The permeability data below have been obtained for a hydrogel material used as a scaffold material for tissue engineering. The original thickness of the specimen is $3$ mm and it is compressed by the displacements provided in the table that follows and the respective permeability calculated using Darcy's Law.

$($a$)$ Plot the permeability data and determine the intrinsic permeability for the hydrogel.

$($b$)$ An exponential function can be used to describe the strain dependence of the permeability. Determine the coefficient $M$ that signifies the strain dependence of $k.$

$($c$)$ Why might strain$-$dependent permeability benefit the load support mechanisms of a hydrated tissue?

Displacement $($mm$)$ Permeability $\frac{m^4}{N}*s$

$-0\mathrm{.}1$

$2\mathrm{.}39{10}^{-14}$

$-0\mathrm{.}2$

$1\mathrm{.}96{10}^{-14}$

$-0\mathrm{.}3$

$1\mathrm{.}62{10}^{-14}$

$-0\mathrm{.}4$

$1\mathrm{.}48{10}^{-14}$

$($d$)$ From a practical perspective, why is a tare load needed to perform the experiment?

$($e$)$ Why is it important that the platens in the permeability device are freeflowing?