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

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SUMMARY

The discussion centers on the governing physical laws for the flow of an incompressible fluid spreading across a closed surface. The Navier-Stokes equations, in conjunction with the continuity equation and potentially the energy equation, provide a comprehensive framework for modeling this phenomenon. The inquiry also addresses whether the fluid will eventually cover the entire surface and if an equilibrium distribution of fluid flow can be achieved when sinks are introduced to maintain a constant volume. These insights establish a clear understanding of fluid dynamics in closed systems.

PREREQUISITES
  • Understanding of Navier-Stokes equations
  • Familiarity with the continuity equation
  • Basic principles of fluid dynamics
  • Knowledge of energy equations in fluid systems
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  • Research the application of Navier-Stokes equations in closed systems
  • Study the continuity equation in the context of incompressible fluids
  • Explore equilibrium states in fluid dynamics
  • Investigate the effects of sinks on fluid distribution on surfaces
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Researchers, physicists, and engineers working in fluid dynamics, particularly those interested in modeling fluid behavior on closed surfaces and understanding equilibrium distributions in incompressible fluid systems.

lavinia
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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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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.
 

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