Question $3.$

$($a$)$ Please describe two common techniques to visualize fluid flows in biological

systems, and compare their limitations$.$

$($b$)$ What are the assumptions of Darcy's law? Based on these assumptions, where in

the human body do you expect Darcy's law not to be valid?

$($c$)$ Solid tumors can be considered as a multiphase structure consisting of different

compartments with different properties. Imagine a spherical tumor with a core$-$

shell structures, where the core and shell parts, with a core radius of $R$ and shell

thickness of L$,$ have hydraulic conductivities of $K_1$ and $K_2,$ respectively.

Therapeutic agents are directly injected at a constant infusion flow rate of $Q$ into

the center of the tumor through a syringe needle with a radius of $r.$

$($i$)$ Please determine the pressure profile in the tumor at steady state.

$($ii$)$ State all assumptions you have made in i$.$

$($iii$)$ For some tumor, the hydraulic permeability can be in the range of ${10}^{-19}{m}^2.$

Explain when Darcy's law and Brinkman equation should be used for this

problem.