Consider a square slab of side , consisting of two parallel layers in perfect thermal contact, initially at uniform temperature . The slab is immersed at time in a well-stirred, insulated tank containing a fluid at constant temperature .

The heat equation describing this system is:

with

at and ,

at and , and

,

where and are the space coordinates, (), is the thermal diffusivity (), is the thermal conductivity () and is the heat transfer coefficient between the slab and the surrounding fluid ().

To solve this problem, it is convenient to introduce the following dimensionless variables:

,

,

and

. Thus the equations become

,

at and ,

at and ,

with

and

,