$($Problem $2-$ use MATLAB$)$

In a cylindrical tube made of milk chocolate $($inside radius

$R=2cm,$ total length $L=50cm),$ a warm milk is flowing

slowly $(v=3\frac{cm}{s})$ in laminar fashion.

The milk entering the tube is chocolate$-$free $(0\frac{g}{c{m}^3}).$

Chocolate dissolving in the milk is carried downstream,

also spreading radially inward by diffusion. The diffusion

coefficient of the chocolate in the milk is $D=0\mathrm{.}10\frac{c{m}^2}{(s)}.$

You realize that the system may be described with the

following PDE $(z$ is the length down the tube, $r$ is the

radius, $c$ is the concentration of chocolate in the milk$)$:

$v(1-\frac{r^2}{R^2})\frac{\text{d}\text{e}\text{l}\text{c}}{\text{d}\text{e}\text{l}\text{z}}=\frac{D}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{\text{d}\text{e}\text{l}\text{c}}{\text{d}\text{e}\text{l}\text{r}})$

You also identify the following boundary conditions:

$($BC #$1)$ At the center of the pipe, the concentration of

chocolate is continuous and is at its minimum. Thus $("$left

side" BC$)$:

$\frac{\text{d}\text{e}\text{l}\text{c}}{\text{d}\text{e}\text{l}\text{r}}=0$

$($BC#$2)$ At the pipe wall, which is solid, the chocolate isalways at a saturation concentration in the milk. Thus

$("$right side" BC$)$:

$c(r=R)=0\mathrm{.}275\frac{g}{c{m}^3}$

Use an m$-$file function to invoke pdepe in MATLAB: solve

for the concentration distribution within the pipe over

the entire radius and length of the pipe.

$($a$)$ Plot this concentration distribution against $z$ and $r$ in

a properly labeled surface plot $($a plot title reporting the

mesh size is acceptable$).$

$($b$)$ What are the following chocolate concentration

values, to four significant digits?

Midpoint of the tube length, at $r=1cm$

End of the tube length, at center of tube

$($c$)$ Display the entire program code used for your

function in MATLAB, and include your name as a

comment