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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.

Incompressible flow equation

In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flow. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is:

where:

(A)

is the fluid flow speed at a point on a streamline, is the acceleration due to gravity,

is the elevation of the point above a reference plane, with the positive

z-direction pointing upward – so in the direction opposite to the gravitational acceleration,

is the pressure at the chosen point, and

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

For conservative force fields, Bernoulli's equation can be generalized as:

where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply: the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;

 friction by viscous forces has to be negligible.



By multiplying with the fluid density , equation (A) can be rewritten as:

or:

where:

is dynamic pressure,

is the piezometric head or hydraulic head (the sum of the

elevation z and the pressure head) and

is the total pressure (the sum of the static pressure p and

dynamic pressure q).

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

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 liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

Real-world application

Condensation visible over the upper surface of a wing caused by the fall in

temperature accompanying the fall in pressure, both due to acceleration of the air. In modern everyday life there are many observations that can be successfully

explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid and a small viscosity often has a large effect on the flow.



Bernoulli's principle can be used to calculate the lift force on an airfoil if the behaviour of the fluid flow in the vicinity of the foil is known. 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 lifting force. 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 – 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 underside. To understand why, it is helpful to

understand circulation, the Kutta condition, and the Kutta–Joukowski theorem.



The Dyson Bladeless Fan (or Air Multiplier) is an implementation that takes advantage of the Venturi effect, Coandă effect and Bernoulli's Principle. 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. 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 Reynolds number and the shape of the orifice.

 In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and

its extension were recently (2009) developed. It was proved that the

depth-averaged specific energy reaches a minimum in converging accelerating free-surface flow over weirs and flumes . Further, in general, a channel control with minimum specific energy in curvilinear flow is not isolated from water waves, as customary state in open-channel hydraulics.

 

The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.

References

^ Clancy, L.J., Aerodynamics, Chapter 3.

^ a b Batchelor, G.K. (1967), Section 3.5, pp. 156–.

^ \"Hydrodynamica\". Britannica Online Encyclopedia.

http://www.britannica.com/EBchecked/topic/6580/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.

^ \"If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the x-direction) and if the particle has a finite size l, then the front of the particle will be ‘seeing’ a different pressure from the rear. More precisely, if the pressure drops in the x-direction (dp/dx < 0) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton’s second law, this force causes an acceleration and the particle’s velocity increases as it moves along the streamline... Bernoulli’s equation describes this mathematically (see the complete derivation in the appendix).\"Babinsky, Holger (November 2003), \"How do wings work?\Physics Education, http://www.iop.org/EJ/article/0031-9120/38/6/001/pe3_6_001.pdf

^ \"Acceleration of air is caused by pressure gradients. Air is accelerated in

direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity.\" Weltner, Klaus; Ingelman-Sundberg, Martin, Misinterpretations of Bernoulli's Law,

http://user.uni-frankfurt.de/~weltner/Mis6/mis6.html

^ \" The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down.

Conversely if the parcel is moving into a region of lower pressure, there will be an higher pressure behind it (higher than the pressure ahead), speeding it up. As

always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton’s laws of motion.\" See How It Flies John S. Denker http://www.av8n.com/how/htm/airfoils.html ^ a b Batchelor, G.K. (1967), §5.1, p. 265.

^ Mulley, Raymond (2004). Flow of Industrial Fluids: Theory and Equations. CRC Press. ISBN 0-8493-2767-9., 410 pages. See pp. 43–44.