# 2D Steady State Heat Equation, Mixed Boundary Conditions - Separation of Variables

## Homework Statement

We have a 2D fin that has length L (x-axis), and thickness t, (y-axis). The left side has a fixed temperature, the right side is insulated, and the top and bottom surfaces are subject to convection. Find an analytical solution for the temperature at steady state.

## Homework Equations

Boundary Conditions:
$(0,y) \quad T=T_b$
$\displaystyle (L,y) \quad \frac{\partial{T}}{\partial{x}}=0$
$\displaystyle (x,t/2) \quad k\frac{\partial{T}}{\partial{y}}+h(T-T_{\infty})=0$
$\displaystyle (x,-t/2) \quad k\frac{\partial{T}}{\partial{y}}+h(T-T_{\infty})=0$

Separation of Variables:
$T(x,y)=X(x)Y(y)$

$X''-\lambda{^2}X=0$
$Y''+\lambda{^2}Y=0$

$X(x)=c_1 \sinh{(\lambda{x})} + c_2 \cosh{(\lambda{x})}$
$Y(y)=c_3 \sin{(\lambda{y})} + c_4 \cos{(\lambda{y})}$

## The Attempt at a Solution

Use symmetry to simplify the boundary conditions in the y-axis direction by cutting the fin along the x-axis at $y=0$ where the flux will be zero:
$\displaystyle (x,0) \quad \frac{\partial{T}}{\partial{y}}=0$

I have tried going through it using separation of variables about half a dozen times now but I can never solve for $\lambda$. Here is my current progress (in Mathematica):

http://img6.imagebanana.com/img/38bbdmje/Selection_006.png

Should I be setting my coordinate system up differently? Do I have to treat the convective (mixed) boundary conditions differently?

Thanks!

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