Laplace Eq. Cylinder and 3D Heat Equation

[splitringtail]

Sorry, it does not seem that Latex is not compiling my code right so I will try my best to be clear.

Homework Statement

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.

Homework Equations

Laplaican for Cylindrical and Rectangular Coordinates.

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.

 

[mighty2000]

Hello, I can try to help clarify some of the concepts and equations for you.

First, in regards to the potential at the curved surface of the cylinder being grounded, this means that the potential at that surface is set to zero. This is a common boundary condition in electrostatics problems.

For the second question, it seems like you are trying to solve the heat equation in cylindrical coordinates. In this case, the appropriate separation constants would depend on the boundary conditions given. If the boundary conditions are in the form of a sinusoidal function, then choosing the separation constants to be in the form of a sine or cosine function would be appropriate. If the boundary conditions are in the form of an exponential function, then choosing the separation constants to be in the form of a hyperbolic sine or cosine function would be appropriate.

It is important to carefully consider the boundary conditions and choose the appropriate separation constants to match those conditions. In some cases, it may be necessary to modify the separation constants to match the boundary conditions, as you have done in your solution attempt. This is a common approach in solving differential equations.

I hope this helps clarify some of the concepts and equations for you. If you have any further questions, please let me know. Good luck with your homework!