# Boundary conditions for the Heat Equation

Leonardo Machado
Hello guys.

I am studying the heat equation in polar coordinates

$$u_t=k(u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{\theta\theta})$$

via separation of variables.
$$u(r,\theta,t)=T(t)R(r)\Theta(\theta)$$

which gives the ODEs

$$T''+k \lambda^2 T=0$$
$$r^2R''+rR+(\lambda^2 r^2-\mu^2)R=0$$
$$\Theta''+\mu^2\Theta=0$$

but i cant properly think about the boundary conditions to this problem. I see every where people resolving it with

$$|u(0,\theta,t)|<\inf \mapsto |R(0)| < \inf$$

and

$$u(r*,\theta,t)=0 \mapsto R(r*)=0$$

being r* the border of the disc.

But i understand the radial condition as a termal bath at zero temperature and i really want to change it for a finite value but i dont know how to procede without the zeros...

Any suggestions?

## Answers and Replies

Staff Emeritus
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The boundary conditions depend on what physical situation you wish to describe. Can you be more specific about this?

Mentor
The boundary condition at r = 0 should be zero radial temperature gradient.

Science Advisor
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Time equation should be
$$T' + k\lambda^2T = 0$$

Staff Emeritus
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The boundary condition at r = 0 should be zero radial temperature gradient.
This is correct only in the case of rotational symmetry of the problem. The general solution will contain contributions from ##J_1##, which has non-zero derivative at ##r = 0##.

Homework Helper
You should have the inner boundary condition:
$$\frac{\partial u}{\partial r}\Bigg|_{r=0}=0$$
This is the proper symmetry condition. The outer boundary condition is physics dependent however and can be absolutely anything.

Leonardo Machado
I have solved it guys!

To operate with inhomogeneous bondary conditions I've used

$$u(r,\theta,t)=v(r,\theta,t)+u_E(r,\theta)$$

being u_E the steady state and "v" the solution of the heat equation.