Problem $5.$ Shear force on moving plate $(20$ points$)$

A thin plate is separated from two fixed walls by a liquid with viscosity $.$ The thin plate is pulled to the right with constant velocity $U_0.$ The plate is evenly spaced between the two fixed walls with gap height $h$ on either side. The velocity distributions of the liquid in the gaps above and below the center plate are linear and are given in the figure below. The vertical distance, $y,$ is measured from the center plate. The plate has length $L$ and width $($into page$)W.$ Calculate the magnitude of the shear force on the moving center plate. Make sure to include contributions from both the top and bottom surfaces. Express your answer symbolically in terms of $U_0,,h,L,$ and $W.$