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Bernoulli's principle

physical science Maturity 11-13 Next article →

Bernoulli's principle is a fundamental concept in fluid dynamics. Fluid dynamics is the study of how liquids and gases move. The principle describes the relationship between pressure, speed, and height in a moving fluid. For a fluid flowing horizontally, the principle states that an increase in speed occurs at the same time as a decrease in pressure. This concept is essential for understanding how everything from aircraft to water systems functions.

VenturiFlow
VenturiFlow

The mechanism of this principle is rooted in the conservation of energy. In a steady flow, the total energy of a fluid remains constant at all points. This total energy is the sum of kinetic energy, potential energy, and internal energy. Kinetic energy is the energy of motion. As a fluid speeds up, its kinetic energy increases. To maintain the energy balance, the sum of the potential energy and internal energy must decrease. This results in a simultaneous drop in the fluid's static pressure.

BernoullisLawDerivationDiagram
BernoullisLawDerivationDiagram

Another way to understand this is through Newton's second law of motion. This law relates force to acceleration. When a fluid flows horizontally from a high-pressure region to a low-pressure region, a net force is created. There is more pressure pushing from behind the fluid than there is in front of it. This difference in pressure accelerates the fluid along its path, or streamline. Consequently, the highest speeds are found where the pressure is lowest. Conversely, the lowest speeds occur where the pressure is highest.

Venturi5
Venturi5

The history of this discovery began with the Swiss mathematician Daniel Bernoulli. He published his findings in his book, Hydrodynamica, in 1738. Bernoulli's original work focused largely on the behavior of liquids. While he deduced the relationship between pressure and speed, the mathematical form we use today came later. In 1752, the scientist Leonhard Euler derived Bernoulli's equation in its standard mathematical form. This equation allows scientists to calculate the exact energy balance within a flow.

Scientists categorize flows to determine how to apply the equation. Most liquid flows and slow-moving gases are considered incompressible flows. In an incompressible flow, the density of the fluid remains constant even if the pressure changes. For these flows, the equation uses several specific variables. These include the fluid flow speed, acceleration due to gravity, and the elevation of a point. It also includes the static pressure and the density of the fluid. If the fluid is moving very fast, such as at high Mach numbers, scientists must use more advanced forms for compressible flows.

Jet gripper air-flow
Jet gripper air-flow

In practical applications, the equation is often simplified. For example, in aerodynamics, the change in height is often so small that it can be ignored. This leads to a version where total pressure is the sum of static pressure and dynamic pressure. Static pressure is the actual pressure associated with the state of the fluid. Dynamic pressure is the pressure caused by the fluid's motion. The sum of these two is known as the total pressure. In many steady flows, such as an aircraft in flight, the total pressure remains constant.

Cloud over A340 wing
Cloud over A340 wing

There are important limits to when this principle applies. The principle is most accurate for isentropic flows. These are flows where irreversible processes like turbulence are small enough to neglect. It also assumes the flow is irrotational, meaning it does not swirl in a way that changes the energy. While it works well for large bodies of fluid moving past solid objects, it does not apply well in boundary layers. For instance, the principle fails in long pipes where friction from viscous forces dominates. In liquids, if the pressure becomes too low, a phenomenon called cavitation can occur.

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🖼️ Images & Media (7)
File:VenturiFlow.png
VenturiFlow.png
Venturi Tube en.webm
File:Venturi5.svg
Venturi5.svg
File:BernoullisLawDerivationDiagram.svg
BernoullisLawDerivationDiagram.svg
File:Cloud over A340 wing.JPG
Cloud over A340 wing.JPG
File:Jet_gripper_air-flow.jpg
Jet_gripper_air-flow.jpg
File:Equal transit-time NASA wrong1 en.svg
Equal transit-time NASA wrong1 en.svg
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