| course |
course_year |
question_number |
tags |
title |
year |
Fluid Dynamics |
IB |
23 |
|
Paper 3, Section II, B |
2014 |
A bubble of gas occupies the spherical region $r \leqslant R(t)$, and an incompressible irrotational liquid of constant density $\rho$ occupies the outer region $r \geqslant R$, such that as $r \rightarrow \infty$ the liquid is at rest with constant pressure $p_{\infty}$. Briefly explain why it is appropriate to use a velocity potential $\phi(r, t)$ to describe the liquid velocity u.
By applying continuity of velocity across the gas-liquid interface, show that the liquid pressure (for $r \geqslant R$ ) satisfies
$$\frac{p}{\rho}+\frac{1}{2}\left(\frac{R^{2} \dot{R}}{r^{2}}\right)^{2}-\frac{1}{r} \frac{d}{d t}\left(R^{2} \dot{R}\right)=\frac{p_{\infty}}{\rho}, \quad \text { where } \dot{R}=\frac{d R}{d t} .$$
Show that the excess pressure $p_{s}-p_{\infty}$ at the bubble surface $r=R$ is
$$p_{s}-p_{\infty}=\frac{\rho}{2}\left(3 \dot{R}^{2}+2 R \ddot{R}\right), \quad \text { where } \ddot{R}=\frac{d^{2} R}{d t^{2}}$$
and hence that
$$p_{s}-p_{\infty}=\frac{\rho}{2 R^{2}} \frac{d}{d R}\left(R^{3} \dot{R}^{2}\right)$$
The pressure $p_{g}(t)$ inside the gas bubble satisfies the equation of state
$$p_{g} V^{4 / 3}=C$$
where $C$ is a constant, and $V(t)$ is the bubble volume. At time $t=0$ the bubble is at rest with radius $R=a$. If the bubble then expands and comes to rest at $R=2 a$, determine the required gas pressure $p_{0}$ at $t=0$ in terms of $p_{\infty}$.
[You may assume that there is contact between liquid and gas for all time, that all motion is spherically symmetric about the origin $r=0$, and that there is no body force. You may also assume Bernoulli's integral of the equation of motion to determine the liquid pressure
$$\frac{p}{\rho}+\frac{\partial \phi}{\partial t}+\frac{1}{2}|\nabla \phi|^{2}=A(t)$$
where $\phi(r, t)$ is the velocity potential.]