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Laplace Eq. Cylinder and 3D Heat Equation 
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#1
Oct2008, 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 3D 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 nonhomogeneous 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 nonintuitive to me. 


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