Consider a saline bag with volume $V=3L$ hung above a patient with the height $H=0\mathrm{.}5m.$ The density and of the saline $=1\mathrm{.}05\frac{g}{c}{m}^3$ and the viscosity of the saline $=1\mathrm{.}2$mPa$*s,$ respectively. The plastic tube AB in circular shape with inner radius $R=0\mathrm{.}2mm$ and length $L=1m$ is used to deliver the saline to the patient, as show in the right figure. The gravity constant $g=9\mathrm{.}8\frac{N}{k}g.$

a$)$ Ignore the height of the saline bag itself, please compute the hydraulic pressure $P_A$ at the turning point A$.$

b$)$ We assume the exit point B keeps the same height as turning point A $.$ At point B the saline is injected into patient's vein, and reduced to the same pressure in vein. The typical vein blood pressure is $10$ mmHg $,$ and $1mmHg=133\mathrm{.}322Pa,$ please determine the exit point pressure $P_B.$

c$)$ The driving pressure of saline flow in plastic tube AB is just the difference between hydraulic pressure $P_A$ and exit point pressure $P_B$ computed in a$)$ and b$).$ For a Poiseuille steady flow, the flow rate $Q$ of the circular tube AB has already been derived by you in Problem $1$ as $Q=f(R,G,),$ where $G$ is the pressure gradient along flow direction in plastic tube AB $,$ and we assume $G$ can be estimated as $G=\frac{P_A-P_B}{L}.$ Please compute the flow rate $Q.$

$1$

d$)$ Based on the computed steady flow rate $Q$ and saline volume $V,$ please compute the drain$-$out time for the saline infusion to the patient.

e$)$ Based on the flow shear stress $=f(r,G,)$ you derive in Q$1-$b$),$ please compute the flow shear stress value at $r=R=0\mathrm{.}2mm$ in the circular plastic tube.