Law of Urination: all mammals empty their bladders over the same duration

Abstract

Many urological studies rely upon animal models such as rats and pigs whose urination physics and correlation to humans are poorly understood. Here we elucidate the hydrodynamics of urination across five orders of magnitude in animal mass. Using high-speed videography and flow-rate measurement at Zoo Atlanta, we discover the "Law of Urination," which states animals empty their bladders over nearly constant duration of 21 ± 13 seconds. This feat is made possible by larger animals having longer urethras, thus higher gravitational force and flow speed. Smaller mammals are challenged during urination due to high viscous and surface tension forces that limit their urine to single drops. Our findings reveal the urethra constitutes as a flow-enhancing device, enabling the urinary system to be scaled up without compromising its function. This study may help in the diagnosis of urinary problems in animals and in inspiring the design of scalable hydrodynamic systems based on those in nature. Significance StatementAnimals eject liquid into environment for waste elimination, communication, and defense from predators. These diverse systems all rely on the fundamental principles of fluid mechanics, which have allowed us to make predictions about urination across a wide range of mammal sizes. In this study, we report a mathematical model that clarifies misconceptions in urology and unifies the results from over 50 independent urological and anatomical studies. The theoretical framework can be extended to study fluid ejection from animals, a topic which has received little attention.

Dimensional Analysis and the Time Required to Urinate

“…We are currently seeking information and data to test this prediction. The factor of 5 is nearly within the 8 − 34s interval cited in [1].…”

“…This conclusion is in disaccord with the argument of Hu et al [1], who stresses the importance of gravity in large animals in explaining experimental observations. We note that astronauts apparently urinate without difficulty even in the absence of gravity [8]; and babies pee in the upward direction as well as down.…”

“…1. As pointed out in [1], some authors ascribe the force which is responsible for expelling the urine to bladder pressure [6][7][8], while others propose a combination of gravity and bladder pressure [9]. For a given driving force and geometry, the flow velocity is determined primarily by fluid inertia and viscosity.…”

“…1. Since C 1 does not change if all lengths are scaled, it follows that, if the dominant contributions were gravity and inertia, the time needed to urinate would be independent of the density of urine but proportional to the square root of the characteristic length of the animal (or proportional to the mass to the one-sixth power, in accordance with [1] for large animals); large animals would take somewhat longer to urinate than small ones, and the required time would not be size independent.…”

“…In a recent publication [1], David Hu and coworkers report the discovery of the "Law of Urination", according to which animals empty their bladders in the nearly constant time interval of 21 ± 13 s regardless of size. This is indeed a remarkable result, given that body masses of the animals in question, from mice to elephants, range over some five orders of magnitute.…”