A 2D rectangular piston of width L = 0.05 in and height H = 0.1 m is sealed inside a slightly wider rectangular container of width L + 26 (6 = 0.0005 m). The container is filled with oil of viscosity p = 0.01 kg m-1 The block is moved upwards at velocity V = 0.5 m s'. You may neglect gravity forces for this problem, and assume that the flow is effectively steady.
a. Derive the formula for the y-velocity in the gap between the block sidewall and the larger container.
b. As the block moves upwards, the incompressible oil is forced downward through the two side gaps to ensure mass conservation. What is the pressure gradient dP/dy due to this constraint?
c. What is the force (per unit depth into the page) required to move the block upward at the specified velocity?