, $\mu m$')
subplot(2,2,2)
semilogy(t,p)
title('Gas pressure')
xlabel('$\omega t$')
ylabel('$p$, atm')
subplot(2,2,3)
plot(t,T(:,1))
title('Gas temperature')
xlabel('$\omega t$')
ylabel('$T$, K')
subplot(2,2,4)
plot(t,abs(S/c))
title('Interface Mach number')
xlabel('$\omega t$')
ylabel('$Ma$, -')
function f = keller_miksis(t,w,r0,T0,rho0,freq,p0,pA,sigmaL,muL,c,gamma)
R = w(1) ;
dR = w(2) ;
p = w(3) ;
T = w(4) ;
dpdt = -3*gamma*p*dR/R ;
dTdt = 3*(1-gamma)*T*dR/R ;
pinf = p0-pA*sin(2*pi*freq*t) ;
pL = (p0+2*sigmaL/r0)*(r0/R)^(3*gamma)-2*sigmaL/R-4*muL*dR/R ;
ddR = ((1+dR/c)*(pL-pinf)/(rho0)-1.5*(1-dR/(3*c))*(dR^2)+...
(R/(rho0*c))*(dpdt+(2*sigmaL*dR+4*muL*dR^2)/(R^2)))/((1-dR/c)*R+4*muL/(rho0*c)) ;
f = [ dR ;...
ddR ;...
dpdt ;...
dTdt ];
end
