A Newtonian fluid with a viscosity of $$ and a density of $$ flows through between two infinitely long, parallel, vertical plates. The plate length is $L$ and the plates are spaced a distance $H$ apart. The pressure gradient is applied across the plate length. As seen in Figure, both plates are moved with a velocity of var${}_0$ at opposite directions.

a$.$ Write the mass and and momentum balances with necessary assumptions and find out a final equation that models this flow system $(20$ points$).$

b$.$ Write the boundary conditions $(10$ points$).$