DONT USE CHATGPT!!!!PROBLEM $3$

Oil is lubricating the space between two plates separated by a distance $h.$ The density of oil is $p$ and its viscosity $.$ The lef plate is moving up at a velocity V $.$ There is no gravity or pressure gradient acting on the oil. The oil can be considered as an incompressible fluid and the system is in steady state.

Part $1$

i$)$ Write and number all assumptions that are reasonable to make in this case.

ii$)$ Write the continuity equation in differential form and in the most general way $($with all terms$).$ Then, simplify the continuity equation indicating at each step the assumptions that were used.

iii$)$ Based on the continuity equation simplified above, and using appropriate boundary conditions, determine the value of the velocity in the $x$ direction $u.$

iv$)$ Write the Navier$-$Stokes equation in the $y-$ direction and simplify it$.$ Again, indicate at each step the assumptions used.

v$)$ Indicate the boundary conditions to be used for the velocity in the $y-$direction $v.$

vi$)$ Solve the equation in $($iv$)$ and use the boundary conditions to find unknown constants. vii$)$ Plot the velocity profile and shear stress profile between the plates.

Part II:

Consider $h=1mm$;$=900k\frac{g}{m^3}$;$=0\mathrm{.}01$ Pa$.$s; $V=1c\frac{m}{s}$;

viii$)$ Calculate the magnitude of the shear stress at the left plate and draw the direction of the force acting on the plate.

ix$)$ Calculate the magnitude of the shear stress at the right plate and draw the direction of the force acting on the plate.

x$)$ Calculate the mass flow rate of oil for a plate width $W=10cm$ into the page.