Problem $1$

In a medical setting we commonly need to infuse drugs and

at times blood in order to treat diseases and save

individual's lives. In this problem we will figure out the velocity profile of blood and a saline $($water$)$ solution in a tube of radius $R$ and

length $L$ at steady state. We will ignore entrance and exit

effects. The saline solution is a Newtonian fluid under constant density and viscosity. The blood can also be considered a constant density and viscosity fluid, however the shear stress relation for the

flow of blood in a cylindrical tube of radius $R$ is the following

$({}_{rz})={}_o^{\frac{1}{2}}(\frac{d{v}_z}{dr}{)}^{\frac{1}{2}}+{}_o^{\frac{1}{2}},$ when $|{}_{rz}|>{}_o,$

$\frac{d{v}_z}{dr}=0,$ when $|{}_{rz}|{}_o$

a$)$ Derive the expressions that describe the velocity profile of saline solution in a horizontal tube of radius $R$ and length $L$ when the pressure difference is $P=P_0-P_L$

b$)$ Derive the expressions that describe the velocity profile of blood in a horizontal tube of radius $R$ and length $L$ when the pressure difference is $P=P_0-P_L($hint: you need to solve the problem at different $r$ ranges $r_{c}r$