Consider incompressible fluid flowing between two fixed concentric cylinders of radii a and b with b>a, and length L. Radial distance is measured by r and axial distance along the cylinders is measured by z. The axis of the cylinder is vertical with z pointing upwards, and both ends are open to the atmosphere.
State appropriate no-slip boundary conditions and use these to obtain the flow $w$ in the form
Progress: I have gotten to integrating the equation with two constants and applied the boundary conditions \begin{align*} w\,(r= a) = &0, \\ w\,(r= b) = &0 \end{align*}
So I got, \begin{align*} w = \dfrac{\rho g}{4\,\mu} r^2 + A\, \ln {r} + B \\ A = - \dfrac{\rho g}{4\,\mu} \, \ln\left(\dfrac{a}{b}\right)\, \left( {a}^{2} - {b}^{2} \right) \\ B = - \dfrac{\rho g}{4\,\mu} \, \left( \ln(a)^{2}\,{b} - \ln(b)^{2}\, {a} \right) \end{align*}
After this, I am not quite sure how I can get the solution that was asked for, I have plugged A and B into the equation but still could not yield the desired result.