Question $2$

a$)$ Non$-$dimensionalize the Euler's equation so that all variables are dimensionless. Indicate

the non$-$dimensional parameter$($s$)$ that arise$($s$)$ as a result. Give the physical meaning of

the resultant parameter$($s$).$

b$)$ Consider a biological tissue with spherical pores of radius $R_p$ and a pore number density

of $n_v.$ The tissue is elastic and the pore size can be changed as the pressure is varied.

Calculate the percentage change in the porosity and specific surface of the tissue as the

pores increase in radius by $20\%.$

c$)$ Drugs have to diffuse across an isolated cylindrical vessel with a radius $R$ and a length $L$

through a constant infusion pressure $P_0.$ The vessel is completely clogged and hence can

be treated as a porous medium with a hydraulic conductivity $K.$ Assume no blood and

lymph circulation in the vessel, and the pressure at the outlet of the vessel is zero.

Determine

i$)$ the pressure and velocity profiles in the vessel at steady state and

ii$)$ the infusion rate.

Comment on whether no$-$slip boundary condition is obeyed in your answer. Please also

discuss how your answer depends on the mesh size of the pores in the problem.