Question $4$

a$)$ Please give two example blood vessels where flow separation may be important.

b$)$ Consider a drug that is used in the brain for the treatment of cancer or degenerative nerve

diseases. The drug is metabolized by a first$-$order reaction with a rate coefficient $k_e$

i$)$ Write down the mass balance for the flow of the drug through a cylindrical blood

vessel with a tadius $R$ in the absence of any reactions. State all assumptions

made.

ii$)$ Rewrite the mass balance in $4$ bi for a reacting drug.

c$)$ State appropriate conditions for which the Brigg Haldane enzyme kinetics should be

adopted, but not the Michaelis$-$Menten approach.

d$)$ The analytical expression of the flow rate $Q$ under oscillatory flow through a cylindrical

pipe is as follows: $Q=\frac{R^2{}^{**}}{i}(1-\frac{2J_1(t^{\frac{3}{2}})}{i^{\frac{3}{2}}{j}_0(t^{\frac{3}{2}})})e^{it}$

Where $R$ is the radius of the channel, $$ is the frequency of the oscillation, $A^{**}$ is the

amplitude of the pressure gradient, $$ is the density of the body fluid and $$ is the

Womersley number.

i$)$ Outline, without actually solving, how the phase lag between the pressure

gradient and the flow rate can be calculated.

ii$)$ Explain whether the phase lag should depend on the flow parameters, such as the

amplitude and frequency of the oscillatory pressure gradient, radius of the

channel and density of the fluid.