A$)$ We found the general solution for the startup flow in a circular tube to be

given by:

$\frac{?}{\text{b}\text{a}\text{r}}(w)\frac{?}{b}=ar(v{)}_z\frac{-}{b}ar(v{)}_{}={\displaystyle \underset{n=1}{\overset{}{}}}[A_n{e}^{\frac{-}{b}ar(t){}_n^2}{J}_0({}_n^{-}r)+B_n{e}^{\frac{-}{b}ar(t){}_n^2}{Y}_0({}_n^{-}r)]$

Do the details of the calculation to arrive at the result:

$\frac{?}{\text{b}\text{a}\text{r}}(w)={\displaystyle \underset{n=1}{\overset{}{}}}\frac{8}{s_n^3{J}_1(s_n)}{e}^{-s_n^{2}t}{J}_0(s_n^{-}r)$

B$)$ Write a program to plot the velocity distribution for the start$-$up flow in a

circular tube as in Fig $4$D$.2($Fig$.4\mathrm{.}1-3$ in first edition$)$ in BSL$.$ You do not

have to include your program but instead describe in a few sentences how it

works. Most important you need to describe how you know that the program

is giving you the correct answers. Show a plot of dimensionless velocity

versus dimensionless position for $\frac{t}{}{R}^2=0\mathrm{.}0001,0\mathrm{.}001,0\mathrm{.}01,0\mathrm{.}05,0\mathrm{.}1,0\mathrm{.}2,$

$0\mathrm{.}5,$ and $t.$