I've been trying to find the exact solution to the advection equation in spherical coordinates given below(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\frac{\partial{\phi}}{\partial{t}} + \frac{u}{r^2}\frac{\partial{}}{\partial{r}}r^2\phi = 0[/tex]

Where the velocity u is a constant. First I tried to expand the second term using product rule and then apply the separation of variables, which gives me the following.

Separation of variables

[tex]\phi = R(r)T(t)[/tex]

[tex] \frac{1}{T}\frac{dT}{dt} = -\frac{u}{R}\frac{dR}{dr} - \frac{2u}{r} = -\lambda^2[/tex]

And the final answer is

[tex] \phi = Ae^{\frac{\lambda}{u}(r-ut)-2}[/tex]

Wonder if this is correct? For Cartsesian coordinates I know the solution is very simple just [tex]\phi=\phi_0(x-ut)[/tex] where [tex]\phi_0[/tex] is the initial profile for phi.

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# Exact solution to advection equation in spherical coordinates

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