A wire is pulled through a circular cylinder of length $L$ that connects two large chambers maintained at a constant pressure $p_0.$ Assume incompressible, Newtonian fluid. Ignore gravity and body effects.

$($a$)$ Derive an expression for the steady$-$state volumetric flow rate between the chambers and an expression for the force required to pull the wire through the cylinder.

$($b$)$ Repeat this calculation when the two chambers are sealed with the total volume equal to that of the fluid. Are the results different? If so$,$ why? Discuss the limiting case $\frac{D_1}{D_2}0.$