NationMaster - Encyclopedia: Bernoulli's principle

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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. Image File history File links This is a lossless scalable vector image. ... A fluid is defined as a substance that continually deforms (flows) under an applied shear stress regardless of the magnitude of the applied stress. ... In physics, mechanical work is the amount of energy transferred by a force. ... This article is about velocity in physics. ... This article is about pressure in the physical sciences. ... Potential energy (U, or Ep), a kind of scalar potential, is energy by virtue of matter being able to move to a lower-energy state, releasing energy in some form. ...

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. BERNOULIS EQUATION ==Incompressible flow== Solid blue lines and broken grey lines represent the streamlines. ...

Liquids are so dense they can be considered to be of constant density, regardless of their pressure. For this reason liquids can be considered to be incompressible and the flow of liquids can be described as incompressible flow. Bernoulli performed his experiments on liquids and his equation is valid only for incompressible flow. Bernoulli's equation is also valid for the flow of gases providing the velocity of the gas is sufficiently low that the variation in density of the gas along each streamline can be ignored. Solid blue lines and broken grey lines represent the streamlines. ...

The original form of Bernoulli's equation is:

${v^2 over 2}+gh+{p over rho}=mathrm{constant}$

where:

v = fluid velocity at a point on a streamline

g = acceleration due to gravity

h = height of the point on the streamline

p = pressure at the point on the streamline

$ρ$ = density of the fluid at all points on the streamline

This can be rewritten as^[1]: This article is about velocity in physics. ... Solid blue lines and broken grey lines represent the streamlines. ... The term g force or gee force refers to the symbol g, the force of acceleration due to gravity at the earths surface. ... Height is the measurement of distance between a specified point and a corresponding plane of reference. ... This article is about pressure in the physical sciences. ... For other uses, see Density (disambiguation). ...

${v^2 rho over 2}+rho g h+p=q+rho g h+p=mathrm{constant}$

where:

q = dynamic pressure

These assumptions must be met for the equation to apply: Velocity pressure is also called fluid dynamic pressure or Q given by the equation. ...

 * The equation applies along a streamline. For constant-density fluids 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. Bernoulli's equation can be used to calculate lift in cases where its assumptions are not violated. A potential flow is characterized by an irrotational velocity field. ... There are multiple definitions of lift: Lift, an aerodynamic force. ...

The above equations clearly use a linear relationship between velocity squared and pressure. For real fluids such as water and air this relationship is only linear for low speeds. The equation suggests there is a velocity at which pressure is zero; and at higher velocities the pressure is negative! Gases and liquids are not capable of negative pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid long before zero pressure is reached.

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

${v^2 over 2}+ phi + w =mathrm{constant}$

$phi ,$ = gravitational potential energy per unit mass, $phi = gh ,$ in the case of a uniform gravitational field

$w ,$ = fluid enthalpy per unit mass, which is also often written as $h ,$ (which conflicts with the use of $h ,$ in this article for "height"). Note that $w = epsilon + frac{p}{rho}$ where $epsilon ,$ 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). t In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or Î”H, or rarely as Ï‡) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... Thermodynamics (from the Greek Î¸ÎµÏÎ¼Î·, therme, meaning heat and Î´Ï…Î½Î±Î¼Î¹Ï‚, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... The fuel value or relative energy density is the quantity of potential energy in fuel, food or other substance. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which no heat is transferred to or from the working fluid. ...

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. Introduction The shock wave is one of several different ways in which a gas in a supersonic flow can be compressed. ... A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ...

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. This article is about the concept of integrals in calculus. ... In fluid dynamics, the Euler equations govern the compressible, Inviscid flow. ... Look up conservation of energy in Wiktionary, the free dictionary. ... For other uses, see Viscosity (disambiguation). ...

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. A Venturi meter is shown in a diagram, the pressure in 1 conditions is higher than 2, and the relationship between the fluid speed in 2 and 1 respectively, is the same as for pressure. ...

The equation of motion for a parcel of fluid on the axis of the pipe is

$m frac{dv}{dt}= -F$

$rho A dx frac{dv}{dt}= -A dp$

$rho frac{dv}{dt}= -frac{dp}{dx}$

In steady flow, $v = v (x)$ so

$frac{dv}{dt}= frac{dv}{dx}frac{dx}{dt} = frac{dv}{dx}v=frac{d}{dx} frac{v^2}{2}$

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

$frac{d}{dx} left( rho frac{v^2}{2} + p right) =0$

$frac{v^2}{2} + frac{p}{rho}= C$

where $C$ is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. We deduce that where the speed is large, pressure is low and vice versa. 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. In physics, a physical constant is a physical quantity of a value that is generally believed to be both universal in nature and not believed to change in time. ...

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

Applying conservation of energy in form of the work-kinetic energy theorem we find that: Image File history File links File links The following pages link to this file: Bernoullis equation ...

the change in KE of the system equals the net work done on the system;

$W=Delta KE. ;$

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 In physics, mechanical work is the amount of energy transferred by a force. ... For other uses, see Force (disambiguation). ... Potential energy can be thought of as energy stored within a physical system. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ...

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

The decrease of potential energy is

$m g h_{1}-m g h_{2}=rho g A _{1} v_{1}Delta t h_{1}-rho g A_{2} v_{2} Delta t h_{2} ;$

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,

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

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

After dividing by $Δ t$ , $ρ$ and $A 1 v 1$ (= rate of fluid flow = $A 2 v 2$ as the fluid is incompressible): 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, as stated in the first paragraph:

$frac{v^{2}}{2}+g h+frac{p}{rho}=C$ (Eqn. 1)

Further division by g produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's Principle: For other uses of this word, see Length (disambiguation). ...

$frac{v^{2}}{2 g}+h+frac{p}{rho g}=C$ (Eqn. 2a)

The middle term, h, can be called an elevation head, although height is used throughout this discussion. $h_{elevation} ,$ represents the internal energy of the fluid due to its height above a reference plane. Height is the measurement of distance between a specified point and a corresponding plane of reference. ...

A free falling mass from a height h (in a vacuum) will reach a velocity For other uses, see Free-fall (disambiguation). ... Look up Vacuum in Wiktionary, the free dictionary. ... This article is about velocity in physics. ...

$v=sqrt{{2 g}{h}},$ or when we rearrange it as a head: $h_{v}=frac{v^{2}}{2 g}$

The term is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion. In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ... It has been suggested that Hydraulic head (hydrology) and Head (hydraulic) be merged into this article or section. ...

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

$p=rho g h ,$ , or when we rearrange it as a head: $psi=frac{p}{rho g}$

The term $frac{p}{rho g}$ is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. Fluid pressure is the pressure at some point within a fluid, such as water or air. ...

When we combine the head due to the velocity and the head due to static pressure with the elevation above a reference point, we obtain a simple relationship useful for incompressible fluids.

$h_{v} + h_{elevation} + psi = C,$ (Eqn. 2b)

If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms:

$frac{rho v^{2}}{2}+ rho g h + p=C$ (Eqn. 3)

We note that the pressure of the system is constant in this form of the Bernoulli Equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system.

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 $Δ t$ , 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 $A 2$ :

$0 = Delta M_1 - Delta M_2 = rho_1 A_1 v_1 , Delta t - rho_2 A_2 v_2 , Delta t$ .

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,

$0 = Delta E_1 - Delta E_2 ,$

where $Δ E 1$ and $Δ E 2$ 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 $p,dV$ 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 $Δ E 2$ may easily be constructed. So now setting $0 = Δ E 1 - Δ E 2$ :

$0 = left[ frac{1}{2} rho_1 v_1^2+ phi_1 rho_1 + epsilon_1 rho_1 + p_1 right] A_1 v_1 , Delta t - left[ frac{1}{2} rho_2 v_2^2 + phi_2rho_2 + epsilon_2 rho_2 + p_2 right] A_2 v_2 , Delta t$

which can be rewritten as:

$0 = left[ frac{1}{2} v_1^2 + phi_1 + epsilon_1 + frac{p_1}{rho_1} right] rho_1 A_1 v_1 , Delta t - left[ frac{1}{2} v_2^2 + phi_2 + epsilon_2 + frac{p_2}{rho_2} right] rho_2 A_2 v_2 , Delta t$

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

$frac{1}{2}v^2 + phi + epsilon + frac{p}{rho} = {rm constant} equiv b$

which is the Bernoulli equation for compressible flow.

Category: Articles lacking in-text citations

Results from FactBites:

Bernoullis Principle (512 words)
	Bernoulli's principle thus says that a rise (fall) in pressure in a flowing fluid must always be accompanied by a decrease (increase) in the speed, and conversely, if an increase (decrease) in, the speed of the fluid results in a decrease (increase) in the pressure.
	Bernouilli's principle also tells us why windows tend to explode, rather than implode in hurricanes: the very high speed of the air just outside the window causes the pressure just outside to be much less than the pressure inside, where the air is still.
	Another example of Bernoulli's principle at work is in the lift of aircraft wings and the motion of ``curve balls'' in baseball.

Bernoullis theory of lift (1208 words)
	The problem is that the physical principles that lie at the foundation of flight are counterintuitive; indeed, the mechanics of flight were ultimately revealed after some fancy manipulation of the physics and mathematics created by Sir Isaac Newton in the late 1600s.
	Bernoulli's principle thus says that a rise (fall) in pressure in a flowing fluid must always be accompanied by a decrease (increase) in the speed, and conversely, if an increase (decrease) in, the speed of the fluid results in a decrease (increase) in the pressure.
	Another example of Bernoulli's principle at work is in the lift of aircraft wings and the motion of ``curve balls'' in baseball.

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