# Bernoulli equation

The higher the flow rate of an ideal fluid, the lower its pressure.

Has it ever occurred to you why planes weighing several hundred tons, having accelerated, take off from the runway and rush upward? If not, first, the next time you are at the airport, take a close look at the section of the plane’s wing. First of all, notice that the wing in section is a combination of two convex lines, and the curvature of the upper contour is greater than the curvature of the lower one, as a result of which the area of ​​the upper surface of the wing is greater than the area of ​​its lower surface. It is this detail of the structure, hardly noticeable to the uninitiated, that allows the plane to take off from the surface of the earth.

And the basic idea here is this: the air flow is cut in two by the leading edge of the wing, and part of it flows around the wing along the upper surface, and the second part along the lower surface. In order for the two streams to close behind the trailing edge of the wing without forming a vacuum, the air flowing around the upper surface of the wing must move faster relative to the aircraft than the air around the lower surface, since it needs to cover a greater distance. And here comes the effect discovered by Daniel Bernoulli, one of the representatives of a real hereditary dynasty of tireless scientific geniuses from Switzerland. (In the authoritative Dictionary of Scientific Biographies, Dictionary of Scientific Biography, at least eight representatives of the Bernoulli surname are mentioned.). Daniel’s father – Johann Bernoulli – was a prominent professor of mathematics at the University of Groningen (later Johann Bernoulli moved to Basel and headed the department of Greek language at the local university, but after his brother’s death he returned to Groningen to replace him as head of the department of mathematics. Book of Daniel “ Hydrodynamics” (Hydrodynamica) was published in 1738, almost simultaneously with Johann Bernoulli’s book “Hydraulics” (Hydraulica), which, by mutual agreement between the son and the father, was, however, specially dated 1732, so that, if something happens, there would be no misunderstandings in the family regarding the priorities in the discoveries. This is such a family!).

Combining the laws of Newtonian mechanics with the law of conservation of energy (cm. The first law of thermodynamics) and the condition of fluid continuity, Daniel Bernoulli was able to derive an equation according to which the pressure from the fluid medium decreases with an increase in the flow rate of this medium (the concept of “fluid” includes liquid or gas). In the case of an aircraft, the air flows around the wing of the aircraft more slowly from below than from above. And due to this effect of the inverse dependence of pressure on speed, the air pressure from below, directed upwards, is greater than the pressure from above, directed downward. As a result, as the aircraft gains speed, the upward pressure difference increases, and the aircraft wings are affected by an increasing lift… As soon as it begins to exceed the force of the plane’s gravitational attraction to the ground, the plane literally soars into the sky. The same force keeps the plane in level flight: at cruising speed and altitude, lift balances gravity.

If you often fly by plane, you could not help but notice another phenomenon directly related to the Bernoulli effect. The plane at the airport of your hometown on different days accelerates along the runway in opposite directions, and lands on it in one direction or another. The choice of direction is not arbitrary: it depends on the direction of the wind. When moving against the prevailing wind, the speed of the air flow around the wing of the aircraft is equal to the speed of the aircraft relative to the ground a plus the speed of the wind itself relative to the ground. Therefore, when moving towards the wind, separation speed from the ground, at which the lift, described by the Bernoulli equation, begins to exceed the force of gravity, is lower, and the aircraft requires a shorter take-off or braking length after landing. This reduces the risk of going beyond the runway and saves fuel due to the fact that part of the lift is generated by the energy of the headwind.

You can also meet the Bernoulli effect when you sit at home by the fireplace on a rainy evening. During particularly strong gusts of wind, flames soar up into the chimney. And the following happens: when the wind speed at the outlet of the pipe increases, the pressure in this place drops. The higher pressure inside the house literally “pushes” the flame out of the fireplace into the chimney. You’ve probably noticed the spiral blades around the outlets of the factory pipes. They are installed there for the same purpose: by directing the wind around and over the pipe opening, they contribute to a whiter exhaust gas emission.

I myself use the Bernoulli effect in a very unexpected way. To maintain my physical condition, I regularly roller-skate in Washington, DC on a special asphalt track that runs along the Potomac River. I get on the track near the National Airport, and, while parking my car, the first thing I look at is in which direction the airliners take off or land. If they land or take off in the direction where I’m going to ride, then everything is in order, and on the way back I will be helped by a fair wind. If they sit down to meet me, then it is better to shorten the jogging distance, because on the way back the wind will blow in my face, and I don’t like that. Thus, the Bernoulli effect allows me to accurately dose my daily physical activity.

Daniel Bernulli
Daniel Bernoulli, 1700–82

Swiss mathematician, physicist and physiologist. Born in Groningen (Netherlands) in a family of hereditary mathematicians and intellectuals. Initially, he received a medical education, and in 1725 he accepted the invitation of the St. Petersburg Academy of Sciences and took the post of professor of the Department of Physiology. Having discovered in this area many unsolved problems from the field of theoretical physics and, in particular, the dynamics of the movement of fluid (blood) in vessels, he returned to the mathematical description of physical processes and in 1730 headed the Department of Pure Mathematics at the St. Petersburg Academy. In 1733 he returned to his homeland in Basel, where he headed the Department of Anatomy and Botany of the local university, and from 1750 – the Department of Experimental Physics, which he headed until his death. As a result of studying the hydrodynamic dependences, he formulated the so-called Bernoulli’s principle and anticipated the birth of the molecular kinetic theory of gases by a century.