Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook