What is the P.D.E. governing this phenomemon?

[lavinia]

At a point on a surface an incompressible fluid begins to up well at a constant rate and spread across the surface.

Is there a physical law - like the heat equation - that describes the flow?

Will the fluid eventually cover the whole surface?

Once the surface is covered allow sinks to appear to keep the volume of fluid on the surface constant. Will one then get an equilibrium distribution of fluid flow on the surface?

I have in mind closed surfaces with no boundary so that the fluid can't fall of any edges or leak through any holes.

 

[boneh3ad]

I am not visualizing your setup perfectly, but given that it is a fluids problem, the Navier-Stokes equations coupled with continuity and (depending on what you are looking for) perhaps the energy equation will be able to describe your system.