- #1
lostidentity
- 18
- 0
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.
[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.