$($a$)$ Flow of body fluids across biological tissues is often represented mathematically

using theories from flow through porous media. One of the key characteristics of a

porous medium is the porosity. Discuss why different tissues with the same porosity

may exhibit different behaviors in transporting water and solutes across them.

$($b$)$ Assume that a biological tissue has two types of empty spherical pores within a solid

matrix; while both types of pores are spherical in shape, their diameters are different,

with the diameters of pore type $1$ and $2$ being $d_1$ and $d_2$ respectively. What are the

specific surface $s$ and the porosity $6$ of the tissue. Assume that the number of spherical

pores per unit volume is $n_4,$ and the ratio of the number of pores with $d_1$ to that with

$d_2$ is $R.$

$($c$)$ Darcy's law states that the velocity through a porous medium vec$(v)$ is equal to the

negative of the hydraulic conductivity $K,$ multiplied by the pressure gradient vec$(grad)P.$

What does $K$ depend on$?$

$($d$)$ Please discuss how Darcy's law needs to be modified, as the pore size of the

biological tissue increases such that the pore sizes are no longer much smaller than

the overall size of the biological tissues.