Bernoulli’s principle
１．Definition
Bernoulli's principle is the law of conservation of energy for fluid flow. It states that “When energy loss or supply is negligible in a certain stream, the energies of two points on one streamline are equal (conserved)” (Fig. 1).
Fig. 1
２．Conditions for Bernoulli's principle
Bernoulli’s principle requires the following conditions to hold:
(1) It holds for two points on the same streamline, the upstream side and the downstream side (point A and point B in Fig. 1).
(2) There is no energy loss or supply. Even if there is a loss or supply, it should be negligible.
３．Common mistakes
Many people misunderstand the statement “where the flow is fast, the pressure is low (it does not always hold)” as Bernoulli's principle. This mistake is widespread in science introductory books, online writings, television programs, etc. Please note that there could be more incorrect explanations depending on the phenomenon.
４．Instances of mistake
(1) “When air is blown into the atmosphere from a pipe or nozzle, the pressure becomes low according to Bernoulli's principle because the flow velocity is fast at the point where the flow is ejected (Incorrect).”
Examples include when blowing with a straw, blowing from the mouth, blowing out wind with a dryer, etc. As shown in Fig. 2, when comparing points A (in the flow) and B (surrounding stationary place, atmospheric pressure), we believe that the pressure becomes lower than atmospheric pressure because point A flows faster than point B; this is incorrect because it is not on the same streamline, and condition (1) is not satisfied; therefore, Bernoulli's principle does not hold. To be correct, the pressure at point A will also become atmospheric pressure (which can be confirmed theoretically and experimentally). Originally, the flow of point A has more energy than point B based on the amount of energy supplied to blow out.
If point A is lower than atmospheric pressure, the surrounding air (atmospheric pressure) is drawn, and the air gathers as it goes downstream and the flow velocity increases gradually, which is a contradiction.
Fig. 2
(2) Same mistake as in the previous paragraph “When air is blown into the atmosphere from a pipe or nozzle, the pressure becomes low according to Bernoulli's principle because the flow velocity is fast at the point where the flow is ejected (Incorrect).”
In Fig. 2, when comparing points A and C (point on the downstream side of the flow), the flow is slow and the pressure is almost equal to the atmospheric pressure at point C. There is also another method to explain that point A is faster than that and the pressure is lower than point C, i.e., it becomes lower than atmospheric pressure (Incorrect). Points A and C are on the same streamline; however, the energy decreases as they go downstream because of the viscous friction on the way; the above condition (2) is not satisfied and Bernoulli's principle does not hold.
(3) “The flow velocity of the flow from a dryer etc. is faster than the surrounding, and the pressure becomes low according to Bernoulli's principle. Therefore, if you float a ping-pong ball, it will not fly out (Incorrect).”
The explanation is that, as shown in Fig. 3, points A (in the flow) and C (near the ball) have a higher flow velocity than point B (surrounding stationary place), and the pressure becomes lower according to Bernoulli's principle (Incorrect).
Since point B is not on the same streamline, Bernoulli's principle does not hold. A force acts in the direction in which the ball and the flow attract each other by bending the flow near the ball (Coandă effect). The contradiction in the explanation of the mistake can also be confirmed from the experiment of “Circle and Square 1 (published on December 2009)”.
Fig. 3
(4) “If you make a hole in the middle of a straw and blow, the flow velocity inside the straw will be high, so the pressure will become low according to Bernoulli's principle, and the surrounding air will be sucked in through the hole (Incorrect).”
The explanation is that, as shown in Fig. 4, the flow velocity at point A inside the straw is faster than that at point B (atmospheric pressure) outside, therefore, it becomes lower than atmospheric pressure, and the surrounding air is sucked in through the hole (Incorrect). Since points A and B are not on the same streamline, Bernoulli's principle does not hold.
To be correct, the pressure at point A needs to be higher than the atmospheric pressure, and air needs to be blown out from the hole. This is known from the fact that when you blow a recorder (vertical flute), air blows out from the side holes in the middle, and many people have experienced it. At point C (exit), it is atmospheric pressure, and there is energy loss due to the viscous friction between that point and point A; thus, point A has more energy than point C. The pressure at point A on the upstream side becomes higher (higher than atmospheric pressure) as much as this energy loss, and air blows out to the outside, where it is atmospheric pressure.
Fig. 4
５．Supplementary explanation
When considering the energy balance of a fluid, consider the sum of kinetic energy, potential energy, work attributed to pressure (which can be regarded as pressure energy), and internal energy (energy generated by molecular motion and molecular vibration). In the case of fluids with a small volume change such as a liquid, Bernoulli's equation about the conservation of the sum of the kinetic energy, potential energy, and work attributed to pressure is used. If the potential energy is constant (same height), the sum of kinetic energy and work caused by pressure becomes constant, and it can be said that “the pressure is small where the flow velocity is high”. Please be aware that this can only be said if many of the above conditions are met.
| [Reference] | “Illustrated Fluid Dynamics Trivia” by Ryozo Ishiwata, Natsumesha, P218－219, P206－209. The Japan Society of Mechanical Engineers Edition “The Wonders of Flow,” Kodansha Bluebacks, P98－109. |