Steady Heat Eqn between two cylinders

[member 428835]

Homework Statement

A cylinder is well-insulated at the base and sides (radially). No heat sources and assume steady state. All other BCs are free (I'm aware the problem is underconstrained). Then we have

$$\nabla^2 f = 0\\
\partial_rf(r=R_0) = \partial_rf(r=R_1) = 0\\
\partial_z f(z=0) = 0\\
f(\theta = 0) = f(\theta = 2\pi)\\
\partial_\theta f(\theta = 0) = \partial_\theta f(\theta = 2\pi)$$

where the first line of BCs imply well-insulated radially, the second implies well insulated at the base, and the third implies continuous heat distribution.

Homework Equations

Separation of variables, which I guess isn't an equation.

The Attempt at a Solution

I'll make the ansatz $f_1 = A R(r)\cos(\omega \theta)\cos(k z)$ and $f_2 = BR(r)\sin(\omega \theta)\cos(k z)$ where $\omega = n / 2\pi:n\in\mathbb N$. $k$ it seems is free. Plugging $f_1,f_2$ into $\nabla^2 f = 0$ gives two linear dependent ODE's in $R(r)$. Solutions to both ODEs (Mathematica) are Bessell functions with the imaginary number $i$ in the argument.

I am unable to enforce the radial flux condition. Any ideas?

 

[mighty2000]

I would first point out that the problem is indeed underconstrained, as stated in the original post. This means that there are potentially infinite solutions that could satisfy the given boundary conditions. Therefore, any proposed solution should be carefully examined to ensure that it is physically meaningful and realistic.

In terms of finding a solution, I agree with the approach of using separation of variables. However, in order to satisfy the radial flux condition, we need to consider the full Fourier series expansion for the radial component of the solution. This means that instead of just considering the cosine and sine terms in the ansatz, we should also include the constant term and the full set of cosine and sine terms with different frequencies (i.e. $\cos(n\theta)$ and $\sin(n\theta)$ where $n$ is a positive integer).

By doing this, we can ensure that the radial flux condition is satisfied at both boundaries ($r=R_0$ and $r=R_1$). However, we will also need to consider the other boundary conditions ($f(\theta=0) = f(\theta=2\pi)$ and $\partial_\theta f(\theta=0) = \partial_\theta f(\theta=2\pi)$) in order to determine the appropriate coefficients for each term in the Fourier series.

Ultimately, the solution will involve a combination of Bessel functions and trigonometric functions, and the specific form will depend on the values of the coefficients determined by the boundary conditions. It is important to carefully consider the physical implications of each term in the solution and ensure that it makes sense in the context of the problem.

In summary, the key to enforcing the radial flux condition is to consider the full Fourier series expansion for the radial component of the solution and use the boundary conditions to determine the appropriate coefficients for each term.