# 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.

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Here's a simpler question: What describes the spread of a constant amount of fluid across a surface? –  Ricky Demer Jan 29 '12 at 9:53

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.

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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.

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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.

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i was not thinking of gravity but some simple assumption that restricts the fluid to the surface.On Riemann surfaces,incompressible flows derive from meromorphic functions.If the fluid spreads symmetrically at first then the flow lines are radial geodesics until they cross. On the sphere starting at the north pole flow follows great circles. A sink would be introduced to at the South pole. On a torus flow lines would cross and the fluid would collide with itself and spread somehow, a ripple of some kind might send it towards dry areas. –  marc Feb 1 '12 at 19:19