Example $6\mathrm{.}7$ Steady$-$state radial diffusion through a blood vessel

Balloon angioplasty is a procedure that is used to open blocked coronary arteries.

A balloon catheter is inserted into the femoral artery and guided to the constrict$-$

ed artery, where it is inflated. In some cases, the vessel closes again due to the for$-$

mation of a clot or the excessive growth of smooch muscle cells, a condition

called restenosis, A drug that treats restenosis after balloon angioplasty is infused

into the circulation long enough for the concentration of the drug in the blood to

be constant at $C_{b.}.$ Measurements in coronary arteries have found that the drug

concentration in the vestel is $C_p$ in the outer part of the artery $($known as the adu.

enfitia$).$ If the inner radius of the artery is $R_b$ and the outer radius of the vessel is

$R_g$ determine the concentration profile and flux of the drug into the artery at $R_b$

Neglect chemical teactions.

Solution At steady state, the derivative with respect to time is zero. If chemical re$-$

actions do not occur, the second term on the right$-$hand side of Equation $(6\mathrm{.}7\mathrm{.}32)$ is

sero and the equation reduces to

$\frac{1}{r}\frac{d}{dr}(r\frac{d{C}_i}{dr})=0.$

To find $C_{in}$ we integrate Equation $(6,7,33)$ twice to yield

$C_4=$Alnr$+B.$

Applying the boundary conditions that $C_5=C_6$ arr $=R_4$ and $C_5-C_6$ at $r=R_4$

Ieads to the following solution for the constants of inegration:

$A=\frac{C_6-C_6}{ln(\frac{R_6}{R_6})},B=C_6-\frac{ln{R}_6(C_6-C_6)}{ln(\frac{R_6}{R_6})}.$

The concentration is then

$C=C_b-\frac{ln(\frac{R_b}{r})(C_b-C_b)}{ln(\frac{R_4}{R_b})}$

and the flux into the artery is

$N_i=-D_0\frac{d{C}_i}{dr}=-\frac{D_1(C_i-C_0)}{rln(\frac{R_1}{R_0})}.$

I. For the problem described above

Write down the weak form of the BVPs$.$ of $2$D Laplace Eq in Cartesian

Write down the matrix forms of the finite element equations.

Let $R_b=2820m,R_o=4220m,C_b=240m\frac{g}{d}L,C_o=540m\frac{g}{d}L$

Use three sets of mesh provided to calculate the solution and compare the solution with

the following analytical solution. Plot each point in your numerical solutions as dots $($e$.$g$.$

rvs$.$ C$).$

Hints: Although the node lists at the inner and outer surfaces are not provided, the geometry is

circular so that these nodes can be easily identified by an $\text{'}$if command, e$.$g$.$ if $(($sqrt $(x^{**}x+y^{**}y)-$

$Rb)$