Question $1[15$ marks$]$

A large plate is pulled at a constant speed of $U=4\frac{m}{s}$ over a fixed plate on a $5-mm$ thick engine oil film at $20C($dynamic viscosity, $=0\mathrm{.}8374$ Pa $*$ s$),$ as shown in Fig. $1.$ Assuming a half parabolic velocity profile in the oil film, as sketched, determine the shear stress developed on the upper plate and its direction. What would happen if a linear velocity profile is assumed? A typical parabolic equation can be expressed as $y^2=kx$ where k is an arbitrary constant.

Fig. $1$