I was reading 'an introduction to fluid dynamics' by middleman.
they introduced an example, but I can't understand how they solve this PDE analytically.
h is dep on time and radius.
although, there is no initial condition and R is dep on time.
As I guess, the formula about V(volume) will be related to it.
This is not a separation of variables solution. It isa long-time similarity solution which is found by seeking a similarity form for the solution. Let h(t,r)=t^\alpha f(\xi), where \xi=r/t^\beta. Substituting these expressions into the PDE one then finds the relationship between \alpha and \beta that yields a ODE for the function f(\xi). which can be integrated. One can then use the volume constraint to pin down values for \alpha and \beta. Finally one finds an expression for the position of the contact line R(t) of the spreading drop.
I have attached some notes that show how this is done. Please check carefully. I do not get the same coefficient that Middleman gives. Likely I have a minor error. Everything else looks OK though.
Ok i found my error in my original solution which I posted. The new version now agrees with Middleman's solution!
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