Liquid oscillating in a U-tube of variable cross section

Abstract

A liquid is set into oscillations in a vertical manometer of smoothly varying cross-sectional area open at both ends to the atmosphere. Ignoring viscous damping, the displacement of either free surface and the pressure at any point in the liquid can be calculated using the unsteady Bernoulli equation which is shown to be consistent with conservation of mechanical energy. If the cross section is constant, the equations are analytically solvable; simple harmonic motion is found for the surface displacement as a function of time, and the pressure variation as a function of depth is in accordance with Newton’s second law. More generally, however, numerical solution of the equations is necessary, as illustrated for the example of a U-tube whose radius is linearly tapered from one end to the other. The level of presentation is appropriate for an upper level undergraduate course in fluid mechanics.

Filling and emptying a tank of liquid

“…neglecting the second-order-small term dm d ẋ. This approach correctly treats the variable-mass problem [2] of liquid in volume 1; equation (6) would not be obtained if one instead wrongly substituted p = m ẋ into equation (5).…”

“…Specifically, the effective area a of a sharp-edged round hole is 61% of its geometrical area [4].) However, the fluid velocity at any fixed height inside the tank is time-dependent and so it is described by the unsteady Bernoulli equation [5],…”