8.8. Consider blood flow in a vessel (i.e., a pipe with a porouswall that is permeable to blood). The radius and length of thevessel are R and L, respectively. In general, the flow isaxisymmetric, the fluid velocity has both radial and axialcomponents that are usually determined numerically. However, thereare two approximate solutions to this problem. One is to uselubrication theory to determine the relationship between the flowrate and the pressure gradient in the vessel is R<1/2 (du/dr)= -a*u
where k is the specific hydraulic permeability of the wall, u isthe axial velocity of the fluid, r is the radial coordinate, and ais a dimensionless quantity that depends on the microstructure ofthe porous wall. The value of a usually varies between 0.1 and 10depending on the size of the pores in the pipe wall.
(a) Determine the axial velocity profile in the vessel using thesecond approach.
(b) Find the flow rate through the vessel
(c) Estimate the slip effect a on the pressure drop of the flowrate through the pipe
