(Solved): 3. The velocity of water (viscosity \( \mu \) ) between two flat stationary parallel plates as sho ...

3. The velocity of water (viscosity \( \mu \) ) between two flat stationary parallel plates as shown is given as \[ u=U_{c}\left(1-\frac{y^{2}}{b^{2}}\right), \] where \( U_{c} \) is the centerline velocity. \[ 2 \] (a) Find the expression of the shear stress at any location \( y \). Your expression should be in terms of the viscosity, \( y \), and \( U_{c} \). (b) Find the shear stress expression at \( y=\pm b \). What is the shear stress at the center \( y=0 \) ? (c) Sketch the variation of shear stress as a function of \( y \). Hand sketch is sufficient. (d) If the length of the plates is \( L \) and width into paper is \( w \), find an expression for the net force exerted by the fluid on the plates. In which direction does this force act? Assume the velocity profile at all \( x \) locations is the same.