Water cataracts and the “shortest-time descent” curve (Brachistochrone) as one natural phenomenon

The rainwater on the ground strikes us with images that suggest the presence of flow obstacles: cataracts on gravel (dams and pools) and roll waves on a smooth pavement (Fig. 1). They do this repeatedly, with a noticeable periodicity. Such flows are common even though they are diverse. They all flow downhill but exhibit abrupt falls. Cataracts are comparable to waterfalls, but their dimensions are much smaller. The biggest waterfalls are associated with tectonic breaks in the crust. Small cataracts are associated with the erosion of the surface and the debris removed from it.^1,2

Cataracts and roll waves are different, but different does not mean unrelated. If we know the relation, we can predict the single phenomenon to which the different manifestations belong. A hint comes from the classical mathematics problem of the shortest-time descent without friction, the brachistochrone.^3–5 Earlier, the descent is steep, and later, it is shallow. The y(x) curve (the solution) is smooth (cf. Fig. 2).

Surely, a cataract is not that way. Its initial fall is steep, an abrupt step, with water falling over a dam. The presence of dissipation (irreversibility) is obvious. Furthermore, the brachistochrone is a solution to a mathematical exercise, which is an idea (a human-made object), a milestone in the history of mathematics, not a phenomenon in nature.

Yet, pictures do not lie. The connection is in the similarity between the two sides of Fig. 1 and in the fact that stepwise flows occur naturally. Therefore, it is appropriate to question whether flow access is easier stepwise, compared with on an incline.

First, here is a brief introduction to the brachistochrone curve of Bernoulli^3,4 and Newton.⁵ The present article is not a review of the brachistochrone literature. The classical treatment of the shortest-time descent is in terms of a body of mass M falling on a frictionless track from point A to point C (Fig. 2). The height difference between the two points is H. The unknown is the shape of the track such that the travel time is minimal. The parametric solution given by Bernoulli in 1696 (cf. Ref. 3) is y(x), where x = β − sin β and y = 1 − cos β, where the parameter β is the angle between the track at point (x, y) and the gravitational acceleration g, with y measured downward. At the start of the motion, β = 0 and x = y = 0.

The solid curve y(x) plotted in Fig. 2 corresponds to the assumption that the end of the slide occurs when M reaches the lowest point on the curve, namely at β = π/2, y = 2, and x = π. At the bottom level, the kinetic energy $12MVend2$ matches its initial gravitational potential energy, MgH, meaning that the end velocity of frictionless descent is the Galilean velocity V_end = (2gH)^1/2.

The brachistochrone continues to attract attention as a mathematics problem stated in more realistic terms, such as descent with friction^6–8 and how to use this problem to strengthen mathematics education.⁷ Descent with irreversibility (not with dry friction) is a feature in the water flow model used in Secs. III and IV. The objective of the analysis is to determine in elementary terms whether the movement is faster when the path is stepped, not straight.

A feature of the smooth y(x) curve plotted in Fig. 2 is that its slope at x = 0 is infinite. Initially, the body is in vertical freefall. In the limit β → 0, which means x → 0, the y(x) curve approaches y → (9/2)^1/3 x^2/3. Another feature is that along most of the track, the curve resembles the horizontal segment, as the angle α approaches 0°.

These features are matched by a simple model of water flow over cataracts. The start is modeled as a short freefall from A to B. The horizontal segment BC is the surface of the pool, which is modeled thermodynamically as a “reservoir,”⁹ large enough so that the flows into and out of it do not alter its state.

The y(x) curve fits inside the ABC triangle. The approximation provided as two segments (AB and BC) is more realistic in the limit L ≫ H, which corresponds to most of the natural water flows that occur on land (e.g., Fig. 1). In this limit, the time of flowing from A to B is t_y = (2H/g)^1/2.

Next is the model of the water pool in steady state. The stream that falls in the pool at B is balanced by another stream that spills over the next dam at C. The pool is maintained in steady state (hydrostatically stratified) by new water that arrives at B and old water that spills from C. Any of us can observe this in an overflowing sink. The water that falls from the faucet is not the same water that spills over the edge of the sink.

The steady state is clear if the system (the pool) is viewed from thermodynamics.⁹ The system persists in time irreversibly (with internal entropy generation). The spot where irreversibility occurs is at B, where the falling water impacts the pool and comes to rest by mixing locally, subsurface. As a whole, the pool does not have thin horizontal layers (laminar or turbulent) rubbing against each other.

The simultaneous flow of water into B and out of C has the same effect as the baton in the 4 × 100 m² relay at the Olympics. Because of passing the baton from the arriving runner to the departing runner, the run time in the 4 × 100 m² race is shorter than quadrupling the duration of a single 100 m run. In the model of Fig. 2, the water arriving at B “passes the baton” to the water departing from C.

One alternative for water is to flow on the straight incline AC, which has the length L/cos  α, where tan α = H/L. There is no “friction coefficient” because the irreversibility (dissipation of power and entropy generation rate¹⁰) is distributed throughout the volume of liquid. The water film on an incline is not a solid block or a particle. It is a continuum, macroscopic, which derives its speed and film thickness from its viscosity and the mass flow rate dictated by the rate of rainfall.

To get an idea of the scales involved, we assume that the layer of water has a thickness D ∼ 1 cm and terminal speed U ∼ 0.5 m/s and find that the Reynolds number UD/ν is 5000. This indicates turbulent flow, which is faster than the flow in the laminar–turbulent transition. Therefore, the following analysis is for turbulent water layer flow on an incline close to horizontal such that L/cos α ∼ L and sin α ∼ α.

Inequality (4) holds in common circumstances. For example, when α ∼ 0.03 (or 2°), L ∼ 1 m, and $m′∼0.05kgs⋅m$ (that is when the scales are U ∼ 0.5 m/s and D ∼ 1 cm), the right-hand side of (4) is 1.75 × 10⁶, and the inequality is satisfied.

The contribution of this article is the previously not recognized connection between common phenomena in nature and multiple invocations and uses of a classic mathematical problem. The stepped flows (Fig. 1) reinforce the universal tendency (evolution) toward configurations for greater flow access in design in nature. This conclusion is considerably more inclusive than the message derived from questioning the two images of Fig. 1. Roll waves (Fig. 1, right) are the same phenomenon as cataracts (Fig. 1, left). The only difference is the substrate, which under roll waves is not erodible.

The phenomenon is what we see, the flow configuration that morphs and looks like other flow configurations. The phenomenon is not a mathematical formula, or a mathematical extremum (min, max, opt, and fastest). What we see in nature has configuration, freedom to change, morphing, and rhythmic flow, stepwise, in and out, cascades, respiration, heartbeat, turbulence, periodic scraping, and renewal, which are reviewed in Refs. 12–29. Such phenomena are contrary to the smooth, continuous, and steady models with which our education in science begins.

The broad view is not more complicated. It is not negating the laminar flow model of the capillary blood vessel, and the river channel with turbulent flow on a rough bottom. The holistic view offered in this article is unifying. It is also a source of intellectual pleasure: when we see a stunning event, such as the opening of the spillway gates at the Oroville Dam (Fig. 3),³⁰ we feel the power of theory, which is the ability to predict the future of an event and to rationalize its natural occurrence.

H. Almahmoud drew Fig. 2. The photographs (Fig. 1) were taken by A. Bejan. This work was supported by a grant from CaptiveAire Systems.

The author has no conflicts to disclose.

Adrian Bejan: Conceptualization (lead); Writing – original draft (lead); Writing – review & editing (lead).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.