- #1
shultz
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Homework Statement
In a one dimensional problem, there is a liquid where x>0 and a wall in x=0. In time t=0 an ideal mass M is injected to the liquid at the point x=x0. The diffusion coefficient of the molecules injected is D.
a. Find the concentration of the material as a function of time and space C(x,t) with the boundry condition that the material cannot pass the wall.
b. Now, let's assume that the material particles get glued to the wall such that the concentration of the material in the liquid is zero on the wall. Find the flow of the molecules on the wall as a function of time and calculate the total mass of the molecules glued to the wall from time t=0 until t=t0.
Clue: In order to meet the right boundery conditions, put an imaginary point mass at x=-x0. The sign of this mass needs to be positive or negative according to the boundary condition.
Homework Equations
[tex]
\[
\frac{{\partial c\left( {x,t} \right)}}{{\partial t}} = D\frac{{\partial ^2 c\left( {x,t} \right)}}{{\partial ^2 x}}
\]
[/tex]
The Attempt at a Solution
First of all, I am not sure how to translate the given data to exact mathematical boundary conditions.
Are the boundary condition in part a are:
[tex]
\[
\left\{ \begin{array}{l}
c\left( {x,0} \right) = \delta \left( {x - x_0 } \right) \\
c_x \left( {0,t} \right) = 0 \\
\end{array} \right.
\]
[/tex]
If it is correct, how do I continue from here?
I know I can solve the equation for infinite region (from both sides) using Fourier transform but I am not sure how to do it on semi infinite region using the given clue.
Thanks a lot!
P.S. I am new to this forum and I am not sure where to put this question (math homework, physics homework, maybe math differential equations etc.) so please forgive me and transfer the question to its more appropriate place.