Problem $3(45$ points$)$

Fluid$-$lubricated bearings are machine parts in which viscous fluid is forced into a converging channel. In Fig. $3,$ a solid bearing is shown in which a lower plate at $z=0$ moves in the positive $x$ direction at a constant speed $U_0.$ The lower boundary of the bearing, located at $z=h(x)$ is fixed and tilted at small angle $,$ which means the length of the plate $L$ is very large. Here, the external pressure is atmospheric pressure $p_a.$ For this geometry, the fluid particles move in the $x$ direction parallel to the lower plate, and there is no velocity in the $y$ and $z$ direction and $w=0.$ Assume that the flow is steady and incompressible. There is no gravity effect.

Fig. J

$($a$)$ Show that the velocity in $x$ direction $(u)$ is a function of $z$; i$.$e$.,u=u(z).$

$($b$)$ Write the boundary conditions for $u$ where $0xL$ and for $p$ beyond the bearing $(x0$ and $xL).$

$($c$)$ Write the momentum equations in $x$ and $z$ directions.

$($d$)$ Derive the velocity distribution from $($c$).$

$($e$)$ Derive the volume flow rate per unit width $Q.$

$(f)$ The flow rate $Q$ is constant because the flow is steady. Then the derivative of $Q$ is zero del$\frac{Q}{d}$elx$=0.$ This equation is the Reynolds lubrication equation. Given $h(x),$ it is an ordinary differential equation for $p.$ Derive the pressure distribution $(p)$ from the Reynolds lubrication equation.

Hint: The solution form is $p(x)-p_a=C_1{b}_1(x)+C_2{b}_2(x)$ where $C_1$ and $C_2$ are constant.

$(g)$ The maximum pressure $(p_m)$ exists between