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Exact solution to advection equation in spherical coordinates

  1. Sep 23, 2010 #1
    I've been trying to find the exact solution to the advection equation in spherical coordinates given below

    [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.
     
  2. jcsd
  3. Sep 23, 2010 #2
    Hi. You need to remember always if needed and no complications, you can back-substitute your presumptive solution into the DE. If it satisfies the DE, then it is a correct solution, barring any initial, boundary or other requirements. Do you mean the general solution? If so than I think it would be some arbitrary function of a function such as C(g(x,y)) as normally encountered in first order PDEs.
     
    Last edited: Sep 23, 2010
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