Hydrostatic Pressure on Solid/Liquid interface during solidification
 

Hi,
The problem I am trying to tackle is as follows:
" A 2-D horizontal rectangular melt pool of steel is getting solidified due to extraction of heat through its outer surfaces. The 2 smaller sides are fixed to a support so that they can't deform due to weight of the remaining melt pool. Thus only longer sides are free to deform/bulge due to hydrostatic pressure of the melt pool."

My objective is to get deformation of the longer sides due to hydrostatic pressure of melt pool.

The obvious solution is to apply hydrostatic pressure on the solid/liquid interface of longer sides of slab and get the deformation. But the difficulty lies in the fact that the solid/liquid interface is not stationary but moves inwards as solification proceeds.

I am stuck at this step. I don't know how to apply pressure on an interface which is moving with time.
Please help.
P.S.- I am also attaching a prototype of what I am trying to solve.