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Laplace Eq. Cylinder and 3D Heat Equation

by splitringtail
Tags: cylinder, equation, heat, laplace
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splitringtail
#1
Oct20-08, 12:21 AM
P: 59
Sorry, it does not seem that Latex is not compiling my code right so I will try my best to be clear.

1. The problem statement, all variables and given/known data

The curved surface of a cylinder of radius a is grounded while the end caps at z = L/2 are
maintained at opposite potentials ψ (r,θ, L/2)= V(r,θ).

Suppose that a simple solid brick with uniform initial temperature
is immersed at time t = 0 in a heat bath. The temperature ψ (r,t) within the material
satisfies

∂ψ = ∇^2 ψ
k∂t

with ψ(r,0)=T1 and ψ(at surface,t)=T2

I really just want to clear up some interpretations.

2. Relevant equations

Laplaican for Cylindrical and Rectangular Coordinates.

3. The attempt at a solution

I want to assume that grounded means the potential at the curved surface is zero, but I cannot find it in my notes or book or even online. After that it's just putting it in the equation, I have to solve both exterior and interior cases, nothing new.

The second question is really confusing me in choosing the appropriate separation constants. Now we solved the 3-D Laplace equation in rectangular coordinates. The separation constants I choose were

a^2 b^2 c^2 = 0 this implies that a^2 = b^2 + c^2

this gives you a sin/cos solutions in two directions and a sinh/cosh in one solution one direction. I choose the sinh/cosh solution for the direction w/ the non-homogeneous boundary, which you repeat six times to get a complete solution. Now for the question given, I want to choose

a^2 b^2 c^2 d= 0 this implies that

d= a^2 b^2 c^2

so, all my solutions are in the sin/cos form except T??

We have T' + k (d) T = 0, since a,b,c are > 0 (other cases are trivial solutions), then d < 0 and k > 0, so I can have an exp(), solution.

I was really wanting a way to work in a sinh/cosh solution, so that I could use what I got from the previous homework, but I guess it is just a little modification. I guess that makes since b/c it is some form of a wave equation. It seems non-intuitive to me.
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