$3..$ When a pipe flow suddenly expands from $A_1$ to $A_2,$ low$-$speed low$-$friction eddies appear in the corner and the flow gradually expands to $A_2$ downstream. $($a$)$ Find the downstream pressure $p_2$ as function of upstream pressure, velocity and the area ratio of the contraction. Neglect wall friction. $($b$)$ Repeat the problem for the case when the corner annular ring has an average pressure $\frac{p_1+p_2}{2}.$ What is the percentage

difference in the change $p_1-p_2?($c$)$ If the contraction is very gradual, what is the percentage change in $p_1-p_2$ compared to previous cases?