8. Heat Equation with Circular Symmetry. Assume that the temperature is circularly symmetric:
u u(r,t), where r^2 x^2 | y^2. Consider any circular annulus a ≤ r ≤ b.
a) Show that the total heat energy is r π f^b_a cpurdr.
b) Show that the flow of heat energy per unit time out of the annulus at r b is: (see attachment for equation).
A similar results holds at r = a.
c) Assuming the thermal properties are spatially homogenous, use parts (a) and (b) to derive the circularly symmetric heat equation without sources: (see attachment for equation)
d) Find the equilibrium temperature distribution inside the circular annulus a ≤ r < b if the outer radius is insulated and the inner radius is at temperature T.
Right now I do not have access to a scanner to work this problem through but essentially what you are doing in the proof is looking at Gausses divergence theorem and its equivalent form to a surface in the first part of the proof. I found some lecture notes that go through this step by step plus a derivation of the heat equation for polar coordinates. I will have access to a scanner tomorrow and can work the problem by ...
Heat Equation with Circular Symmetry, Total Heat Energy, Flow of Heat Energy and Equilibrium Temperature Distribution are investigated.