I am doing research with a professor and he asked me to graph some solutions to a 1D diffusion equations of the following form:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

C_e \frac{\partial T_e}{\partial t}=\frac{\partial}{\partial z}\left( \kappa \frac{\partial T_e}{\partial z} \right) - g(T_e-T_i)+\alpha (1-R)If(t)\exp (-\alpha z)

[/tex]

[tex]

C_i \frac{\partial T_i}{\partial t}=-g(T_e-T_i)

[/tex]

Let me describe to you what this equation is used for. My professor wishes to find the heat distribution of a piece of metal after a laser pulse hits it.

This equation deals with one dimension, meaning it only takes into account the depth of the metal slab, not the height or width.

Bad Visual Ahead:

The first object (on the left) is the laser. The "^" is a laser pulse. The line of o-o-o's is a latice. Note the 1 dimensioned nature.Code (Text):|== ^ o-o-o-o-o-o-o-o-o-o-o-o

The first differential equation deals with the temperature of the electrons and the second is the temperature of the lattice (the equations are coupled). This is called the "two-temperature model" because it accounts for the time delay between the transfer of energy between the electrons, which initially absorb the energy of the laser, and the lattice, which sends a vibration throughout the crystal.

Here's an explanation of the constants (i.e. mathematically, you can imagine them all to be, say, 0.5):

The electron heat diffusion coefficient is given by [tex]\kappa[/tex]. The heat capacities of the electron and lattice are given by [tex]C_e[/tex] and [tex]C_i[/tex], respectively. The coupling strength is given by [tex]g[/tex]. The damping coefficient and reflectivity of the metal in question is given by [tex]\alpha[/tex] and [tex]R[/tex], respectively. The maximum intensity of the pulse is given by [tex]I[/tex]

Now, I'll explain the last term in the first equation, [tex]\alpha (1-R)If(t)\exp (-\alpha z)[/tex].

The function f(t) is the intensity of the pulse at a given instant. Here's a visual: Imagine a line of cars at a traffic light. Instead of the light turning green in an instant, the green light's intensity slowly increases, reaches a peak, and then dims again (like a travelling Gaussian distro. wave). This is how the laser pulse is modelled - therefore, f(t) is given by the laser used, but it will be some sort of Gaussian with respect to time.

The exponential function with respect to depth describes the intensity decay due to atoms being "blocked" by the atoms in front of them. Again, using the (poor) line of cars analogy, imagine that your visibility of the green light is partially blocked because of the cars in front of you and said blocking is dependant on how deep you are in the line of cars.

Now for the numerical computing part . I'm no expert at this at all so I've been trying to research this a bit. I've solved the diffusion equation through Fourier series, but this is different in that the boundary conditions are not what I'm use to.

Assumptions that could make no sense:

T(0) and T(infinity) = room temperature (assuming one pulse)

T'(0) and T'(infinity) = 0 (assuming one pulse)

These assumption are for both electron and lattice temperatures.

(If you're still reading THANK YOU!)

Ok - here's how I started:

To solve for T_e and T_i numerically, the method of finite difference was used (I know, it's not the best, but I'm learning). For a general diffusion equation, T(z,t),

[tex]\frac{\partial T}{\partial t}=\frac{\partial^2 T}{\partial z^2}[/tex],

the numerical solution, given in terms of finite differentials, delta t and delta z, and at a given point (i,j) is:

[tex]\frac{T(i,j+\delta t) - T(i,j)}{\delta t} = \frac{T(i-\delta z,j)-2T(i,j)+T(i+\delta z,j)}{(\delta z)^2}[/tex].

Therefore, the function T can be found by interating over i and j given some initial boundary conditions:

[tex]T(i,j+\delta t) = T(i,j) + \delta t \frac{T(i-\delta z,j)-2T(i,j)+T(i+\delta z,j)}{(\delta z)^2}[/tex].

By substituting into the original equations:

[tex]T_e(i,j+\delta t)=T_e(i,j) + \frac{\delta t}{C_e} \biggl[\kappa \frac{T(i-\delta z,j)-2T(i,j)+T(i+\delta z,j)}{(\delta z)^2} - g\bigl(T_e(i,j)-T_i(i,j)\bigl)+\alpha (1-R)If(j)\exp (-\alpha i)\biggl][/tex].

And for the second one:

[tex]T_i(i,j+\delta t)=T_i(i,j)-\frac{\delta t}{C_i}g\bigl(T_e(i,j)-T_i(i,j)\bigl)[/tex]

Now, before people start asking for consultation fees, I wanted to know if this much made sense. If so, goodie. This is sort of my num-comp 'theory.' If this is wrong, there's no sense in displaying my also-would-be-wrong MatLab code.

Thank you VERY MUCH in advance!

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# 1D Diffusion Equations (Numerical Computing) [kind of long]

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