1.II.15C

State the unsteady form of Bernoulli's theorem.

A spherical bubble having radius $R_{0}$ at time $t=0$ is located with its centre at the origin in unbounded fluid. The fluid is inviscid, has constant density $\rho$ and is everywhere at rest at $t=0$. The pressure at large distances from the bubble has the constant value $p_{\infty}$, and the pressure inside the bubble has the constant value $p_{\infty}-\triangle p$. In consequence the bubble starts to collapse so that its radius at time $t$ is $R(t)$. Find the velocity everywhere in the fluid in terms of $R(t)$ at time $t$ and, assuming that surface tension is negligible, show that $R$ satisfies the equation

$R \ddot{R}+\frac{3}{2} \dot{R}^{2}=-\frac{\triangle p}{\rho}$

Find the total kinetic energy of the fluid in terms of $R(t)$ at time $t$. Hence or otherwise obtain a first integral of the above equation.

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