Is there a PDE for this phenomenon? 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.
 A: I might be wrong but this seems to be somewhat related to the phenomena of sinks and sources, in that case the stream functions $\psi$ obey the law $d \psi = v_r ds$ where $v_r$ is radial velocity, and $ds$ is arclength measure.
A: I believe that the behavior of this fluid can still be described by the Navier-Stokes equations, with sources and sinks included. What I know for sure is that this problem, being a free boundary problem, is certainly an active area of study. Predicting the shape of the boundary of the "puddle" that is formed is non-trivial. 
A: Maybe I am being too literal and 3D here, but it seems that to   make sense of your  problem you need  gravity, otherwise  nothing holds the fluid to the surface.  So introduce   gravity. Is your constant rate' greater than the escape velocity at the surface? Ifyes' , then
sorry, the answer is no:  all the fluid shoots  out into space.  If the ratio
of (flow rate)^2/ (gravitation force on surface) is small enough, if  the fluid
is not too viscous, and if  the surface is `nice' (eg, compact, bounding a compact region) then certainly the answer is yes.  To see that viscosity plays a role, imagine
an incredibly visous honey shooting out of your hole in the earth. You will build a taller and taller volcano with your flow of honey. As the viscosity tends to infinity,
it seems you may have to wait forever for your goo to cover the earth. 
How to turn this into a math problem?  It is a free boundary Navier Stokes
equation with gravitational force on the right hand side.  Not the simplest thing.
It seems that there might be some kind of `thin film' limit that might
be geometrically more pleasing, and more what you had in mind. A good question to pass to a professional fluids guy. 
