A solid cylinder of diameter Dc moves downward at velocity V

A solid cylinder of diameter D_c moves downward at velocity V_c into a tube of diameter D_T filled with an incompressible liquid of density rho. This causes the liquid to rise into the annular space between the tube and the solid cylinder, as shown in the figure below. Starting with an appropriate control volume sketched on the figure below, use Raynold\'s Transport Theorem to derive an expression for average velocity of liquid as it rises in the tube in terms of the variables given.

Solution

Use a control volume surrounding the falling cylinder:

The volume change due to the falling cylinder becomes Vcyl* Pi/4 *(Dc)^2

Area of the liquid in the annular space contained in the control volume: Pi/4*( Dt^2 -Dc^2) =Vann

volume rate of change [Annular volume* V1]

Equating the two: V1 = Dc^2/(Dt^2-Dc^2)* Vcyl

If control volume moves with the falling cylinder, the equation changes to V1= Dt^2/( Dt^2-Dc^2)*Vcyl