Consider two fluids placed between two parallel flat plates of infinite length and width. The flow is steady, incompressible, parallel, and laminar. The upper plate moves with velocity V to the right, and the lower plate is at rest. Gravity acts in the $-$z direction $($down in the figure$).$ There is no forcing pressure gradient pushing the fluid through the channel: the flow is established solely by the viscous effects created by the moving upper plate. You can ignore surface tension effects and assume the interface is horizontal. The pressure at the bottom of the flow $($z $=0)$ is equal to P$0.\backslash$table$[[$Moving wall,V$],[$h$\_(2),$Interface,Fluid $2,\backslash$rho $\_(2),\backslash$mu $\_(2)],[$h$\_(1),,$Fluid $1,\backslash$rho $\_(1)+\backslash$mu $\_(1)]]$

a$)$ List all the appropriate boundary conditions for both velocity and pressure

Solve for the velocity field $($hint: split the solution into two portions, one for each

fluid. Generate expressions for U$1$ as a function of z and U$2$ as a function of z$).$

b$)$ Let fluid $1$ and fluid $2$ be unused motor oil, both at $80\backslash$deg C$.$ Also let h$1=5\mathrm{.}0$ mm$,$ h$2=8\mathrm{.}0$ mm and V $=10\mathrm{.}0$ m$/$s$.$ Comment on the results.