A liquid moving in a frictionless U-tube whose liquid column is displaced a distance Z upward and then released, see Figure 2.

a) Neglect the short horizontal leg and combine control-volume analysis for the left and right legs to derive a single differential equation for V(t) of the liquid column.

b) Now, including an average wall shear stress resistance of the form 8V/D, where is the fluid viscosity and D is the tube diameter. Find the differential equation for dV/dt and then solve for V (t), assuming an initial displacement z = z0, V = 0 at t = 0. The result should be a damped oscillation tending toward z = 0. Hint: Use the Navier-Stokes equations

Figure 2