Consider a vertical cylindrical tube (narrow diameter D, length L) connected to a wide open funnel filled with a substantial quantity of water (density p, dynamic viscosity u): see Figure.

Consider a steady state in which the water level in the funnel is maintained at a depth H. The result is a constant flow of water, at a volumetric flow rate Q, through the tube into the open air (pressure po) again. The conditions are such that the flow in the tube is laminar. You may ignore entrance effects at the rather smooth transition from funnel to tube. ) Carry out a dimensional analysis to find out how Q depends on, among other variables, H and D. Starting from a force balance on a thin 'slice', derive expressions for the shear stress profile and the velocity profile for the flow through the tube ) Make a sketch of the shear stress profile and the velocity profile in a single figure, By integrating the velocity profile, find an expression for the flow rate Q. Compare this expression with your answer for question a and comment.