THE 456th FIGHTER INTERCEPTOR SQUADRON 
THE PROTECTORS OF S. A. C. 

Bernoulli's Principle 
+ Larger Font   Smaller Font
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.
Incompressible flow
The original form, for incompressible flow in a uniform gravitational field, is:
where:
 v = fluid velocity along the streamline
 g = acceleration due to gravity
 h = height of the fluid
 p = pressure along the streamline
 ρ = density of the fluid
These assumptions must be met for the equation to apply:
 Inviscid flow − viscosity (internal friction) = 0
 Steady flow
 Incompressible flow − ρ = constant along a streamline. Density may vary from streamline to streamline, however.
 Generally, the equation applies along a streamline. For constantdensity potential flow, it applies throughout the entire flow field.
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.
A common example used to illustrate the effect of Bernoulli's principle is when air flows around an airplane wing; the velocity of the air is higher and the pressure is lower on the top surface of the wing when compared to the bottom surface. This pressure differential creates an upwards lift force on the wings making flight possible.
This can be rewritten as:
or:
 q + ρgh + p = constant
where:
 q = dynamic pressure
Compressible flow
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.
Derivations of Bernoulli equation
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 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 workenergy principle is invoked. Rather, the workenergy principle was inherently derived by a simple manipulation of the momentum equation. The derivation that follows includes gravity and applies to a curved trajectory, but a workenergy principle must be assumed.
Click on Picture to enlarge
A streamtube of fluid moving to the right. Indicated are pressure, height, velocity, distance (s), and crosssectional area. Applying conservation of energy in form of the workkinetic energy theorem we find that:
 the change in KE of the system equals the net work done on the system;
Therefore,
 the work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy.
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 A_{1}v_{1} (= rate of fluid flow = A_{2}v_{2} 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
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 A_{1} is equal to the amount of mass passing outwards through the boundary defined by the area A_{2}:
 .
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 A_{1} and A_{2} 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 ΔE_{1} and ΔE_{2} are the energy entering through A_{1} and leaving through A_{2}, respectively.
The energy entering through A_{1} 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:
A similar expression for ΔE_{2} may easily be constructed. So now setting 0 = ΔE_{1} − ΔE_{2}:
which can be rewritten as:
Now, using the previouslyobtained result from conservation of mass, this may be simplified to obtain
which is the Bernoulli equation for compressible flow.
Wikipedia
Last Updated 
02/10/2014 
Powered By 
456FIS.ORG 