Consider a line source that extracts energy at time at a rate per unit length along the origin of a cylindrical region that is surrounded by water at temperature at time . The moving surface of separation between the solid and the liquid is at radius ; and are the temperatures of the ice and water, respectively. These equations describe the system for :

for ,

for , with boundary conditions at :

,

,

and initial conditions:

,

,

.

Here and are thermal diffusivites of ice and water, is the freezing point, and stand for the thermal conductivities of ice and water, is the latent heat and is the density of both water and ice (this assumption neglects the decrease of density in freezing). An exact solution to this problem is discussed by Carslaw and Jaeger [1]:

, ,

, ,

where ,

and is a root of

.

The solution is shown with the thermal constants for ice and water in cgs units. These problems are important in contexts such as water freezing around cylindrical pipes. Simple analytical solutions are still used today to validate more sophisticated numerical methods.