# How do I find the impact pressure of cavitation due to implosion of a super critical fluid bubble?

I have solved the Rayleigh-Plesset equation for a time varying bubble of a super critical fluid. I am trying to find if the impact pressure caused by SCF bubbles is lesser as compared to that caused by water bubbles. Here is the equation I arrived at: $$x x'' + \frac{3}{2}(x')^2 + A\frac{x'}{x} = W$$

Here $A$ and $W$ are constants that depend on the viscosity, temperature and specific heat of the fluid continuum. I have used Weierstrass' elliptic function to find an analytical solution for this second order, nonlinear equation. I have used the Chapman-Enskog theory for finding the viscosity of the SCF and its distribution.This is the equation I arrived at: $$\boxed{ R \ddot{R} + \frac{3}{2} (\dot{R})^2 + \frac{4\mu_0}{\rho_L}\left[1 + b_0 \rho_L \left(\frac{1}{y} + 0.8 + 0.7614\right)\right]\frac{\dot{R}}{R} = W }$$ where $$W = \frac{\exp(\gamma)\left(\frac{p_2}{4} - 1\right) + \frac{p_2}{4} + \frac{p_c}{2}}{\rho_L} \,.$$ My question is how do I go about finding the impact pressure of such a system? And how do I model their distribution by a mathematical distribution?

Impact pressure in general would be $$\frac{P_{t}}{P}=\left(1+M^{2}\frac{\gamma-1}{2}\right)^{\frac{\gamma}{\gamma-1}}$$, for isentropic flow. $P_{t}$ is the total pressure and $P$ is the static pressure; the impact pressure is $$P_{I}=P_{t}-P$$. M is the Mach number and in this case it would be <0.8 as the bursting of a bubble in a fluid would be subsonic.$\gamma$ is the ratio of the specific heats.

• I've latex-ed the images. please check whether the equations are correct. – Biswajit Banerjee Oct 20 '16 at 1:22
• @BiswajitBanerjee Yeah they are correct, Thanks a lot! – Spoilt Milk Oct 20 '16 at 14:19

The quantity that you are interested in is related to the momentum transfer from the bubble to an object.

You have a solution for the rate of change of the radius as a function of time. Convert that into a velocity and compute the momentum of the bubble mass as a function of time. Use Newton's equation (momentum equation) to compute the resulting force. Convert that force into a pressure by dividing by the appropriate surface area.

That will give you an estimate of the impact pressure.

Note that the equations you have solved assume that the bubble is isolated and in an infinite fluid. Therefore, your estimates will only be qualitative.

• I have solved it for the radius and have used Weierstrass elliptic functions for the same, I did not convert that into the momentum as you have mentioned as that would take – Spoilt Milk Oct 20 '16 at 14:21
• The equation into a a order higher and make it a nonlinear third order non homogeneous ODE. I later found a relationship between the density and the variation of surface tension, and hence used this to find the impact pressure. Just wanted to know if this is a good idea. Thanks! – Spoilt Milk Oct 20 '16 at 14:24
• What is your definition of impact pressure? – Biswajit Banerjee Oct 20 '16 at 18:51
• Also, I don't see why the equations become a nonlinear third order ODE. The expression $f = d/dt(m v)$ should be easy to evaluate given closed form expressions for $m$ and $v$. – Biswajit Banerjee Oct 20 '16 at 18:56
• Sorry, but I don't think I understand the problem. The "hydrodynamic impact pressure" that I know of depends on the properties of a solid and a fluid. Your version just depends on the fluid and can be computed if you know $\gamma$. Also, there is no collapse model described in your question. I suggest you provide details of the physical process instead of just equations that may or may not be relevant. – Biswajit Banerjee Oct 21 '16 at 5:06