Semi-infinite slab, surface heating on a radius r=a; T=?

Tags:
1. Sep 6, 2016

leialee

1. The problem statement, all variables and given/known data

We are heating a semi-infinite slab with a laser (radius of a stream is $a$), which presents us with a steady surface heating (at $z=0$), everywhere else on the surface the slab is isolated.
How does the temperature change with time?
Look at the limit cases: at $t \to \infty$, at $z=0$, in the direction of the laser beam, etc.

2. Relevant equations

So initialy the problem is 3D and we have cylindrical symmetry, and everything is the same for any given $\phi$, so problem becomes 2D and the only coordinates relevant are $r$ and $z$.

We have full diffusion equation
$$D\nabla^2 T=\frac{\partial T}{\partial t}-\frac{q}{\rho c_p}$$

where $q$ is not a function of time, but it is Heaviside function of radius and delta function of $z$; something like this: $q(r,z)=H(a-r)\delta (z-0)$ ??

Boundrary conditions are $T(r\to \infty, z\to \infty, t)=0$ and $\partial T/ \partial z (z=0, a<r<\infty)=0$
Initial condition is $T(r,z,t=0)=0$

I'm thinking to solve this with Green's functions; so that i take the solution for heating infinite space with a point source, but considering slab is semi-infinite I would multiply Green's function for infinite space by 2 (as in we have two sources each on one side of the surface so that there is no conducting over the surface??).

So I've got Greens function for infinite 3D object:
$$G(\textbf{r}-\textbf{r}_0;t)=(4\pi Dt)^{-\frac{3}{2}} e^{-\frac{(\textbf{r}-\textbf{r}_0)^2}{4Dt}}$$
and its solution
$$T(\textbf{r},t)=\int_{-\infty}^{t+}dt_0 \int d^3r_0~G(\textbf{r}-\textbf{r}_0;t-t_0)\frac{q(\textbf{r}_0,t_0)}{\rho c_p}$$

I'm really not very sure how to take on this problem, so any discussion and comment would be most welcome.

3. The attempt at a solution

2. Sep 8, 2016

Staff: Mentor

I think that this problem is solved in Carslaw and Jaeger, Conduction of Heat in Solids.

Can you guess what the long-time steady state solution looks like?

3. Sep 8, 2016

leialee

Yeah, i checked that book out and its a bit to advanced for my knowleadge.

4. Sep 8, 2016

Staff: Mentor

And yet you're using Green's functions.

5. Sep 8, 2016

leialee

Okay, so i went to look into that book again; but i still dont understand. So maybe you would be so kind to explain to me some stuff from the chapter 8.2..
Like in 8.2, why does he take equation (1) for the temperature; where does that come from? Later on in (2) i would guess this goes by the fact that medium isn't finite, so all $\lambda$ are allowed.
Also, why does (4) satisfie (6), i dont see how.

6. Sep 8, 2016

leialee

Ah I see, its the diffusion equation in cylindrical coordinates and for stacionary case.

7. Sep 8, 2016

leialee

Could you please explain to me at least the thought process, how Jaeger get the solution in 10.5 (4) ?

8. Sep 8, 2016

Staff: Mentor

I don't have a copy of Carslaw and Jaeger. I just remember that the solution to this problem is in there.