Q$1($a$)$ Biological tissues are typically characterized by the presence of complicated

matrices with pores. Therefore, flow through porous media needs to be understood.

The tissues often consist of pores with different shapes. For simplicity, in this

problem, we assume the pores to be a mixture of cylindrical and spherical ones. The

cylindrical pores have a radius $r$ and length $L,$ where $L$ is longer than the longest

dimension of the tissue of interest. The spherical pores have a radius of $R.$

$($i$)$ What are the specific surface and the porosity of the tissue, in terms of $r,L,R,$

$n$ and $x,$ where $n$ is the number of cylinders per unit cross$-$sectional area and $x$

is the ratio of the numbers of cylindrical pores to spherical pores? Please state

all key assumptions you have used.

$($ii$)$ What are the hydraulic conductivity $K$ and specific hydraulic permeability $k$ of

the tissue? Hint: For a single cylinder, volumetric flow rate $q=\frac{-r^4}{8}\frac{dP}{dx},$ where

$\frac{dP}{dx}$ is the pressure gradient.

$($iii$)$ Propose a situation when Brinkman equation should be used to describe flow

through this particular biological tissue?

$($b$)$ The activity of an enzyme $E$ follows the following reaction with the substrate $S$ to

form the product $P$:$E+S{?}_{k_{-1}}^{{}^{k_1}}ES{k}_{2}E+P.$ To measure the Michaelis constant $K_M,$

the reaction rate $r$ and the substrate concentration $S$ are often plotted in the so$-$

called "Lineweaver$-$Burke plot", where the $x-$axis is $\frac{1}{S}$ and $y-$axis is $\frac{1}{r}.$ The unit

of concentration in the problem is $mM$ and the unit of time is sec$.$

$($i$)$ Derive the expression of $r$ with respect to the concentrations of the substrate,

enzyme and product, as well as the corresponding rate constants.

$($ii$)$ Express $K_M$ in terms of $k_1,k_1$ and $k_2.$

$($iii$)$ If the data in the Lineweaver$-$Burke plot fits a line with the equation

$y=3\mathrm{.}7{10}^{5}x+2\mathrm{.}311{10}^5,$ what are the values of $K_M$ and the maximum reaction

rate $r_{\text{m}\text{a}\text{x}}?$ Hint: what is $r_{\text{m}\text{a}\text{x}}$ based on your answer to $1(b)(i)?$