A thin $20cm20cm$ flat plate is pulled at $2\frac{m}{s}$ horizontally through a $3\mathrm{.}2mm-$thick oil layer sandwiched between two walls, one stationary and the other moving at a constant velocity of $0\mathrm{.}2\frac{m}{s},$ as shown below. The dynamic viscosity of the oil is $0\mathrm{.}03N*\frac{s}{m^2}.$ Assume velocity in each oil layer to vary linearly.

$(1).$ Sketch the velocity profile between the two walls. Note important points on the sketch.

$(2).$ Find the location where the oil velocity is zero.

$(3).$ Find the force that needs to be applied on the plate to maintain its motion.

$(4).(6$ extra points$)$ Calculate the volume flow rates between the walls per unit width $($width is along the direction perpendicular to the paper$).$