A volume $V$ of very viscous fluid of density $\rho$ and dynamic viscosity $\mu$ is released at the origin on a rigid horizontal boundary at time $t=0$. Using lubrication theory, determine the velocity profile in the gravity current once it has spread sufficiently that the axisymmetric thickness $h(r, t)$ of the current is much less than the radius $R(t)$ of the front.

Derive the differential equation

$$\frac{\partial h}{\partial t}=\frac{\beta}{r} \frac{\partial}{\partial r}\left(r h^{3} \frac{\partial h}{\partial r}\right)$$

where $\beta$ is to be determined.

Write down the other equations that are needed to determine the appropriate similarity solution for this problem.

Determine the similarity solution and calculate $R(t)$.