# Numerical Solution to the Rayleigh Plesset Equation

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• KDPhysics
In summary, the conversation discusses the process of numerically solving the Rayleigh Plesset equation using the odeint function in Python. The code is given and the parameters and initial conditions are defined. The conversation also mentions a mistake in the code and provides a corrected version. The results are plotted and compared to past literature.
KDPhysics
TL;DR Summary
I have been trying to find a numerical solution to the Rayleigh Plesset equation (for a sonoluminescing bubble) using a Python code. Read up here if you need to refresh on this (surprisingly) unknown field: https://en.m.wikipedia.org/wiki/Sonoluminescence
However, the python code seems to give a strange output. Please, gods of computational physics help me...
I have been trying to numerically solve the Rayleigh Plesset equation:

$R\ddot{R} + \frac{3}{2}(\dot{R})^2=\frac{p_g-p_0-p(t)}{\rho_l}-4\mu\frac{\dot{R}}{R}-\frac{2\gamma}{\rho_lR}$

using the odeint python function. The code is given below:
Python:
import numpy as np
from matplotlib import pyplot as plt
from scipy.integrate import odeint

# define equations
def equation(y0, t):
R, u = y0
return u, (P_g-P_0-1317000*np.sin(2*np.pi*26500*t)-2*sigma/R)/(R*rho)-4*miu*u/(R**2)-3*u**2/(2*R)

def plot_results(time, R):
plt.plot(time, R)

s = " for R0=" + str(R_0*1000) + "mm"
plt.title("Oscillations in Bubble Radius" + s)
plt.xlabel("Time")
plt.grid(True)
plt.show()

# parameters
time = np.arange(0, 0.0005, 0.000000025)
rho = 1000
sigma = 0.0725
miu = 8.9*10**(-4)
P_g = 2330
P_0 = 10000

# initial conditions
R_0 = 0.0001
u_0 = 0

R_1 = odeint(equation, [R_0, u_0], time)

R = R_1[:,0]

plot_results(time, R)

Wh
en i run the code, I get a very strange output.

It should look more like this:

[1]: https://i.stack.imgur.com/aTc3j.png
[2]: https://i.stack.imgur.com/fQ9J8.gif

EDIT:
Ok, I've been working on the code for a while, and I think I have found the mistake. I was using an approximate form of the equation. For anyone interested in the correct code, here it is (feel free to edit it if you want, I am bad at writing efficient code XD).

Python:
import numpy as np
from matplotlib import pyplot as plt
from scipy.integrate import odeint

# define equations
def equation(y0, t):
R, u = y0
return u, (P_g-P_0-1317000*np.cos(2*np.pi*26500*t)-2*sigma/R-4*miu*u/R+(2*sigma/R_0+P_0-P_g)*(R_0/R)**(3*k))/(R*rho)-3*u**2/(2*R)

#define plot
def plot_results(mtimes, R):
plt.plot(mtimes, R)

s = " for R0=" + str(R_0*10**6) + "$\mu$m"
plt.title("Oscillations in Bubble Radius" + s)
plt.xlabel("T/$\mu$s")
plt.ylabel("R/$\mu$m")
plt.grid(True)
plt.show()

# parameters
time = np.arange(0, 0.00002, 0.00000000025)
rho = 1000
sigma = 0.0725
miu = 8.9*10**(-4)
P_g = 2330
P_0 = 10000
k = 1.33

# initial conditions
R_0 = 0.0000026
u_0 = 0

R_1 = odeint(equation, [R_0, u_0], time)

R = R_1[:,0]*10**6
mtimes = time*10**6

#plot the results
plot_results(mtimes, R)

I rescaled the axes so that the marks wouldn't overlap, and the results should coincide with past literature.

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• 327819_1_En_7_Fig5_HTML.gif
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Last edited:
Thanks for your help!Great job on finding and fixing the mistake in your code! It's always important to double check and make sure that your equations are accurate, especially when working with numerical solutions. Keep up the good work!

## 1. What is the Rayleigh Plesset Equation?

The Rayleigh Plesset Equation is a mathematical model that describes the dynamics of a gas bubble in a liquid. It takes into account factors such as the gas pressure inside the bubble, the surface tension of the liquid, and the viscosity of the liquid.

## 2. Why is numerical solution necessary for the Rayleigh Plesset Equation?

The Rayleigh Plesset Equation is a non-linear differential equation that cannot be solved analytically. Therefore, numerical methods must be used to approximate the solution.

## 3. What are the most commonly used numerical methods for solving the Rayleigh Plesset Equation?

The most commonly used numerical methods for solving the Rayleigh Plesset Equation are the Runge-Kutta method and the finite difference method. These methods involve breaking the equation into smaller, simpler steps and using iterative calculations to approximate the solution.

## 4. What are the limitations of numerical solutions to the Rayleigh Plesset Equation?

Numerical solutions to the Rayleigh Plesset Equation are subject to error and may not always accurately represent the behavior of a gas bubble in a liquid. Additionally, these solutions may be computationally expensive and time-consuming.

## 5. How are numerical solutions to the Rayleigh Plesset Equation used in scientific research?

Numerical solutions to the Rayleigh Plesset Equation are used in various fields of science, such as biomedical engineering, oceanography, and acoustics. They can help researchers understand the behavior of gas bubbles in different environments and inform the design of technologies such as ultrasound imaging and drug delivery systems.

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