Bernoulli effect

Dictionary: Bernoulli effect

n.
The phenomenon of internal pressure reduction with increased stream velocity in a fluid.

[After Daniel BERNOULLI.]

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Britannica Concise Encyclopedia: Bernoulli's principle

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Principle that relates pressure, velocity, and height for a nonviscous fluid with steady flow. A consequence is that, for horizontal flow, as the speed of a fluid increases, the pressure it exerts decreases. Derived by Daniel Bernoulli (see Bernoulli family), the principle explains the lift of an airplane in motion. As the speed of the plane increases, air flows faster over the curved top of the wing than underneath. The upward pressure exerted by the air under the wing is thus greater than the pressure exerted downward above the wing, resulting in a net upward force, or lift. Race cars use the principle to keep their wheels pressed to the ground as they accelerate. A race car's spoiler — shaped like an upside-down wing, with the curved surface at the bottom — produces a net downward force.

For more information on Bernoulli's principle, visit Britannica.com.

Science of Everyday Things: Bernoulli's Principle

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Concept

Bernoulli's principle, sometimes known as Bernoulli's equation, holds that for fluids in an ideal state, pressure and density are inversely related: in other words, a slow-moving fluid exerts more pressure than a fast-moving fluid. Since "fluid" in this context applies equally to liquids and gases, the principle has as many applications with regard to airflow as to the flow of liquids. One of the most dramatic everyday examples of Bernoulli's principle can be found in the airplane, which stays aloft due to pressure differences on the surface of its wing; but the truth of the principle is also illustrated in something as mundane as a shower curtain that billows inward.

How It Works

The Swiss mathematician and physicist Daniel Bernoulli (1700-1782) discovered the principle that bears his name while conducting experiments concerning an even more fundamental concept: the conservation of energy. This is a law of physics that holds that a system isolated from all outside factors maintains the same total amount of energy, though energy transformations from one form to another take place.

For instance, if you were standing at the top of a building holding a baseball over the side, the ball would have a certain quantity of potential energy—the energy that an object possesses by virtue of its position. Once the ball is dropped, it immediately begins losing potential energy and gaining kinetic energy—the energy that an object possesses by virtue of its motion. Since the total energy must remain constant, potential and kinetic energy have an inverse relationship: as the value of one variable decreases, that of the other increases in exact proportion.

The ball cannot keep falling forever, losing potential energy and gaining kinetic energy. In fact, it can never gain an amount of kinetic energy greater than the potential energy it possessed in the first place. At the moment before the ball hits the ground, its kinetic energy is equal to the potential energy it possessed at the top of the building. Correspondingly, its potential energy is zero—the same amount of kinetic energy it possessed before it was dropped.

Then, as the ball hits the ground, the energy is dispersed. Most of it goes into the ground, and depending on the rigidity of the ball and the ground, this energy may cause the ball to bounce. Some of the energy may appear in the form of sound, produced as the ball hits bottom, and some will manifest as heat. The total energy, however, will not be lost: it will simply have changed form.

Bernoulli was one of the first scientists to propose what is known as the kinetic theory of gases: that gas, like all matter, is composed of tiny molecules in constant motion. In the 1730s, he conducted experiments in the conservation of energy using liquids, observing how water flows through pipes of varying diameter. In a segment of pipe with a relatively large diameter, he observed, water flowed slowly, but as it entered a segment of smaller diameter, its speed increased.

It was clear that some force had to be acting on the water to increase its speed. Earlier, Robert Boyle (1627-1691) had demonstrated that pressure and volume have an inverse relationship, and Bernoulli seems to have applied Boyle's findings to the present situation. Clearly the volume of water flowing through the narrower pipe at any given moment was less than that flowing through the wider one. This suggested, according to Boyle's law, that the pressure in the wider pipe must be greater.

As fluid moves from a wider pipe to a narrower one, the volume of that fluid that moves a given distance in a given time period does not change. But since the width of the narrower pipe is smaller, the fluid must move faster in order to achieve that result. One way to illustrate this is to observe the behavior of a river: in a wide, unconstricted region, it flows slowly, but if its flow is narrowed by canyon walls (for instance), then it speeds up dramatically.

The above is a result of the fact that water is a fluid, and having the characteristics of a fluid, it adjusts its shape to fit that of its container or other solid objects it encounters on its path. Since the volume passing through a given length of pipe during a given period of time will be the same, there must be a decrease in pressure. Hence Bernoulli's conclusion: the slower the rate of flow, the higher the pressure, and the faster the rate of flow, the lower the pressure.

Bernoulli published the results of his work in Hydrodynamica (1738), but did not present his ideas or their implications clearly. Later, his friend the German mathematician Leonhard Euler (1707-1783) generalized his findings in the statement known today as Bernoulli's principle.

The Venturi Tube

Also significant was the work of the Italian physicist Giovanni Venturi (1746-1822), who is credited with developing the Venturi tube, an instrument for measuring the drop in pressure that takes place as the velocity of a fluid increases. It consists of a glass tube with an inward-sloping area in the middle, and manometers, devices for measuring pressure, at three places: the entrance, the point of constriction, and the exit. The Venturi meter provided a consistent means of demonstrating Bernoulli's principle.

Like many propositions in physics, Bernoulli's principle describes an ideal situation in the absence of other forces. One such force is viscosity, the internal friction in a fluid that makes it resistant to flow. In 1904, the German physicist Ludwig Prandtl (1875-1953) was conducting experiments in liquid flow, the first effort in well over a century to advance the findings of Bernoulli and others. Observing the flow of liquid in a tube, Prandtl found that a tiny portion of the liquid adheres to the surface of the tube in the form of a thin film, and does not continue to move. This he called the viscous boundary layer.

Like Bernoulli's principle itself, Prandtl's findings would play a significant part in aerodynamics, or the study of airflow and its principles. They are also significant in hydrodynamics, or the study of water flow and its principles, a discipline Bernoulli founded.

Laminar Vs. Turbulent Flow

Air and water are both examples of fluids, substances which—whether gas or liquid—conform to the shape of their container. The flow patterns of all fluids may be described in terms either of laminar flow, or of its opposite, turbulent flow.

Laminar flow is smooth and regular, always moving at the same speed and in the same direction. Also known as streamlined flow, it is characterized by a situation in which every particle of fluid that passes a particular point follows a path identical to all particles that passed that point earlier. A good illustration of laminar flow is what occurs when a stream flows around a twig.

By contrast, in turbulent flow, the fluid is subject to continual changes in speed and direction—as, for instance, when a stream flows over shoals of rocks. Whereas the mathematical model of laminar flow is rather straightforward, conditions are much more complex in turbulent flow, which typically occurs in the presence of obstacles or high speeds.

Turbulent flow makes it more difficult for two streams of air, separated after hitting a barrier, to rejoin on the other side of the barrier; yet that is their natural tendency. In fact, if a single air current hits an airfoil—the design of an airplane's wing when seen from the end, a streamlined shape intended to maximize the aircraft's response to airflow—the air that flows over the top will "try" to reach the back end of the airfoil at the same time as the air that flows over the bottom. In order to do so, it will need to speed up—and this, as will be shown below, is the basis for what makes an airplane fly.

When viscosity is absent, conditions of perfect laminar flow exist: an object behaves in complete alignment with Bernoulli's principle. Of course, though ideal conditions seldom occur in the real world, Bernoulli's principle provides a guide for the behavior of planes in flight, as well as a host of everyday things.

Real-Life Applications

Flying Machines

For thousands of years, human beings vainly sought to fly "like a bird," not realizing that this is literally impossible, due to differences in physiognomy between birds and homo sapiens. No man has ever been born (or ever will be) who possesses enough strength in his chest that he could flap a set of attached wings and lift his body off the ground. Yet the bird's physical structure proved highly useful to designers of practical flying machines.

A bird's wing is curved along the top, so that when air passes over the wing and divides, the curve forces the air on top to travel a greater distance than the air on the bottom. The tendency of airflow, as noted earlier, is to correct for the presence of solid objects and to return to its original pattern as quickly as possible. Hence, when the air hits the front of the wing, the rate of flow at the top increases to compensate for the greater distance it has to travel than the air below the wing. And as shown by Bernoulli, fast-moving fluid exerts less pressure than slow-moving fluid; therefore, there is a difference in pressure between the air below and the air above, and this keeps the wing aloft.

Only in 1853 did Sir George Cayley (1773-1857) incorporate the avian airfoil to create history's first workable (though engine-less) flying machine, a glider. Much, much older than Cayley's glider, however, was the first manmade flying machine built "according to Bernoulli's principle"—only it first appeared in about 12,000

B.C., and the people who created it had had little contact with the outside world until the late eighteenth century A.D. This was the boomerang, one of the most ingenious devices ever created by a stone-age society—in this case, the Aborigines of Australia.

Contrary to the popular image, a boomerang flies through the air on a plane perpendicular to the ground, rather than parallel. Hence, any thrower who properly knows how tosses the boomerang not with a side-arm throw, but overhand. As it flies, the boomerang becomes both a gyroscope and an airfoil, and this dual role gives it aerodynamic lift.

Like the gyroscope, the boomerang imitates a top; spinning keeps it stable. It spins through the air, its leading wing (the forward or upward wing) creating more lift than the other wing. As an airfoil, the boomerang is designed so that the air below exerts more pressure than the air above, which keeps it airborne.

Another very early example of a flying machine using Bernoulli's principles is the kite, which first appeared in China in about 1000 B.C. The kite's design, particularly its use of lightweight fabric stretched over two crossed strips of very light wood, makes it well-suited for flight, but what keeps it in the air is a difference in air pressure. At the best possible angle of attack, the kite experiences an ideal ratio of pressure from the slower-moving air below versus the faster-moving air above, and this gives it lift.

Later Cayley studied the operation of the kite, and recognized that it—rather than the balloon, which at first seemed the most promising apparatus for flight—was an appropriate model for the type of heavier-than-air flying machine he intended to build. Due to the lack of a motor, however, Cayley's prototypical airplane could never be more than a glider: a steam engine, then state-of-the-art technology, would have been much too heavy.

Hence, it was only with the invention of the internal-combustion engine that the modern airplane came into being. On December 17, 1903, at Kitty Hawk, North Carolina, Orville (1871-1948) and Wilbur (1867-1912) Wright tested a craft that used a 25-horsepower engine they had developed at their bicycle shop in Ohio. By maximizing the ratio of power to weight, the engine helped them overcome the obstacles that had dogged recent attempts at flight, and by the time the day was over, they had achieved a dream that had eluded men for more than four millennia.

Within fifty years, airplanes would increasingly obtain their power from jet rather than internal-combustion engines. But the principle that gave them flight, and the principle that kept them aloft once they were airborne, reflected back to Bernoulli's findings of more than 160 years before their time. This is the concept of the airfoil.

As noted earlier, an airfoil has a streamlined design. Its shape is rather like that of an elongated, asymmetrical teardrop lying on its side, with the large end toward the direction of airflow, and the narrow tip pointing toward the rear. The greater curvature of its upper surface in comparison to the lower side is referred to as the airplane's camber. The front end of the airfoil is also curved, and the chord line is an imaginary straight line connecting the spot where the air hits the front—known as the stagnation point—to the rear, or trailing edge, of the wing.

Again, in accordance with Bernoulli's principle, the shape of the airflow facilitates the spread of laminar flow around it. The slower-moving currents beneath the airfoil exert greater pressure than the faster currents above it, giving lift to the aircraft. Of course, the aircraft has to be moving at speeds sufficient to gain momentum for its leap from the ground into the air, and here again, Bernoulli's principle plays a part.

Thrust comes from the engines, which run the propellers—whose blades in turn are designed as miniature airfoils to maximize their power by harnessing airflow. Like the aircraft wings, the blades' angle of attack—the angle at which airflow hits it. In stable flight, the pilot greatly increases the angle of attack (also called pitched), whereas at takeoff and landing, the pitch is dramatically reduced.

Drawing Fluids Upward: Atomizers and Chimneys

A number of everyday objects use Bernoulli's principle to draw fluids upward, and though in terms of their purposes, they might seem very different—for instance, a perfume atomizer vs. a chimney—they are closely related in their application of pressure differences. In fact, the idea behind an atomizer for a perfume spray bottle can also be found in certain garden-hose attachments, such as those used to provide a high-pres-sure car wash.

The air inside the perfume bottle is moving relatively slowly; therefore, according to Bernoulli's principle, its pressure is relatively high, and it exerts a strong downward force on the perfume itself. In an atomizer there is a narrow tube running from near the bottom of the bottle to the top. At the top of the perfume bottle, it opens inside another tube, this one perpendicular to the first tube. At one end of the horizontal tube is a simple squeeze-pump which causes air to flow quickly through it. As a result, the pressure toward the top of the bottle is reduced, and the perfume flows upward along the vertical tube, drawn from the area of higher pressure at the bottom. Once it is in the upper tube, the squeeze-pump helps to eject it from the spray nozzle.

A carburetor works on a similar principle, though in that case the lower pressure at the top draws air rather than liquid. Likewise a chimney draws air upward, and this explains why a windy day outside makes for a better fire inside. With wind blowing over the top of the chimney, the air pressure at the top is reduced, and tends to draw higher-pressure air from down below.

The upward pull of air according to the Bernoulli principle can also be illustrated by what is sometimes called the "Hoover bugle"—a name perhaps dating from the Great Depression, when anything cheap or contrived bore the appellation "Hoover" as a reflection of popular dissatisfaction with President Herbert Hoover. In any case, the Hoover bugle is simply a long corrugated tube that, when swung overhead, produces musical notes.

You can create a Hoover bugle using any sort of corrugated tube, such as vacuum-cleaner hose or swimming-pool drain hose, about 1.8 in (4 cm) in diameter and 6 ft (1.8 m) in length. To operate it, you should simply hold the tube in both hands, with extra length in the leading hand—that is, the right hand, for most people. This is the hand with which to swing the tube over your head, first slowly and then faster, observing the changes in tone that occur as you change the pace.

The vacuum hose of a Hoover tube can also be returned to a version of its original purpose in an illustration of Bernoulli's principle. If a piece of paper is torn into pieces and placed on a table, with one end of the tube just above the paper and the other end spinning in the air, the paper tends to rise. It is drawn upward as though by a vacuum cleaner—but in fact, what makes it happen is the pressure difference created by the movement of air.

In both cases, reduced pressure draws air from the slow-moving region at the bottom of the tube. In the case of the Hoover bugle, the corrugations produce oscillations of a certain frequency. Slower speeds result in slower oscillations and hence lower frequency, which produces a lower tone. At higher speeds, the opposite is true. There is little variation in tones on a Hoover bugle: increasing the velocity results in a frequency twice that of the original, but it is difficult to create enough speed to generate a third tone.

Spin, Curve, and Pull: the Counterintuitive Principle

There are several other interesting illustrations—sometimes fun and in one case potentially tragic—of Bernoulli's principle. For instance, there is the reason why a shower curtain billows inward once the shower is turned on. It would seem logical at first that the pressure created by the water would push the curtain outward, securing it to the side of the bathtub.

Instead, of course, the fast-moving air generated by the flow of water from the shower creates a center of lower pressure, and this causes the curtain to move away from the slower-moving air outside. This is just one example of the ways in which Bernoulli's principle creates results that, on first glance at least, seem counterintuitive—that is, the opposite of what common sense would dictate.

Another fascinating illustration involves placing two empty soft drink cans parallel to one another on a table, with a couple of inches or a few centimeters between them. At that point, the air on all sides has the same slow speed. If you were to blow directly between the cans, however, this would create an area of low pressure between them. As a result, the cans push together. For ships in a harbor, this can be a frightening prospect: hence, if two crafts are parallel to one another and a strong wind blows between them, there is a possibility that they may behave like the cans.

Then there is one of the most illusory uses of Bernoulli's principle, that infamous baseball pitcher's trick called the curve ball. As the ball moves through the air toward the plate, its velocity creates an air stream moving against the trajectory of the ball itself. Imagine it as two lines, one curving over the ball and one curving under, as the ball moves in the opposite direction.

In an ordinary throw, the effects of the airflow would not be particularly intriguing, but in this case, the pitcher has deliberately placed a "spin" on the ball by the manner in which he has thrown it. How pitchers actually produce spin is a complex subject unto itself, involving grip, wrist movement, and other factors, and in any case, the fact of the spin is more important than the way in which it was achieved.

If the direction of airflow is from right to left, the ball, as it moves into the airflow, is spinning clockwise. This means that the air flowing over the ball is moving in a direction opposite to the spin, whereas that flowing under it is moving in the same direction. The opposite forces produce a drag on the top of the ball, and this cuts down on the velocity at the top compared to that at the bottom of the ball, where spin and airflow are moving in the same direction.

Thus the air pressure is higher at the top of the ball, and as per Bernoulli's principle, this tends to pull the ball downward. The curve ball—of which there are numerous variations, such as the fade and the slider—creates an unpredictable situation for the batter, who sees the ball leave the pitcher's hand at one altitude, but finds to his dismay that it has dropped dramatically by the time it crosses the plate.

A final illustration of Bernoulli's often counterintuitive principle neatly sums up its effects on the behavior of objects. To perform the experiment, you need only an index card and a flat surface. The index card should be folded at the ends so that when the card is parallel to the surface, the ends are perpendicular to it. These folds should be placed about half an inch (about one centimeter) from the ends.

At this point, it would be handy to have an unsuspecting person—someone who has not studied Bernoulli's principle—on the scene, and challenge him or her to raise the card by blowing under it. Nothing could seem easier, of course: by blowing under the card, any person would naturally assume, the air will lift it. But of course this is completely wrong according to Bernoulli's principle. Blowing under the card, as illustrated, will create an area of high velocity and low pressure. This will do nothing to lift the card: in fact, it only pushes the card more firmly down on the table.

Where to Learn More

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

"Bernoulli's Principle: Explanations and Demos." (Web site). <http://207.10.97.102/physicszone/lesson/02forces/bernoull/bernoull.html> (February 22, 2001).

Cockpit Physics (Department of Physics, United States Air Force Academy. Web site.). <http://www.usafa.af.mil/dfp/cockpit-phys/> (February 19, 2001).

K8AIT Principles of Aeronautics Advanced Text. (Web site). <http://wings.ucdavis.edu/Book/advanced.html> (February 19, 2001).

Schrier, Eric and William F. Allman. Newton at the Bat: The Science in Sports. New York: Charles Scribner's Sons, 1984.

Smith, H. C. The Illustrated Guide to Aerodynamics. Blue Ridge Summit, PA: Tab Books, 1992.

Stever, H. Guyford, James J. Haggerty, and the Editors of Time-Life Books. Flight. New York: Time-Life Books, 1965.

Sports Science and Medicine: Bernoulli effect

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A hydrodynamic effect due to the relationship between relative velocity and relative pressure, which acts on an object as it moves through a fluid. The pressure exerted on the object by the fluid decreases as the velocity of the fluid increases. Thus, regions of relative high velocity fluid flow are associated with regions of relative low pressure and conversely regions of relative low velocity fluid flow are associated with regions of relative high pressure. The Bernoulli effect acts on balls and other projectiles in flight. When a region of relative high pressure is created on the under surface of a projectile and a region of relative low pressure on the upper surface, the result is a lift force directed perpendicular to the projectile from the high pressure side to the low pressure side.

Columbia Encyclopedia: Bernoulli's principle

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Bernoulli's principle, physical principle formulated by Daniel Bernoulli that states that as the speed of a moving fluid (liquid or gas) increases, the pressure within the fluid decreases. The phenomenon described by Bernoulli's principle has many practical applications; it is employed in the carburetor and the atomizer, in which air is the moving fluid, and in the aspirator, in which water is the moving fluid. In the first two devices air moving through a tube passes through a constriction, which causes an increase in speed and a corresponding reduction in pressure. As a result, liquid is forced up into the air stream (through a narrow tube that leads from the body of the liquid to the constriction) by the greater atmospheric pressure on the surface of the liquid. In the aspirator air is drawn into a stream of water as the water flows through a constriction. Bernoulli's principle can be explained in terms of the law of conservation of energy (see conservation laws, in physics). As a fluid moves from a wider pipe into a narrower pipe or a constriction, a corresponding volume must move a greater distance forward in the narrower pipe and thus have a greater speed. At the same time, the work done by corresponding volumes in the wider and narrower pipes will be expressed by the product of the pressure and the volume. Since the speed is greater in the narrower pipe, the kinetic energy of that volume is greater. Then, by the law of conservation of energy, this increase in kinetic energy must be balanced by a decrease in the pressure-volume product, or, since the volumes are equal, by a decrease in pressure.

Veterinary Dictionary: Bernoulli principle

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A principle which relates to the flow of fluids through tubes: the total hydraulic energy of fluid moving along a tube is constant so that if the tube dilates and causes the velocity to decrease, the kinetic energy involved in moving the fluid is reduced resulting in an increase in the lateral pressure on the vessel wall. This accounts for the tendency for aneurysms to enlarge and for vessels to dilate below a constriction.

Wikipedia: Bernoulli's principle

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This article is about Bernoulli's principle and Bernoulli's equation in fluid dynamics. For an unrelated topic in ordinary differential equations, see Bernoulli differential equation .

A flow of air into a venturi meter. The kinetic energy increases at the expense of the fluid pressure, as shown by the difference in height of the two columns of water.

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.^[1]^[2] Bernoulli's principle is named for the Dutch-Swiss mathematician and physicist Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.^[3]

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle can be derived from the principle of conservation of energy. This states that in a steady flow the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. If the fluid is flowing out of a reservoir the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit mass (the sum of pressure and gravitational potential ρ g h) is the same everywhere.^[4]

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

1 Incompressible flow equation
2 Compressible flow equation
- 2.1 Compressible flow in fluid dynamics
- 2.2 Compressible flow in thermodynamics
3 Derivations of Bernoulli equation
4 Real world application
5 Misunderstandings about the generation of lift
6 References
7 Notes
8 Further reading
9 See also
10 External links

Incompressible flow equation

In most flows of liquids, and of gases at low Mach number, the mass density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. For this reason the fluid in such flows can be considered to be incompressible and these flows can be described as incompressible flow. Bernoulli performed his experiments on liquids and his equation in its original form is valid only for incompressible flow.

The original form of Bernoulli's equation^[5] -- valid at any point along a streamline -- is:

${v^2 \over 2}+\Psi+{p\over\rho}=\text{constant}$

where:

v is the fluid flow speed at a point on a streamline,

Ψ is the gravitational potential,

p is the pressure at the point, and

ρ is the density of the fluid at all points in the fluid.

The following assumptions must be met for the equation to apply:

The fluid must be incompressible - even though pressure varies, the density must remain constant.
The streamline must not enter a boundary layer. (Bernoulli's equation is not applicable where there are viscous forces, such as in a boundary layer.)

If the acceleration due to gravity (g) does not change over the length scale of the problem (eg the situation occurs at an elevation which is low compared with the radius of the Earth) then Ψ = g z, where z is the elevation of the point above some reference plane (positive z-direction points upward, in the opposite direction to the gravitational acceleration). This will be assumed for the rest of this section.

By multiplying with the mass density ρ, the above equation can be rewritten as:

$\tfrac12\, \rho\, v^2\, +\, \rho\, g\, z\, +\, p\, =\, \text{constant}\,$

or:

$q\, +\, \rho\, g\, h\, =\, p_0\, +\, \rho\, g\, z\, =\, \text{constant}\,$

where:

$q\, =\, \tfrac12\, \rho\, v^2$ is dynamic pressure,

$h\, =\, z\, +\, \frac{p}{\rho g}$ is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head)^[6]^[7] and

$p_0\, =\, p\, +\, q\,$ is the total pressure (the sum of the static pressure p and dynamic pressure q).^[8]

The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head H:

$H\, =\, z\, +\, \frac{p}{\rho g}\, +\, \frac{v^2}{2\,g}\, =\, h\, +\, \frac{v^2}{2\,g},$ so divide the above constant by ρ and g to get the total head H in terms of metres of fluid column.^[7]^[6]

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids -- when the pressure becomes too low -- cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquids, the changes in mass density become significant so that the assumption of constant density is invalid.

Simplified form

In many applications of Bernoulli's equation, the change in the ρ g z term along the streamline is so small compared with the other terms it can be ignored. For example, in the case of aircraft in flight, the change in height z along a streamline is so small the ρ g z term can be omitted. This allows the above equation to be presented in the following simplified form:

$p + q = p_0\,$

where p₀ is called total pressure, and q is dynamic pressure^[9]. Many authors refer to the pressure p as static pressure to distinguish it from total pressure p₀ and dynamic pressure q. In Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."^[10]

The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:

static pressure + dynamic pressure = total pressure^[10]

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure p and dynamic pressure q. Their sum p + q is defined to be the total pressure p₀. The significance of Bernoulli's principle can now be summarized as total pressure is constant along a streamline.

If the fluid flow is irrotational, the total pressure on every streamline is the same and Bernoulli's principle can be summarized as total pressure is constant everywhere in the fluid flow.^[11] It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in open bodies of water. However, it is important to remember that Bernoulli's principle does not apply in the boundary layer or in fluid flow through long pipes.

Applicability of incompressible flow equation to flow of gases

Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation -- in its incompressible flow form -- can not be assumed to be valid. However if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas, (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature, however this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle, or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.

Unsteady potential flow

The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics.

For an irrotational flow, the flow velocity can be described as the gradient ∇φ of a velocity potential φ. In that case, and for a constant density ρ, the momentum equations of the Euler equations can be integrated to:^[12]

$\frac{\partial \varphi}{\partial t} + \tfrac{1}{2} v^2 + \frac{p}{\rho} + gz = f(t),$

which is a Bernoulli equation valid also for unsteady -- or time dependent -- flows. Here ∂φ/∂t denotes the partial derivative of the velocity potential φ with respect to time t, and v = |∇φ| is the flow speed. The function f(t) depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment t does not only apply along a certain streamline, but in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case f is a constant.^[12]

Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation

$\Phi=\varphi-\int_{t_0}^t f(\tau)\, \operatorname{d}\tau$ resulting in $\frac{\partial \Phi}{\partial t} + \tfrac{1}{2} v^2 + \frac{p}{\rho} + gz=0.$

Note that the relation of the potential to the flow velocity is unaffected by this transformation: ∇Φ = ∇φ.

The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian (not to be confused with Lagrangian coordinates).

Lift and Drag curves for a typical airfoil

Compressible flow equation

Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and compressible fluids at very low speeds (perhaps up to 1/3 of the sound speed in the fluid). It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.

Compressible flow in fluid dynamics

For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,

$\frac {v^2}{2}+ \int_{p_1}^p \frac {d\tilde{p}}{\rho(\tilde{p})}\ + \Psi = \text{constant}$ ^[13] (constant along a streamline)

where:

p is the pressure

ρ is the density

v is the flow speed

Ψ is the potential associated with the conservative force field, often the gravitational potential

In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation becomes

$\frac {v^2}{2}+ gz+\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \text{constant}$ ^[14] (constant along a streamline)

where, in addition to the terms listed above:

γ is the ratio of the specific heats of the fluid

g is the acceleration due to gravity

z is the elevation of the point above a reference plane

In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term gz can be omitted. A very useful form of the equation is then:

$\frac {v^2}{2}+\left( \frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}$

where:

p₀ is the total pressure

ρ₀ is the total density

Compressible flow in thermodynamics

Another useful form of the equation, suitable for use in thermodynamics, is:

${v^2 \over 2} + \Psi + w =\text{constant}.$ ^[15]

Here w is the enthalpy per unit mass, which is also often written as h (not to be confused with "head" or "height").

Note that $w = \epsilon + \frac{p}{\rho}$ where ε is the thermodynamic energy per unit mass, also known as the specific internal energy or "sie."

The constant on the right hand side is often called the Bernoulli constant and denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of energy, b is constant along any given streamline. More generally, when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When the change in Ψ can be ignored, a very useful form of this equation is:

${v^2 \over 2}+ w = w_0$

where w₀ is total enthalpy.

When shock waves are present, in a reference frame moving with a shock, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

Derivations of Bernoulli equation

Bernoulli equation for incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects.

The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe.

The equation of motion for a parcel of fluid, having a length dx, mass density ρ, mass m = ρ A dx and flow velocity v = dx / dt, moving along the axis of the horizontal pipe, with cross-sectional area A is

$m \frac{\operatorname{d}v}{\operatorname{d}t}= F$

$\rho A \operatorname{d}x \frac{\operatorname{d}v}{\operatorname{d}t}= -A \operatorname{d}p$

$\rho \frac{\operatorname{d}v}{\operatorname{d}t}= -\frac{\operatorname{d}p}{\operatorname{d}x}$

In steady flow, v = v(x) so

$\frac{\operatorname{d}v}{\operatorname{d}t}= \frac{\operatorname{d}v}{\operatorname{d}x}\frac{\operatorname{d}x}{\operatorname{d}t} = \frac{\operatorname{d}v}{\operatorname{d}x}v=\frac{d}{\operatorname{d}x} \left( \frac{v^2}{2} \right).$

With density ρ constant, the equation of motion can be written as

$\frac{\operatorname{d}}{\operatorname{d}x} \left( \rho \frac{v^2}{2} + p \right) =0$

$\frac{v^2}{2} + \frac{p}{\rho}= C$

where C is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. The deduction is: where the speed is large, pressure is low and vice versa.

In the above derivation, no external work-energy principle is invoked. Rather, Bernoulli's principle was inherently derived by a simple manipulation of the momentum equation.

A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (s), and cross-sectional area. Note that in this figure elevation is denoted as h, contrary to the text where it is given by z.

Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.^[16] In the form of the work-energy theorem, stating that^[17]

the change in the kinetic energy E_kin of the system equals the net work W done on the system;

$W = \Delta E_\text{kin}. \;$

Therefore,

the work done by the forces in the fluid = increase in kinetic energy.

The system consists of the volume of fluid, initially between the cross-sections A₁ and A₂. In the time interval Δt fluid elements initially at the inflow cross-section A₁ move over a distance s₁ = v₁ Δt, while at the outflow cross-section the fluid moves away from cross-section A₂ over a distance s₂ = v₂ Δt. The displaced fluid volumes at the inflow and outflow are respectively A₁ s₁ and A₂ s₂. The associated displaced fluid masses are -- when ρ is the fluid's mass density -- equal to density times volume, so ρ A₁ s₁ and ρ A₂ s₂. By mass conservation, these two masses displaced in the time interval Δt have to be equal, and this displaced mass is denoted by Δm:

$\begin{align} \rho A_1 s_1 &= \rho A_{1} v_{1} \Delta t = \Delta m, \\ \rho A_2 s_2 &= \rho A_{2} v_{2} \Delta t = \Delta m. \end{align}$

The work done by the forces consists of two parts:

The work done by the pressure acting on the area's A₁ and A₂

$W_\text{pressure}=F_{1,\text{pressure}}\; s_{1}\, -\, F_{2,\text{pressure}}\; s_{2}=p_{1} A_{1} s_1 - p_{2} A_{2} s_{2} = \Delta m\, \frac{p_1}{\rho} - \Delta m\, \frac{p_2}{\rho}. \;$

The work done by gravity: the gravitational potential energy in the volume A₁ s₁ is lost, and at the outflow in the volume A₂ s₂ is gained. So, the change in gravitational potential energy ΔE_pot,gravity in the time interval Δt is

$\Delta E_\text{pot,gravity} = \Delta m\, g z_{2} - \Delta m\, g z_{1}. \;$

Now, the work by the force of gravity is opposite to the change in potential energy, W_gravity = −ΔE_pot,gravity: while the force of gravity is in the negative z-direction, the work--gravity force times change in elevation--will be negative for a positive elevation change Δz = z₂ − z₁, while the corresponding potential energy change is positive.^[18] So:

$W_\text{gravity} = -\Delta E_\text{pot,gravity} = \Delta m\, g z_{1} - \Delta m\, g z_{2}. \;$

And the total work done in this time interval $Δ t$ is

$W = W_\text{pressure} + W_\text{gravity}. \,$

The increase in kinetic energy is

$\Delta E_\text{kin} = \frac{1}{2} \Delta m\, v_{2}^{2}-\frac{1}{2} \Delta m\, v_{1}^{2}.$

Putting these together, the work-kinetic energy theorem W = ΔE_kin gives:^[16]

$\Delta m\, \frac{p_{1}}{\rho} - \Delta m\, \frac{p_{2}}{\rho} + \Delta m\, g z_{1} - \Delta m\, g z_{2} = \frac{1}{2} \Delta m\, v_{2}^{2} - \frac{1}{2} \Delta m\, v_{1}^{2}$

$\frac12 \Delta m\, v_{1}^{2} + \Delta m\, g z_{1} + \Delta m\, \frac{p_{1}}{\rho} = \frac12 \Delta m\, v_{2}^{2} + \Delta m\, g z_{2} + \Delta m\, \frac{p_{2}}{\rho}.$

After dividing by the mass Δm = ρ A₁ v₁ Δt = ρ A₂ v₂ Δt the result is:^[16]

$\frac12 v_{1}^{2}+g z_{1}+\frac{p_{1}}{\rho}=\frac12 v_{2}^{2}+g z_{2}+\frac{p_{2}}{\rho}$

or, as stated in the first paragraph:

$\frac{v^{2}}{2}+g z+\frac{p}{\rho}=C$ (Eqn. 1)

Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle:

$\frac{v^{2}}{2 g}+z+\frac{p}{\rho g}=C$ (Eqn. 2a)

The middle term, z, represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, z is called the elevation head and given the designation z_elevation.

A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed

$v=\sqrt{{2 g}{z}},$ when arriving at elevation z=0. Or when we rearrange it as a head: $h_{v}=\frac{v^{2}}{2 g}$

The term v² / (2 g) is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion.

The hydrostatic pressure p is defined as

$p=p_0-\rho g z \,$ , with p₀ some reference pressure, or when we rearrange it as a head: $\psi=\frac{p}{\rho g}$

The term p / (ρg) is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container.

When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head.

$h_{v} + z_\text{elevation} + \psi = C\,$ (Eqn. 2b)

If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms:

$\frac{\rho v^{2}}{2}+ \rho g z + p=C$ (Eqn. 3)

We note that the pressure of the system is constant in this form of the Bernoulli Equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system.

Bernoulli equation for compressible fluids

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time Δt, the amount of mass passing through the boundary defined by the area A₁ is equal to the amount of mass passing outwards through the boundary defined by the area A₂:

$0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t$ .

Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by A₁ and A₂ is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero,

$0= \Delta E_1 - \Delta E_2 \,$

where ΔE₁ and ΔE₂ are the energy entering through A₁ and leaving through A₂, respectively.

The energy entering through A₁ is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical p dV work:

$\Delta E_1 = \left[\frac{1}{2} \rho_1 v_1^2 + \Psi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t$

where Ψ = gz is a force potential due to the Earth's gravity, g is acceleration due to gravity, and z is elevation above a reference plane.

A similar expression for $Δ E 2$ may easily be constructed. So now setting $0 = Δ E 1 - Δ E 2$ :

$0 = \left[\frac{1}{2} \rho_1 v_1^2+ \Psi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t - \left[ \frac{1}{2} \rho_2 v_2^2 + \Psi_2 \rho_2 + \epsilon_2 \rho_2 + p_2 \right] A_2 v_2 \, \Delta t$

which can be rewritten as:

$0 = \left[ \frac{1}{2} v_1^2 + \Psi_1 + \epsilon_1 + \frac{p_1}{\rho_1} \right] \rho_1 A_1 v_1 \, \Delta t - \left[ \frac{1}{2} v_2^2 + \Psi_2 + \epsilon_2 + \frac{p_2}{\rho_2} \right] \rho_2 A_2 v_2 \, \Delta t$

Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

$\frac{1}{2}v^2 + \Psi + \epsilon + \frac{p}{\rho} = {\rm constant} \equiv b$

which is the Bernoulli equation for compressible flow.

Real world application

In modern every-day life there are many observations that can be successfully explained by application of Bernoulli's principle.

Bernoulli's Principle can be used to calculate the lift force on an airfoil if you know the behavior of the fluid flow in the vicinity of the foil. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface then Bernoulli's principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lift force.^{[nb 1]}^[19] Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli's equations^[20] -- established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. Bernoulli's principle does not explain why the air flows faster past the top of the wing and slower past the under-side. To understand why, it is helpful to understand circulation, the Kutta condition and the Kutta-Joukowski theorem. Another way to explain why the air speeds up is to observe that the air is deflected downward in the vicinity of the wing. Newton's 1st & 3rd law imply that there is an upward force on the wing, which implies an area of low pressure on the top of the wing. When the air flows through this region of low pressure, it speeds up according to Bernoulli's principle.^[21] Of course, this does not explain why the air deflects downward.

The carburetor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburetor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli's principle - in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure.

The pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to the airspeed indicator which determines the dynamic pressure of the airflow past the aircraft. Dynamic pressure is the difference between stagnation pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure.^[22]

The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect.

The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation, and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli's law, showing that Torricelli's law is compatible with Bernoulli's principle. Viscosity lowers this drain rate. This is reflected in the discharge coefficient which is a function of the Reynold's number and the shape of the orifice.^[23]

The principle also makes it possible for sail-powered craft to travel faster than the wind that propels them (if friction can be sufficiently reduced). If the wind passing in front of the sail is fast enough to experience a significant reduction in pressure, the sail is pulled forward, in addition to being pushed from behind. Although boats in water must contend with the friction of the water along the hull, ice sailing and land sailing vehicles can travel faster than the wind.^[24]^[25]

Misunderstandings about the generation of lift

Main article: Lift (force)

Many explanations for the generation of lift (on airfoils, propeller blades, etc.) can be found; but some of these explanations can be misleading, and some are false. This has been a source of heated discussion over the years. In particular, there has been debate about whether lift is best explained by Bernoulli's principle or Newton's laws of motion. Modern writings agree that Bernoulli's principle and Newton's laws are both relevant and correct.^[26]^[27]

Several of these explanations use Bernoulli's principle to connect the flow kinematics to the flow-induced pressures. In case of incorrect (or partially correct) explanations of lift, also relying at some stage on Bernoulli's principle, the errors generally occur in the assumptions on the flow kinematics, and how these are produced. It is not Bernoulli's principle itself that is questioned because this principle is well established^[28]^[29]^[30]^[21].

References

^ Clancy, L.J., Aerodynamics, Chapter 3.
^ Batchelor, G.K. (1967), Section 3.5, pp. 156-64.
^ "Hydrodynamica". Britannica Online Encyclopedia. http://www.britannica.com/EBchecked/topic/658890/Hydrodynamica#tab=active~checked%2Citems~checked&title=Hydrodynamica%20--%20Britannica%20Online%20Encyclopedia. Retrieved 2008-10-30.
^ Streeter, V.L., Fluid Mechanics, Example 3.5, McGraw–Hill Inc. (1966), New York.
^ Clancy, L.J., Aerodynamics, Section 3.4.
^ ^a ^b Mulley, Raymond (2004), Flow of Industrial Fluids: Theory and Equations, CRC Press, ISBN 0849327679 , 410 pages. See pp. 43-44.
^ ^a ^b Chanson, Hubert (2004), Hydraulics of Open Channel Flow: An Introduction, Butterworth-Heinemann, ISBN 0750659785 , 650 pages. See p. 22.
^ Oertel, Herbert; Prandtl, Ludwig; Böhle, M.; Mayes, Katherine (2004), Prandtl's Essentials of Fluid Mechanics, Springer, pp. 70–71, ISBN 0387404376
^ "Bernoulli's Equation". NASA Glenn Research Center. http://www.grc.nasa.gov/WWW/K-12/airplane/bern.htm. Retrieved 2009-03-04.
^ ^a ^b Clancy, L.J., Aerodynamics, Section 3.5.
^ Clancy, L.J. Aerodynamics, Equation 3.12
^ ^a ^b Batchelor, G.K. (1967), p. 383
^ Clarke C. and Carswell B., Astrophysical Fluid Dynamics
^ Clancy, L.J., Aerodynamics, Section 3.11
^ Van Wylen, G.J., and Sonntag, R.E., (1965), Fundamentals of Classical Thermodynamics, Section 5.9, John Wiley and Sons Inc., New York
^ ^a ^b ^c Feynman, R.P.; Leighton, R.B.; Sands, M. (1963), The Feynman Lectures on Physics, ISBN 0-201-02116-1 , Vol. 2, §40-3, p. 40-6 -- 40-9.
^ Tipler, Paul (1991), Physics for Scientists and Engineers: Mechanics (3rd extended ed.), W. H. Freeman, ISBN 0-87901-432-6 , p. 138.
^ Feynman, R.P.; Leighton, R.B.; Sands, M. (1963), The Feynman Lectures on Physics, ISBN 0-201-02116-1 , Vol. 1, §14-3, p. 14-4.
^ Resnick, R. and Halliday, D. (1960), Physics, Section 18-5, John Wiley & Sons, Inc., New York ("[streamlines] are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.")
^ Eastlake, Charles N. Eastlake, "An Aerodynamicist’s View of Lift, Bernoulli, and Newton", THE PHYSICS TEACHER Vol. 40, March 2002, http://www.df.uba.ar/users/sgil/physics_paper_doc/papers_phys/fluids/Bernoulli_Newton_lift.pdf "The resultant force is determined by integrating the surface-pressure distribution over the surface area of the airfoil."
^ ^a ^b Weltner, Klaus; Ingelman-Sundberg, Martin, Physics of Flight - reviewed, http://user.uni-frankfurt.de/~weltner/Flight/PHYSIC4.htm "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause."
^ Clancy, L.J., Aerodynamics, Section 3.8
^ Mechanical Engineering Reference Manual Ninth Edition
^ Ice Sailing Manual, p. 2
^ Wind Sports – Ice sailing hand held sails
^ "Newton vs Bernoulli". http://www.grc.nasa.gov/WWW/K-12/airplane/bernnew.html.
^ Ison, David. Bernoulli Or Newton: Who's Right About Lift? Retrieved on 2008-05-21
^ Phillips, O.M. (1977). The dynamics of the upper ocean (2^nd edition ed.). Cambridge University Press. ISBN 0 521 29801 6. Section 2.4.
^ Batchelor, G.K. (1967). Sections 3.5 and 5.1
^ Lamb, H. (1994). §17-§29

Notes

^ Clancy, L.J., Aerodynamics, Section 5.5 ("When a stream of air flows past an airfoil, there are local changes in flow speed round the airfoil, and consequently changes in static pressure, in accordance with Bernoulli's Theorem. The distribution of pressure determines the lift, pitching moment and form drag of the airfoil, and the position of its centre of pressure.")

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Bernoulli effect

Contents

Incompressible flow equation

Simplified form

Applicability of incompressible flow equation to flow of gases

Unsteady potential flow

Compressible flow equation

Compressible flow in fluid dynamics

Compressible flow in thermodynamics

Derivations of Bernoulli equation

Real world application

Misunderstandings about the generation of lift

References

Notes

Further reading

See also

External links