Question

8.8. Consider blood flow in a vessel (i.e., a pipe with a porous wall that is permeable to blood). The radius and length of the vessel are R and L, respectively. In general, the flow is axisymmetric, the fluid velocity has both radial and axial components that are usually determined numerically. However, there are two approximate solutions to this problem. One is to use lubrication theory to determine the relationship between the flow rate and the pressure gradient in the vessel is R<<L. The other is to estimate the axial velocity component by solving a standard problem of unidirectional flow in a pipe with an impermeable wall, but with a different kind of no-slip condition at the wall. The conventional no-slip condition is replaced by the equation: k^1/2 (du/dr)= -a*u

where k is the specific hydraulic permeability of the wall, u is the axial velocity of the fluid, r is the radial coordinate, and a is a dimensionless quantity that depends on the microstructure of the porous wall. The value of a usually varies between 0.1 and 10 depending on the size of the pores in the pipe wall.

(a) Determine the axial velocity profile in the vessel using the second approach.

(b) Find the flow rate through the vessel

(c) Estimate the slip effect a on the pressure drop of the flow rate through the pipe

Answer 1

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