Exact solution to advection equation in spherical coordinates

In summary, the conversation discusses the process of finding the solution to the advection equation in spherical coordinates, with a constant velocity u. The speaker first tries to expand the equation and apply separation of variables, resulting in a final solution of \phi = Ae^{\frac{\lambda}{u}(r-ut)-2}. They question whether this solution is correct and mention that for Cartesian coordinates, the solution is much simpler (\phi=\phi_0(x-ut)). The other speaker suggests back-substituting the solution into the equation to check its validity.
  • #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.
 
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  • #2
lostidentity said:
Wonder if this is correct?

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.
 
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Related to Exact solution to advection equation in spherical coordinates

1. What is the advection equation in spherical coordinates?

The advection equation in spherical coordinates is a partial differential equation that describes the transport of a quantity (such as temperature or concentration) in a spherical coordinate system. It takes into account the effects of both convection and diffusion on the quantity being transported.

2. What is an exact solution to the advection equation in spherical coordinates?

An exact solution to the advection equation in spherical coordinates is a mathematical expression that accurately describes the behavior of the transported quantity over time and space. It takes into account the initial conditions, boundary conditions, and physical parameters of the system.

3. How is the exact solution to the advection equation in spherical coordinates derived?

The exact solution to the advection equation in spherical coordinates is typically derived using separation of variables, where the solution is expressed as a product of functions that each depend on only one variable (time, radial distance, polar angle, or azimuthal angle). These functions are then solved for using the boundary and initial conditions.

4. What are some common applications of the advection equation in spherical coordinates?

The advection equation in spherical coordinates has many applications in various fields of science and engineering, including atmospheric and oceanic modeling, chemical and biological transport, and astrophysics. It is also commonly used in the study of fluid dynamics and heat transfer in cylindrical or spherical systems.

5. Can the exact solution to the advection equation in spherical coordinates be used in real-world scenarios?

Yes, the exact solution to the advection equation in spherical coordinates can be used in real-world scenarios to accurately predict the behavior of a transported quantity. However, it may require simplifying assumptions and idealized conditions to be applicable to a specific system, as well as accurate input data and numerical methods for solving the equation.

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