Consider a siphon that moves water $($kinematic viscosity coefficient $v)$ from an upper jug to a lower cylinder as shown in the figure. The length of the tube connecting the jug to the cylinder is $l$ and the acceleration of gravity is $g.$ Answer the following questions.

$(1)$ If the water surface height of the jug is $H$ and that of the cylinder is $h,$ find the velocity of the water moving through the tube. Note that the flow of water in the tube is laminar.

$(2)$ What is the velocity distribution in the tube in $(1)?$ Find the shear stress acting on the wall of the straight section of the tube. What is the kinetic energy correction factor for this case?

$(3)$ The water surface in the jug descends and the water surface in the cylinder rises. Show the ordinary differential equation for the relative distance at which the water surfaces approach each other.

$(4)$ Set the initial water surface height of the jug at $H_0$ and the liquid surface height in the cylinder at $h_0,$ and solve the differential equation in problem $(2).$ Also, illustrate the approximate shape of the change in relative distance with respect to time.

$(5)$ From the results in $(4),$ find the final respective water surface heights.