An incompressible, viscous fluid is placed between horizontal, infinite, parallel plates as is shown in Figure. The two plates move in opposite directions with constant velocities, and as shown. The pressure gradient in the $x$ direction is zero, and the only body force is due to the fluid weight. Use the Navier$-$Stokes equations to derive an expression for the velocity distribution between the plates. Assume laminar flow.

$u=\frac{U_1+U_2}{2}y-U_1$

$u=\frac{U_1+U_2}{2}y+U_2$

$u=\frac{U_1+U_2}{2}y-U_2$

$u=\frac{U_1-U_2}{2}y-U_2$