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.