3D Boundary Value Heat Transfer with constant flux and nonzero boundary

[bnay]

Hello,

I'm having trouble getting started on this problem. Here's the question:

[PLAIN]http://img810.imageshack.us/img810/2464/ee323assn3q3.jpg

My issue is in setting up the governing partial differential equation in 3 dimensions. What I've tried so far is setting du/dt equal to the Laplacian of u(x,y,z,t) + g(x,y). I'm not entirely sure if I can do this, though. It leads me to assume u(x,y,z,t) = X(x)Y(y)Z(z)T(t), but I'm not sure how to cancel out the inhomogeneous boundary conditions.

I know that if g were a constant I could let v = u - g, but since it isn't its derivative with respect to x and y won't go away and v will stay inhomogeneous.

Sorry if that sounds like word vomit - its late and I've been up reading textbooks that only seem to have one dimensional examples. If anybody could shed some light on how to tackle this problem I would be very thankful - I'm mostly looking for a starting point and hopefully I can take it from there.

 

[timthereaper]

If you partition u(x,y,z,t) into the sum of a "steady-state" and a "variable" portion, you might be able to make the steady state portion satisfy the nonhomogeneous boundary conditions and the variable portion satisfy the homogeneous ones. You can use a modified form for 3 dimensions of this:

http://en.wikiversity.org/wiki/Topic:Heat_equation/Solution_to_the_2-D_Heat_Equation

See if that helps.

 

[bnay]

If you partition u(x,y,z,t) into the sum of a "steady-state" and a "variable" portion, you might be able to make the steady state portion satisfy the nonhomogeneous boundary conditions and the variable portion satisfy the homogeneous ones. You can use a modified form for 3 dimensions of this:

http://en.wikiversity.org/wiki/Topic:Heat_equation/Solution_to_the_2-D_Heat_Equation

See if that helps.

Thank you for your help.

Setting the steady state solution as X(x)Y(y)Z(z) has worked quite well for X and Z, but for Y it seems that I don't have enough information to solve it, since I can't eliminate X(x) and Y(y) due to the inhomogenous conditions.

I know X(x)Y(y)Z(0) = g(x,y), which implies that g(x,y)/XY is a constant, but since I don't know what g(x,y) is I can't assume X or Y are constants.

I think it's safe to say that Y(y) can't be a constant since X(x)Y'(0)Z(z) = -q(x,z)/k and Y'(b) = 0

This led me to try an exponential solution for Y, but I seem to have too many variables to eliminate.

Is there any way I can simplify this to get rid of the X and Z dependence?