State the unsteady form of Bernoulli's theorem.
A spherical bubble having radius at time is located with its centre at the origin in unbounded fluid. The fluid is inviscid, has constant density and is everywhere at rest at . The pressure at large distances from the bubble has the constant value , and the pressure inside the bubble has the constant value . In consequence the bubble starts to collapse so that its radius at time is . Find the velocity everywhere in the fluid in terms of at time and, assuming that surface tension is negligible, show that satisfies the equation
Find the total kinetic energy of the fluid in terms of at time . Hence or otherwise obtain a first integral of the above equation.