Bernoulli effect
n.
The phenomenon of internal pressure reduction with increased stream velocity in a fluid.
[After Daniel BERNOULLI.]
|
Results for Bernoulli effect
|
On this page:
|
The phenomenon of internal pressure reduction with increased stream velocity in a fluid.
[After Daniel BERNOULLI.]
For more information on Bernoulli's principle, visit Britannica.com.
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.
| This article or section includes a list of references or a list of external
links, but its sources remain unclear because it lacks in-text citations. You can improve this article by introducing more precise citations. |
Bernoulli's Principle states that for an ideal fluid (low speed air is a good approximation), with no work being performed on the fluid, an increase in velocity occurs simultaneously with decrease in pressure or a change in the fluid's gravitational potential energy.
This principle is a simplification of Bernoulli's equation, which states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path. It is named after the Dutch/Swiss mathematician/scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others. In fluid flow with no viscosity, and therefore, one in which a pressure difference is the only accelerating force, the principle is equivalent to Newton's laws of motion.
Bernoulli's Principle was often cited as the primary cause of lift in aircraft wings. While it has some effect, the simple description (eg. "equal transit time" or "longer path" theories) provided in many elementary textbooks is incorrect. [1]
The original form, for incompressible flow in a uniform

where:
These assumptions must be met for the equation to apply:
An increase in velocity and the corresponding decrease in pressure, as shown by the equation, is often called Bernoulli's principle. The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler.
Bernoulli's equation can be used to calculate lift in cases where its assumptions are not violated. This requires knowledge of fluid streamlines and fluid speeds, which are not provided by Bernoulli's principle. [2]
This can be rewritten as[3]:

where:
A second, more general form of Bernoulli's equation may be written for compressible fluids, in which case, following a streamline:

= gravitational
potential energy per unit mass,
in the case of a uniform gravitational field
= fluid enthalpy per unit mass, which is also often written as
(which conflicts with the use of
in this article for "height"). Note that
where
is the fluid
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 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.
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 on the axis of the pipe is



In steady flow, v = v(x) so

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

or

where C is a constant, sometimes referred to as the Bernoulli constant. We deduce that where the speed is large, pressure is low. In the above derivation, no external work-energy principle is invoked. Rather, the work-energy principle was inherently derived by a simple manipulation of the momentum equation. The derivation that follows includes gravity and applies to angled trajectory, but a work-energy principle must be assumed.
Applying conservation of energy in form of the work-kinetic energy theorem we find that:

Therefore,
The work done by the forces is

The decrease of potential energy is

The increase in kinetic energy is

Putting these together,

or

After dividing by Δt, ρ and A1v1 (= rate of fluid flow = A2v2 as the fluid is incompressible):

or, as stated in the first paragraph:

Further division by g implies

A free falling mass from a height h (in vacuum), will reach a velocity
or
.The term
is called the velocity
head.
The hydrostatic pressure or static head is defined as
, or
.The term
is
also called the pressure head.
A way to see how this relates to conservation of energy directly is to multiply by density and by unit volume (which is allowed
since both are constant) yielding:
and
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:
.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,

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
work:
![\Delta E_1 = \left[ \frac{1}{2} \rho_1 v_1^2 + \phi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t](http://content.answers.com/main/content/wp/en/math/9/3/6/9364fdec271c653b66acf1071156bb5c.png)
A similar expression for ΔE2 may easily be constructed. So now setting 0 = ΔE1 - ΔE2:
![0 = \left[ \frac{1}{2} \rho_1 v_1^2+ \phi_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 + \phi_2\rho_2 + \epsilon_2 \rho_2 + p_2 \right] A_2 v_2 \, \Delta t](http://content.answers.com/main/content/wp/en/math/4/2/c/42ca460091570d3e1e4411e0f23f431c.png)
which can be rewritten as:
![0 = \left[ \frac{1}{2} v_1^2 + \phi_1 + \epsilon_1 + \frac{p_1}{\rho_1} \right] \rho_1 A_1 v_1 \, \Delta t - \left[ \frac{1}{2} v_2^2 + \phi_2 + \epsilon_2 + \frac{p_2}{\rho_2} \right] \rho_2 A_2 v_2 \, \Delta t](http://content.answers.com/main/content/wp/en/math/c/a/3/ca33bead05e8d1d09da0924e542c7923.png)
Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

which is the Bernoulli equation for compressible flow.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
Join the WikiAnswers Q&A community. Post a question or answer questions about "Bernoulli effect" at WikiAnswers.
Copyrights:
![]() | Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more | |
![]() | Britannica Concise Encyclopedia. Britannica Concise Encyclopedia. © 2006 Encyclopædia Britannica, Inc. All rights reserved. Read more | |
![]() | Sports Science and Medicine. The Oxford Dictionary of Sports Science & Medicine. Copyright © Michael Kent 1998, 2006, 2007. All rights reserved. Read more | |
![]() | Columbia Encyclopedia. The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/ Read more | |
![]() | Wikipedia. This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Bernoulli's principle". Read more |