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Deriving a heat equation

  • #1
Consider heat flow in a long circular cylinder where the temperature depends only on t and on the distance r to the axis of the cylinder. Here [tex]r=\sqrt{x^{2}+y^{2}}[/tex] is the cylindrical coordinate. From the three dimensional heat equation derive the equation [tex]u_{t}=k(u_{rr}+\frac{u_{r}}{r})[/tex].

My book describes how the general heat equation is derived, but I'm having trouble incorporating the equation for r and in general understanding these concepts.

The book starts out with:

[tex]H(t)=\int\int\int c\rho u dxdydz[/tex] which I'm somewhat confident I can change to

[tex]H(t)=\int\int\int c\rho u dxdydr[/tex] because of the equation with r

Then the book has
[tex]\frac{dH}{dt}=\int\int\int c\rho u_{t} dxdydr[/tex] which makes sense to me

then after this I am a bit confused. I don't really know what to do next in terms of answering the question.
 

Answers and Replies

  • #2
ideasrule
Homework Helper
2,266
0
This can be done very easily, by rewriting the heat equation in spherical coordinates and taking advantage of the symmetry of the situation. A little bit of re-arranging gives you the equation.
 

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