gas

FIG. $1$: Schematic of between parallel disks with radially stretching lower disk.

A Newtonian fluid with constant density $$ and viscosity $$ flows between two parallel imper$-$

meable disks. The distance between the disks $H$ is constant and the lower disk stretches in

the radial direction with velocity $Vr$ where $V$ is constant. Assume creeping flow and that the

gravitational force can be neglected. Based on these assumptions, the velocity field has the

form: $v_r=v_r(r,z),v_{}=0,v_z=v_z(r,z).$

$($a$)5$ pts Write the simplified continuity equation for this flow.

$($b$)10$ pts Write the simplified components of the Navier$-$Stokes equation for this flow.

$($c$)10$ pts Since $\frac{H}{R}1,$ assume the lubrication approximation holds: $p^L=p^L(r).$ Based

on this, write the differential equation that governs $v_z.$

$($d$)5$ pts Write the boundary conditions for $v_r$ and $v_z.$

$($e$)5$ pts Solve the equation from $($c$)$ to obtain an expression for $v_z(r,z).$

$($f$)5$ pts Derive Reynolds equation for this flow to obtain an expression for $p^L(r).$