- #1

fluidistic

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- Homework Statement
- Solve the steady state heat equation in a rectangle whose bottom surface is kept at a fixed temperature, left and right sides are insulated and top side too, except for a point in a corner where heat is generated constantly through time.

- Relevant Equations
- ##\kappa \nabla ^2 T + g =0##

I have checked several textbooks about the heat equation in a rectangle and I have found none that deals with my exact problem. I have though to use separation of variables first (to no avail), then Green's function (to no avail), then simplifying the problem for example by defining a new function in terms of ##T(x,y)## such that it would satisfy a homogeneous problem instead, but to no avail. (is a problem even called homogeneous when ##dT/dx|_{x_0} = 0## rather than ##T(x=x_0)=0##? I guess not.)

Out of memory, when I went with separation of variables to tackle ##\kappa \left( \frac{\partial ^2 T}{\partial x^2}+ \frac{\partial ^2 T}{\partial y^2}\right) = 0##, I obtained solutions of the form ##X(x)Y(y)## with ##X(x)=A\cosh(\alpha x)+B\sinh(\alpha x)## and ##Y(y)=C\cos(\alpha y)+D\sin(\alpha y)## where ##\alpha## is the separation constant. The boundary conditions are of the type Dirichlet for the bottom surface: ##T(x,y=0)=T_0##. And Neumann elsewhere: ##\frac{\partial T}{\partial x}|_{x=0, y=0}## for ##y\in [0,b)##, ##\frac{\partial T}{\partial x}|_{x=a, y=0}## for ##y\in [0,b]## and ##\frac{\partial T}{\partial y}|_{x, y=b}## for ##x\in (0,a]##. The power generated translates as the Neumann boundary condition ##\nabla T \cdot \hat n## and so ##\frac{\partial T}{\partial x}|_{x=0, y=b}+ \frac{\partial T}{\partial y}|_{x=0, y=b}=p## where ##p## is the power density of the heat source.

I have been stuck there, I could not get to apply and know the constants ##A##, ##B##, ##C## and ##D##, nor ##\alpha##. All of these constants are in fact depending on ##n##, natural numbers, because the separable solutions are eigenfunctions, etc.

Any pointer would be appreciated. Thank you!

Out of memory, when I went with separation of variables to tackle ##\kappa \left( \frac{\partial ^2 T}{\partial x^2}+ \frac{\partial ^2 T}{\partial y^2}\right) = 0##, I obtained solutions of the form ##X(x)Y(y)## with ##X(x)=A\cosh(\alpha x)+B\sinh(\alpha x)## and ##Y(y)=C\cos(\alpha y)+D\sin(\alpha y)## where ##\alpha## is the separation constant. The boundary conditions are of the type Dirichlet for the bottom surface: ##T(x,y=0)=T_0##. And Neumann elsewhere: ##\frac{\partial T}{\partial x}|_{x=0, y=0}## for ##y\in [0,b)##, ##\frac{\partial T}{\partial x}|_{x=a, y=0}## for ##y\in [0,b]## and ##\frac{\partial T}{\partial y}|_{x, y=b}## for ##x\in (0,a]##. The power generated translates as the Neumann boundary condition ##\nabla T \cdot \hat n## and so ##\frac{\partial T}{\partial x}|_{x=0, y=b}+ \frac{\partial T}{\partial y}|_{x=0, y=b}=p## where ##p## is the power density of the heat source.

I have been stuck there, I could not get to apply and know the constants ##A##, ##B##, ##C## and ##D##, nor ##\alpha##. All of these constants are in fact depending on ##n##, natural numbers, because the separable solutions are eigenfunctions, etc.

Any pointer would be appreciated. Thank you!