Bournoli's Theorem

Topics: Fluid dynamics, Bernoulli's principle, Energy Pages: 5 (1202 words) Published: August 27, 2013
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: [pic]
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 constant-density 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[1]:

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