1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Deriving a heat equation

  1. Mar 2, 2010 #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.
  2. jcsd
  3. Mar 2, 2010 #2


    User Avatar
    Homework Helper

    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook