- #1
lostidentity
- 16
- 0
I've been trying to find the exact solution to the advection equation in spherical coordinates given below
\frac{\partial{\phi}}{\partial{t}} + \frac{u}{r^2}\frac{\partial{}}{\partial{r}}r^2\phi = 0
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
\phi = R(r)T(t)
\frac{1}{T}\frac{dT}{dt} = -\frac{u}{R}\frac{dR}{dr} - \frac{2u}{r} = -\lambda^2
And the final answer is
\phi = Ae^{\frac{\lambda}{u}(r-ut)-2}
Wonder if this is correct? For Cartsesian coordinates I know the solution is very simple just \phi=\phi_0(x-ut) where \phi_0 is the initial profile for phi.
\frac{\partial{\phi}}{\partial{t}} + \frac{u}{r^2}\frac{\partial{}}{\partial{r}}r^2\phi = 0
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
\phi = R(r)T(t)
\frac{1}{T}\frac{dT}{dt} = -\frac{u}{R}\frac{dR}{dr} - \frac{2u}{r} = -\lambda^2
And the final answer is
\phi = Ae^{\frac{\lambda}{u}(r-ut)-2}
Wonder if this is correct? For Cartsesian coordinates I know the solution is very simple just \phi=\phi_0(x-ut) where \phi_0 is the initial profile for phi.