- #1
leialee
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Homework Statement
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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.
Homework 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.