Dynamics and frictional dissipation for treading slowly in a puddle

The process of producing a liquid column is common in daily life and industrial applications, such as walking through a puddle and roller printing. While governed by the Navier-Stokes equation, its dynamics are often studied by numerical means, which hinders a full understanding of the rich mixture of physics behind, for instance, the competition of surface and potential energies, and how the pinch off is affected by the kinetic energy and water jet when a large cylinder is used. For pedestrians rushing out of the rain, the water column inevitably involves turbulence and defies simple theoretical analyses. As a result, this work will focus only on cases with a low Reynolds number to enable laminar flow and the existence of reversible and quasistatic stages. Combined with simple models, we elucidate the mechanism that drives the change of morphology and derive analytic expressions for the critical height and upper radius for the liquid column when transiting between three stages. Stage I is characterized by a static and reversible profile for the column whose upper radius r_{t} equals that of the cylinder. The column becomes irreversible and r_{t} starts shrinking upon entering stage II. It is not until r_{t} stops shrinking that the column neck accelerates its contraction and descends toward the pool, the quantitative behavior of which is among the successful predictions of our theory. Pinch off dominates the second half of stage III without its usual signature of self-similarity. This is discussed and explained with an interesting incident involving a water jet similar to that made by a dropping stone.