Solving this (WLOG $const$ = 0) leads us to $h = C\csc\theta$, which indeed corresponds to flat $y = C$ solutions.Īdditionally clearly the PDE also conserves area, as it can be written in flux conservative form.
annulus entirely filled with only one fluid) then, $h_t = 0$ when $(h_\theta \sin \theta Indeed, if we ignore trivial steady states of $h = R_1$ and $h = R_2$ which correspond to the interface being on an annulus boundary (ie. We can check if the physical intuition of flat interface final state truly is a steady state solution ($h_t = 0$) of our PDE. The physical steady state for this system is a flat interface (dense fluid runs downhill until it all collects in the bottom of the annulus with light fluid above). I am trying to solve it via a finite-difference scheme. It describes the movement of a fluid-fluid interface inside an annulus of inner radius $R_1$ and outer $R_2$ under the action of gravity, where one fluid is denser than another.
I am working with the following PDE, which is an advection-diffusion type equation.
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