Heat equation in a sphere surface

In summary, the initial temperature distribution can be written as an infinite sum of spherical functions, and the heat equation for one of those spherical functions can be solved easily because they have the property l(l-1)Y_{l,m}(\theta,\phi)=l(l+1)Y_{l,m}(\theta,\phi) with the spherical Laplace operator. The solution to this particular initial condition is (the constants set equal to 1) T(\theta,\phi,t)=Y_{l,m}(\theta,\phi)e^{-l(l+1)t}. With that and the condition that no heat leaves or "enters" the thin
  • #1
Clausius2
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I was wondering what happens if I want to solve the heat equation in a sphere surface, neglecting its thickness. I have one initial condition for T(t=0), in particular this initial profile can depend on azimuth and zenith angles, it is not uniform. Perhaps I have saying something stupid but I think for large times the temperature would be uniform in all over the surface. The question is I have not any boundary condition, except those of angular periodicity. Or do you think the solution is precisely the initial profile?.
 
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  • #2
I've figured out a way of solving it, I'm not sure if it works. Anyway here I go:

The initial temperature distribution can be written as an infinite sum of spherical functions,
[tex]T(\theta, \phi, t=0) = \sum_{l=0}^\infty \sum_{m=-l}^{+l}a_{l,m} Y_{l,m}(\theta, \phi)[/tex]
The heat equation for one of those spherical functions can be solved easily because they have the property [tex]\Delta_{S^2} Y_{l,m}(\theta,\phi)=l(l+1)Y_{l,m}(\theta,\phi)[/tex] with the spherical Laplace operator [tex]\Delta_{S^2}[/tex] (it might be l(l-1) instead of l(l+1), I'm not sure anymore).
The solution to this particular initial condition is (the constants set equal to 1) [tex]T(\theta,\phi,t)=Y_{l,m}(\theta,\phi)e^{-l(l+1)t}[/tex].

Since the heat equation is linear one can solve it seperately for each part of the infinite sum, and therefore the solution with general initial conditions is
[tex]T(\theta, \phi, t) = \sum_{l=0}^\infty e^{-l(l+1)t} \sum_{m=-l}^{+l}a_{l,m} Y_{l,m}(\theta, \phi)[/tex].

So the problem is practically solved if you have found the series representation of the initial condition.
Perhaps I have saying something stupid but I think for large times the temperature would be uniform in all over the surface.
Yes, that's true.
 
Last edited:
  • #3
:eek: Bufff!

I was hoping such an answer given by a mathmatician. I just was reffering to the physical problem of the non-existence of apparent boundary conditions. Your answer is very technical, although is always welcomed of course. :biggrin:
 
  • #4
The only necessary boundary condition is indeed the initial temperature distribution. With that and the condition that no heat leaves or "enters" the thin surface the whole development in time is determined.

I'm not sure if the angular periodicity can be called a boundary condition...its more something that comes up when you use spherical coordinates. I guess when the initial condition is periodic then the solution to the heat equation is automatically periodic for all times. I'm sure one can derive this somehow directly from the equation, though I don't know how.
 

1. What is the heat equation in a sphere surface?

The heat equation in a sphere surface is a mathematical representation of the flow of heat in a spherical object. It takes into account factors such as temperature, heat conductivity, and the rate of change of temperature over time.

2. How is the heat equation derived for a sphere?

The heat equation for a sphere is derived from the general heat equation, which is a partial differential equation that describes how the temperature of a physical system changes over time. To derive the heat equation for a sphere, the general heat equation is applied to a spherical coordinate system.

3. What are the boundary conditions for the heat equation in a sphere surface?

The boundary conditions for the heat equation in a sphere surface usually include the temperature at the surface of the sphere and the rate of heat transfer at the center of the sphere. These boundary conditions can also be modified to include other factors such as external heat sources or changes in the material properties of the sphere.

4. How is the heat equation in a sphere surface used in real-world applications?

The heat equation in a sphere surface is used in various fields such as engineering, physics, and geology to model and analyze heat transfer in spherical objects. It is particularly useful in designing and optimizing systems that involve heat transfer, such as thermal insulation, heat exchangers, and geothermal energy systems.

5. Are there any limitations to the heat equation in a sphere surface?

While the heat equation in a sphere surface is a powerful tool for studying heat transfer in spherical objects, it does have its limitations. It assumes a uniform material and neglects factors such as convection and radiation, which can significantly affect heat transfer in real-world scenarios. Therefore, it should be used in conjunction with other equations and models to obtain a more accurate representation of the system.

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