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Encyclopedia > Bernoulli's equation

In fluid dynamics, Bernoulli's equation, derived by Daniel Bernoulli, describes the behavior of a fluid moving along a streamline. There are typically two different formulations of the equations; one applies to incompressible fluids and the other applies to compressible fluids. Fluid dynamics is the subdiscipline of fluid mechanics that studies fluids (liquids and gases) in motion. ... Daniel Bernoulli Daniel Bernoulli (Groningen, February 9, 1700 – Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ... A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids. ... In fluid dynamics, a streamline is a line which is everywhere tangent to the velocity of the flow. ...


The original form, for incompressible flow in a uniform graviational field (such as on Earth), is:

v = fluid velocity along the streamline
g = acceleration due to gravity on Earth
h = height from an arbitrary point in the direction of gravity
p = pressure along the streamline
ρ = fluid density

These assumptions must be met for the equation to apply: This article is about velocity in physics. ... g (also gee, g-force or g-load) is a non-SI unit of acceleration defined as exactly 9. ... Earth, also known as Terra, and Tellus mostly in the 19th century, is the third-closest planet to the Sun. ... Height is a measurement of the distance from the bottom to the top of something which is upright. ... It has been suggested that gravitation be merged into this article or section. ... YOU SUCK DICK Pressure (symbol: p) is the force per unit area acting on a surface in a direction perpendicular to that surface. ... Density (symbol: ρ - Greek: rho) is a measure of mass per unit of volume. ...

  • Inviscid flow − viscosity (internal friction) = 0
  • Steady flow
  • Incompressible flowρ = 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.

The decrease in pressure simultaneous with an increase in velocity, as predicted by the equation, is often called Bernoulli's principle. Viscosity is a measure of the resistance of a fluid to deformation under shear stress. ... The pitch drop experiment at the University of Queensland. ... Steady flow is a type of liquid flow in which The liquid flow is smooth and uniform. ... In fluid mechanics, an incompressible fluid is a fluid whose density (often represented by the Greek letter ρ) is constant: it is the same throughout the field and it does not change through time. ... In fluid dynamics, potential flow, also known as irrotational flow (of incompressible fluids) is steady flow defined by the equations (zero rotation) (zero divergence = volume conservation) Equivalently, where: v is the vector fluid velocity Φ is the fluid flow potential, scalar × is curl · is divergence. ... —Bernoullis principle states that in fluid flow, an increase in velocity occurs simultaneously with decrease in pressure. ...


The equation is named for Daniel Bernoulli although it was first presented in the above form by Leonhard Euler. Daniel Bernoulli Daniel Bernoulli (Groningen, February 9, 1700 – Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


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

Here, φ is the velocity potential energy per unit mass, which is just φ = gh in the case of a uniform gravitational field, and w is the fluid enthalpy per unit mass, which is also often written as h (which conflicts with our use of h in these notes for "height"). Note that w = epsilon + frac{p}{rho} where ε is the fluid thermodynamic energy per unit mass, also known as the specific internal energy or "sie". Enthalpy (symbolized H, also called heat content) is the sum of the internal energy of matter and the product of its volume multiplied by the pressure. ...


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. Even more generally when b may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).


When shocks are present, it should be noted that in a reference frame comoving 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.


Derivation

A streamtube of fluid moving to the right. Indicated are pressure, height, velocity, distance (s), and cross-sectional area.
A streamtube of fluid moving to the right. Indicated are pressure, height, velocity, distance (s), and cross-sectional area.

Let us begin with the Bernoulli equation for incompressible fluids. Image File history File links File links The following pages link to this file: Bernoullis equation ...


The equation 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. One has that In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... In fluid dynamics, the Euler equations govern the motion of a compressible, inviscid fluid. ... Conservation of energy is possibly the most important, and certainly the most practically useful of several conservation laws in physics. ... The pitch drop experiment at the University of Queensland. ...

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

The work done by the forces is Look up work in Wiktionary, the free dictionary. ... In physics, a force is an external cause responsible for any change of a physical system. ... Potential energy is stored energy. ... Kinetic energy is energy that a body has as a result of its speed. ...

F_{1} s_{1}-F_{2} s_{2}=p_{1} A_{1} v_ {1}Delta t-p_{2} A_{2} v_{2}Delta t. ;

The decrease of potential energy is

The increase in kinetic energy is

frac{1}{2} m v_{2}^{2}-frac{1}{2} m v_{1}^{2}=frac{1}{2}rho A_{2} v_{2}Delta t v_{2} ^{2}-frac{1}{2}rho A_{1} v_{1}Delta t v_{1}^{2}.

Putting these together, one gets

p_{1} A_{1} v_{1}Delta t-p_{2} A_{2} v_{2}Delta t+rho g A_{1} v_{1}Delta t h_{1}-rho g A_{2} v_{2}Delta t h_{2}=frac{1}{2}rho A_{2} v_{2}Delta t v_{2}^{2}-frac{1}{2}rho A_{1} v_{1}Delta t v_{1}^{2}

or

frac{rho A_{1} v_{1}Delta t v_{1}^{ 2}}{2}+rho g A_{1} v_{1}Delta t h_{1}+p_{1} A_{1 } v_{1}Delta t=frac{rho A_{2} v_{2}Delta t v_{ 2}^{2}}{2}+rho g A_{2} v_{2}Delta t h_{2}+p_{2} A_{2} v_{2}Delta t.

After dividing by Δt, ρ and A1v1 (= rate of fluid flow = A2v2 as the fluid is incompressible) one finds: In fluid dynamics, the rate of fluid flow is the volume of fluid which passes through a given area per unit time. ...

frac{v_{1}^{2}}{2}+g h_{1}+frac{p_{1}}{rho}=frac{v_{2}^{2}}{2}+g h_{2}+frac{p_{2}}{rho}

or frac{v^{2}}{2}+g h+frac{p}{rho}=C (as stated in the first paragraph).



Further division by g implies

frac{v^{2}}{2 g}+h+frac{p}{rho g}=C.

A free falling mass from a height h (in vacuum), will reach a velocity Free-fall or free fall in the strict sense is the condition of acceleration which is due only to gravity. ... For other uses, see vacuum cleaner and Vacuum (musical group). ... This article is about velocity in physics. ...

v=sqrt{{2 g}{h}}, or h=frac{v^{2}}{2 g}.

The term frac{v^2}{2 g} is called the velocity head. In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ...


The hydrostatic pressure or static head is defined as Hydrostatic pressure is the pressure exerted by a fluid due to its weight. ...

p = ρgh, or h=frac{p}{rho g}.

The term frac{p}{rho g} is also called the pressure head. In fluid dynamics, head refers to the constant right hand side in the incompressible steady version of Bernoullis equation. ...


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:


v2ρ + P = constant and


mV2 + P * volume = constant


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.


We apply conservation of energy 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, we have


0 = ΔE1 − ΔE2


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 graviational 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 + phi_1 rho_1 + epsilon_1 rho_1 + p_1 right] A_1 v_1 , Delta t


A similar expression for ΔE2 may easily be constructed. So now setting 0 = ΔE1 − ΔE2 we obtain


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_2rho_2 + epsilon_2 rho_2 + p_2 right] A_2 v_2 , Delta t


Let us rewrite this 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


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



which is the sought solution.


External Links

  • Daniel Bernoulli and the making of the fluid equation the story of what happened.
  • Testing Bernoulli: a simple experiment here is an experiment that you can easily do yourself to test Bernoulli's equation. There are also 2 questions and answers.

  Results from FactBites:
 
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