Human Knee Joint $(22$ pts$)$

In the human knee joint, synovial fluid is provided by a thin layer that separates the cartilage$-$covered bones of the upper and lower leg. When the joint is leaded by standing, fluid contained in the gap between the surfaces spreads outward. This squeeze film supports the load of the body and helps to prevent solid contact between the two cartilage surfaces.

NOW you are requested to design the artificial knee joint to support a clint's body $($weight Mg $).$ As a simple model to the behavior of the knee joint, you consider two circular surfaces $($radius $R)$ each with a layer of cartilage $($thickness $).$ At the instant of interest, the two surfaces are approaching each other atspeed $V=-d\frac{h}{d}t$ where $h$ is the instantaneous gap height $(hR).$ Here, we can model the cartilage as a homogenous, isotropic porous material $($with many small pores as like $($ponge$).$ Flow in the cartilage is assumed to borprimarity in the radial direction and is described by Darcy's Law:

$u_c=-\frac{K}{}\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{r}}$

where $K$ is the permeability that is inversely proportional to the pore size of the cartilage, $u_c$ is the radial fluid velocity in the cartilage. The vigcogity of synovial fluid is $$ and gravitational effects is neglected. The atmosphere is at a constant pressure $\frac{?}{\text{b}\text{a}\text{r}}(P_a).$

Synovial joint, sagitulal section

A$.$ Describe the conditions you can assume this radial flow in the gap as a thin film flow $($i$.$e$.$ locally fully developed flow$)$ with characteristic film thickness $h,$ diameter $R,$ velocity $V$ and time $=\frac{R}{V}[$Hint: if you cannot remember, you will be able to derive the conditions from $N-S$ eqns.$](3$ pts$).$

B$.$ Develop the goveming equations for synovial fluid flows between the cartifages with all of your assumptions to simplify the equations $($it is okay to consider only continuity & N$-$S eqn. in r$-$direction$)(3$ pts$).$

C$.$ Set two boundary conditions to solve the N$-$S eqn. of B $(2$ pts$)$

D$.$ Find velocity profile $u(z)$ between the cartilages as a function of $z,h,,\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{r}},K($Draw the velocity profile between the bones$)[$Hint: you know the velocity profile in the cartilages already$](3$ pts$).$

E$.$ Find volume flow rate $Q(r)$ between the bones as a function of $r,h,,\frac{\text{d}\text{e}\text{l}\text{P}}{\text{d}\text{e}\text{l}\text{r}},K,(2$ pts$).$

F$.$ Present the downward velocity $V$ in terms of $r,Q[$Hint: consider the mass conservation for the control volume of synovial fluid in the gap $($red dotted box$).](1$ pts$)$

G$.$ Using the answers of E and F $,$ derive the radial pressure distribution $P(r)$ with a proper boundary condition in terms of $P_a,h,,V,r,R,K,(2$ pts$)$

H$.$ When $h,$ calculate the supporting force