- #1
nettle404
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Homework Statement
Consider heat flow in a long circular cylinder where the temperature depends only on [itex]t[/itex] and on the distance [itex]r[/itex] to the axis of the cylinder. Here [itex]r=\sqrt{x^2+y^2}[/itex] is the cylindrical coordinate. From the three-dimensional heat equation derive the equation [itex]U_t=k(U_{rr}+2U_r/r)[/itex].
Homework Equations
The standard heat equation is
[tex]c\rho\frac{\partial}{\partial t}U(x,y,z,t)=\kappa\nabla^2U(x,y,z,t)[/tex]
The Attempt at a Solution
Attempted to work backwards from [itex]U_t=k(U_{rr}+U_r/r)[/itex] with the chain rule, but that did not produce anything of value. I can also probably solve the problem by deriving the heat equation starting in cylindrical coordinates, but the question asks to specifically "transplant" cylindrical coordinates onto the Cartesian coordinate heat equation by some tricky variable substitution algebra I can't imagine performing. That is: Where should I start?