Please explain in step by step and make sure that notation used are mentioned properly and mathematical symbols are shown properly in solution. thanks

$(20)$ Question $4$

The lubrication of bearings is a problem in fluid mechanics. The oil or lubricant between

the bearing and the slider is a viscous fluid and most bearings operate in the laminar range

with a very small Reynolds number. The spacing between the bearing and the slider $($the

flow gap$)$ is much smaller than the length of the slider so that the flow becomes fully

developed throughout most of the gap. Because the Reynolds number is so small, the

inertia of the fluid is negligible compared with the pressure and viscous forces.

Consider a step bearing moving at velocity $U,$ as shown in Figure $3.$ The spacings $h_1$

and $h_2$ are much less than $L_1$ and $L_2$ and the width of the bearing in the $z$ direction is

assumed to be very large, so that leakage in the $z$ direction can be neglected. To facilitate

the analysis, we attach coordinate systems to the slider with the origins at the beginning

of each bearing section and we invert the direction of $y,$ such that the relative velocity is

$u=0$ when $y=0,$ and $u=-U$ when $y=h$ in the respective sections.

Figure $3$: The step bearing and its pressure distribution.

$(5)$

$(5)$

$($a$)$ Assuming incompressible flow of a Newtonian fluid, simplify the equations of motion

$($the Navier$-$Stokes equations$)$ in $x$ and in $y$ directions.

$($b$)$ Find an expression for the total flow rate and for the pressure change between any

two points in a single section of the bearing.

$($c$)$ Find an expression for the relative pressure distribution in the step bearing, assuming

a zero gauge pressure at the inlet and outlet, as shown in Figure $3.$

$($d$)$ Calculate the total load carrying capacity per unit length of $z$ for a step bearing with

the following conditions:

$U=0\mathrm{.}5\frac{m}{s}$;${}_{\text{o}\text{i}\text{l}}=3\mathrm{.}85\frac{Ns}{m^2}$;${}_{\text{o}\text{i}\text{l}}=900k\frac{g}{m^3}$

$h_1=1mm$;$L_1=30mm$

$h_2=2mm$;$L_2=100mm$