Bernoulli's principle

A flow of water into a venturi meter. The kinetic energy increases at the expense of the fluid pressure, as shown by the difference in height of the two columns of water.

Continuum mechanics

Laws

Conservations
Energy Mass Momentum
Inequalities
Clausius–Duhem (entropy)

Solid mechanics

Contact mechanics (frictional)

Fluid mechanics

Fluids
Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure
Liquids
Surface tension Capillary action
Gases
Atmosphere Boyle's law Charles's law Gay-Lussac's law Combined gas law
Plasma

Rheology

Smart fluids
Viscoelasticity Rheometry Rheometer
Magnetorheological Electrorheological Ferrofluids

Scientists

In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy.^[1]^[2] The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738.^[3]

Bernoulli's principle can be applied to various types of fluid flow, resulting in various forms of Bernoulli's equation; there are different forms of Bernoulli's equation for different types of flow. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy, potential energy and internal energy remains constant.^[2] Thus an increase in the speed of the fluid – implying an increase in both its dynamic pressure and kinetic energy – occurs with a simultaneous decrease in (the sum of) its static pressure, potential energy and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same on all streamlines because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ρ g h) is the same everywhere.^[4]

Bernoulli's principle can also be derived directly from Isaac Newton's Second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline.^[5]^[6]^[7]

Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.^[8]

Incompressible flow equation

In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible and these flows are called incompressible flows. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline, is:

${\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}={\text{constant}}$

(A)

where:

v

is the fluid flow speed at a point on a streamline,

g

is the acceleration due to gravity,

z

is the elevation of the point above a reference plane, with the positive

z

-direction pointing upward – so in the direction opposite to the gravitational acceleration,

p

is the pressure at the chosen point, and

ρ

is the density of the fluid at all points in the fluid.

The constant on the right-hand side of the equation depends only on the streamline chosen, whereas $v$ , $z$ and $p$ depend on the particular point on that streamline.

The following assumptions must be met for this Bernoulli equation to apply:^[9]

the flow must be steady, i.e. the fluid velocity at a point cannot change with time,
the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
friction by viscous forces has to be negligible.

For conservative force fields (not limited to the gravitational field), Bernoulli's equation can be generalized as:^[9]

{\frac {v^{2}}{2}}+\Psi +{\frac {p}{\rho }}={\text{constant}}

where $Ψ$ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity $Ψ = gz$ .

By multiplying with the fluid density $ρ$ , equation (A) can be rewritten as:

{\tfrac {1}{2}}\rho v^{2}+\rho gz+p={\text{constant}}

or:

q+\rho gh=p_{0}+\rho gz={\text{constant}}

where

$q = 1 / 2 ρv 2$ is dynamic pressure,
$h = z + p / ρg$ is the piezometric head or hydraulic head (the sum of the elevation $z$ and the pressure head)^[10]^[11] and
$p 0 = p + q$ is the total pressure (the sum of the static pressure $p$ and dynamic pressure $q$ ).^[12]

The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head $H$ :

H=z+{\frac {p}{\rho g}}+{\frac {v^{2}}{2g}}=h+{\frac {v^{2}}{2g}},

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids – when the pressure becomes too low – cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

Simplified form

In many applications of Bernoulli's equation, the change in the $ρgz$ term along the streamline is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height $z$ along a streamline is so small the $ρgz$ term can be omitted. This allows the above equation to be presented in the following simplified form:

p+q=p_{0}

where $p 0$ is called "total pressure", and $q$ is "dynamic pressure".^[13] Many authors refer to the pressure $p$ as static pressure to distinguish it from total pressure $p 0$ and dynamic pressure $q$ . In Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."^[14]

The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:^[14]

static pressure + dynamic pressure = total pressure

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure $p$ and dynamic pressure $q$ . Their sum $p + q$ is defined to be the total pressure $p 0$ . The significance of Bernoulli's principle can now be summarized as "total pressure is constant along a streamline".

If the fluid flow is irrotational, the total pressure on every streamline is the same and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow".^[15] It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in open bodies of water. However, it is important to remember that Bernoulli's principle does not apply in the boundary layer or in fluid flow through long pipes.

If the fluid flow at some point along a streamline is brought to rest, this point is called a stagnation point, and at this point the total pressure is equal to the stagnation pressure.

Applicability of incompressible flow equation to flow of gases

Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation – in its incompressible flow form – cannot be assumed to be valid. However, if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas, (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature, however this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle, or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.

Unsteady potential flow

The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics.

For an irrotational flow, the flow velocity can be described as the gradient $\nabla φ$ of a velocity potential $φ$ . In that case, and for a constant density $ρ$ , the momentum equations of the Euler equations can be integrated to:^[16]

{\frac {\partial \varphi }{\partial t}}+{\tfrac {1}{2}}v^{2}+{\frac {p}{\rho }}+gz=f(t),

which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here $\partial φ / \partial t$ denotes the partial derivative of the velocity potential $φ$ with respect to time $t$ , and $v = | \nabla φ |$ is the flow speed. The function $f (t)$ depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment $t$ does not only apply along a certain streamline, but in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case $f$ is a constant.^[16]

Further $f (t)$ can be made equal to zero by incorporating it into the velocity potential using the transformation

\Phi =\varphi -\int _{t_{0}}^{t}f(\tau )\,\mathrm {d} \tau ,

resulting in

\displaystyle {\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}v^{2}+{\frac {p}{\rho }}+gz=0.

Note that the relation of the potential to the flow velocity is unaffected by this transformation: $\nabla Φ = \nabla φ$ .

The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian (not to be confused with Lagrangian coordinates).

Compressible flow equation

Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and compressible fluids up to approximately Mach number 0.3.^[17] It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.

Compressible flow in fluid dynamics

For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,^[18]

{\frac {v^{2}}{2}}+\int _{p_{1}}^{p}{\frac {\mathrm {d} {\tilde {p}}}{\rho ({\tilde {p}})}}+\Psi ={\text{constant (along a streamline)}}

where:

$p$ is the pressure
$ρ$ is the density
$v$ is the flow speed
$Ψ$ is the potential associated with the conservative force field, often the gravitational potential

In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation becomes^[19]

{\frac {v^{2}}{2}}+gz+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}={\text{constant (along a streamline)}}

where, in addition to the terms listed above:

$γ$ is the ratio of the specific heats of the fluid
$g$ is the acceleration due to gravity
$z$ is the elevation of the point above a reference plane

In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term gz can be omitted. A very useful form of the equation is then:

{\frac {v^{2}}{2}}+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}=\left({\frac {\gamma }{\gamma -1}}\right){\frac {p_{0}}{\rho _{0}}}

where:

$p 0$ is the total pressure
$ρ 0$ is the total density

Compressible flow in thermodynamics

The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is:^[2]^[20]^[21]

{\frac {v^{2}}{2}}+\Psi +w={\text{constant}}.

Here $w$ is the enthalpy per unit mass, which is also often written as $h$ (not to be confused with "head" or "height").

Note that $w = ε + p / ρ$ where $ε$ is the thermodynamic energy per unit mass, also known as the specific internal energy. So, for constant internal energy $ε$ the equation reduces to the incompressible-flow form.

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 the change in $Ψ$ can be ignored, a very useful form of this equation is:

{\frac {v^{2}}{2}}+w=w_{0}

where $w 0$ is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature.

When shock waves are present, in a reference frame in which the shock is stationary and the flow is steady, 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 the Bernoulli equation

Bernoulli equation for incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by either integrating Newton's second law of motion or by applying the law of conservation of energy between two sections along a streamline, ignoring viscosity, compressibility, and thermal effects.

Derivation through integrating Newton's Second Law of Motion

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.

Define a parcel of fluid moving through a pipe with cross-sectional area $A$ , the length of the parcel is $d x$ , and the volume of the parcel $A d x$ . If mass density is $ρ$ , the mass of the parcel is density multiplied by its volume $m = ρA d x$ . The change in pressure over distance $d x$ is $d p$ and flow velocity $v = d x / d t$ .

Apply Newton's second law of motion (force = mass × acceleration) and recognizing that the effective force on the parcel of fluid is $- A d p$ . If the pressure decreases along the length of the pipe, $d p$ is negative but the force resulting in flow is positive along the $x$ axis.

{\begin{aligned}m{\frac {\mathrm {d} v}{\mathrm {d} t}}&=F\\\rho A\mathrm {d} x{\frac {\mathrm {d} v}{\mathrm {d} t}}&=-A\mathrm {d} p\\\rho {\frac {\mathrm {d} v}{\mathrm {d} t}}&=-{\frac {\mathrm {d} p}{\mathrm {d} x}}\end{aligned}}

In steady flow the velocity field is constant with respect to time, $v = v (x) = v (x (t))$ , so $v$ itself is not directly a function of time $t$ . It is only when the parcel moves through $x$ that the cross sectional area changes: $v$ depends on $t$ only through the cross-sectional position $x (t)$ .

{\begin{aligned}{\frac {\mathrm {d} v}{\mathrm {d} t}}&={\frac {\mathrm {d} v}{\mathrm {d} x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}\\&={\frac {\mathrm {d} v}{\mathrm {d} x}}v\\&={\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {v^{2}}{2}}\right).\end{aligned}}

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

{\frac {\mathrm {d} }{\mathrm {d} x}}\left(\rho {\frac {v^{2}}{2}}+p\right)=0

by integrating with respect to $x$

{\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. The deduction is: where the speed is large, pressure is low and vice versa.

In the above derivation, no external work–energy principle is invoked. Rather, Bernoulli's principle was derived by a simple manipulation of Newton's second law.

A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (

s

), and cross-sectional area. Note that in this figure elevation is denoted as

h

, contrary to the text where it is given by

z

Derivation by using conservation of energy

Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.^[22] In the form of the work-energy theorem, stating that^[23]

the change in the kinetic energy

E kin

of the system equals the net work $W$ done on the system;

W=\Delta E_{\text{kin}}.

Therefore,

the work done by the forces in the fluid equals increase in kinetic energy.

The system consists of the volume of fluid, initially between the cross-sections $A 1$ and $A 2$ . In the time interval $Δ t$ fluid elements initially at the inflow cross-section $A 1$ move over a distance $s 1 = v 1 Δ t$ , while at the outflow cross-section the fluid moves away from cross-section $A 2$ over a distance $s 2 = v 2 Δ t$ . The displaced fluid volumes at the inflow and outflow are respectively $A 1 s 1$ and $A 2 s 2$ . The associated displaced fluid masses are – when $ρ$ is the fluid's mass density – equal to density times volume, so $ρA 1 s 1$ and $ρA 2 s 2$ . By mass conservation, these two masses displaced in the time interval $Δ t$ have to be equal, and this displaced mass is denoted by $Δ m$ :

{\begin{aligned}\rho A_{1}s_{1}&=\rho A_{1}v_{1}\Delta t=\Delta m,\\\rho A_{2}s_{2}&=\rho A_{2}v_{2}\Delta t=\Delta m.\end{aligned}}

The work done by the forces consists of two parts:

The work done by the pressure acting on the areas $A 1$ and $A 2$
$W_{\text{pressure}}=F_{1,{\text{pressure}}}s_{1}-F_{2,{\text{pressure}}}s_{2}=p_{1}A_{1}s_{1}-p_{2}A_{2}s_{2}=\Delta m{\frac {p_{1}}{\rho }}-\Delta m{\frac {p_{2}}{\rho }}.$
The work done by gravity: the gravitational potential energy in the volume $A 1 s 1$ is lost, and at the outflow in the volume $A 2 s 2$ is gained. So, the change in gravitational potential energy $Δ E pot,gravity$ in the time interval $Δ t$ is
$\Delta E_{\text{pot,gravity}}=\Delta m\,gz_{2}-\Delta m\,gz_{1}.\;$

Now, the work by the force of gravity is opposite to the change in potential energy,

W gravity = - ΔE pot,gravity

: while the force of gravity is in the negative

z

-direction, the work—gravity force times change in elevation—will be negative for a positive elevation change

Δ z = z 2 - z 1

, while the corresponding potential energy change is positive.^[24] So:

W_{\text{gravity}}=-\Delta E_{\text{pot,gravity}}=\Delta m\,gz_{1}-\Delta m\,gz_{2}.

And therefore the total work done in this time interval $Δ t$ is

W=W_{\text{pressure}}+W_{\text{gravity}}.

The increase in kinetic energy is

\Delta E_{\text{kin}}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}-{\tfrac {1}{2}}\Delta m\,v_{1}^{2}.

Putting these together, the work-kinetic energy theorem $W = Δ E kin$ gives:^[22]

\Delta m{\frac {p_{1}}{\rho }}-\Delta m{\frac {p_{2}}{\rho }}+\Delta m\,gz_{1}-\Delta m\,gz_{2}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}-{\tfrac {1}{2}}\Delta m\,v_{1}^{2}

{\tfrac {1}{2}}\Delta m\,v_{1}^{2}+\Delta m\,gz_{1}+\Delta m{\frac {p_{1}}{\rho }}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}+\Delta m\,gz_{2}+\Delta m{\frac {p_{2}}{\rho }}.

After dividing by the mass $Δ m = ρA 1 v 1 Δ t = ρA 2 v 2 Δ t$ the result is:^[22]

{\tfrac {1}{2}}v_{1}^{2}+gz_{1}+{\frac {p_{1}}{\rho }}={\tfrac {1}{2}}v_{2}^{2}+gz_{2}+{\frac {p_{2}}{\rho }}

or, as stated in the first paragraph:

{\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}=C

(Eqn. 1), Which is also Equation (A)

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:

{\frac {v^{2}}{2g}}+z+{\frac {p}{\rho g}}=C

(Eqn. 2a)

The middle term, $z$ , represents the potential energy of the fluid due to its elevation with respect to a reference plane. Now, $z$ is called the elevation head and given the designation $z elevation$ .

A free falling mass from an elevation $z > 0$ (in a vacuum) will reach a speed

v={\sqrt {{2g}{z}}},

when arriving at elevation $z = 0$ . Or when we rearrange it as a head:

h_{v}={\frac {v^{2}}{2g}}

The term $v 2 / 2 g$ is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion.

The hydrostatic pressure p is defined as

p=p_{0}-\rho gz,

with $p 0$ some reference pressure, or when we rearrange it as a head:

\psi ={\frac {p}{\rho g}}.

The term $p / ρg$ 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. When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head.

h_{v}+z_{\text{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 gz+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.

Bernoulli equation for 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 internal energy per unit of mass ( $ε 1$ ) entering, and the energy entering in the form of mechanical $p d V$ work:

\Delta E_{1}=\left({\tfrac {1}{2}}\rho _{1}v_{1}^{2}+\Psi _{1}\rho _{1}+\epsilon _{1}\rho _{1}+p_{1}\right)A_{1}v_{1}\,\Delta t

where $Ψ = gz$ is a force potential due to the Earth's gravity, $g$ is acceleration due to gravity, and $z$ is elevation above a reference plane. A similar expression for $Δ E 2$ may easily be constructed. So now setting $0 = Δ E 1 - Δ E 2$ :

0=\left({\tfrac {1}{2}}\rho _{1}v_{1}^{2}+\Psi _{1}\rho _{1}+\epsilon _{1}\rho _{1}+p_{1}\right)A_{1}v_{1}\,\Delta t-\left({\tfrac {1}{2}}\rho _{2}v_{2}^{2}+\Psi _{2}\rho _{2}+\epsilon _{2}\rho _{2}+p_{2}\right)A_{2}v_{2}\,\Delta t

which can be rewritten as:

0=\left({\tfrac {1}{2}}v_{1}^{2}+\Psi _{1}+\epsilon _{1}+{\frac {p_{1}}{\rho _{1}}}\right)\rho _{1}A_{1}v_{1}\,\Delta t-\left({\tfrac {1}{2}}v_{2}^{2}+\Psi _{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

{\tfrac {1}{2}}v^{2}+\Psi +\epsilon +{\frac {p}{\rho }}={\text{constant}}\equiv b

which is the Bernoulli equation for compressible flow.

An equivalent expression can be written in terms of fluid enthalpy ( $h$ ):

{\tfrac {1}{2}}v^{2}+\Psi +h={\text{constant}}\equiv b

Applications

Condensation visible over the upper surface of an Airbus A340 wing caused by the fall in temperature accompanying the fall in pressure.

In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid^[25] and a small viscosity often has a large effect on the flow.

Bernoulli's principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli's principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lifting force.^[26]^[27] Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli's equations^[28] – established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. Bernoulli's principle does not explain why the air flows faster past the top of the wing and slower past the underside. See the article on aerodynamic lift for more info.

The carburettor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburettor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli's principle; in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure.
An injector on a steam locomotive (or static boiler).
The pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to the airspeed indicator, which determines the dynamic pressure of the airflow past the aircraft. Dynamic pressure is the difference between stagnation pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure.^[29]
The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently, Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect.
The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation, and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli's law, showing that Torricelli's law is compatible with Bernoulli's principle. Viscosity lowers this drain rate. This is reflected in the discharge coefficient, which is a function of the Reynolds number and the shape of the orifice.^[30]

The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.

Misunderstandings about the generation of lift

Many explanations for the generation of lift (on airfoils, propeller blades, etc.) can be found; some of these explanations can be misleading, and some are false,^[31] in particular the idea that air particles flowing above and below a cambered wing should reach simultaneously the trailing edge. This has been a source of heated discussion over the years. In particular, there has been debate about whether lift is best explained by Bernoulli's principle or Newton's laws of motion. Modern writings agree that both Bernoulli's principle and Newton's laws are relevant and either can be used to correctly describe lift.^[32]^[33]^[34]

Several of these explanations use the Bernoulli principle to connect the flow kinematics to the flow-induced pressures. In cases of incorrect (or partially correct) explanations relying on the Bernoulli principle, the errors generally occur in the assumptions on the flow kinematics and how these are produced. It is not the Bernoulli principle itself that is questioned because this principle is well established (the airflow above the wing is faster, the question is why it is faster).^[35]^[36]^[37]^[38]

Misapplications of Bernoulli's principle in common classroom demonstrations

There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle.^[39] One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it. As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure".^[40]^[41]^[42]

One problem with this explanation can be seen by blowing along the bottom of the paper: were the deflection due simply to faster moving air one would expect the paper to deflect downward, but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom.^[43] Another problem is that when the air leaves the demonstrator's mouth it has the same pressure as the surrounding air;^[44] the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is equal to the pressure of the surrounding air.^[45]^[46] A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation since the air above and below are different flow fields and Bernoulli's principle only applies within a flow field.^[47]^[48]^[49]^[50]

As the wording of the principle can change its implications, stating the principle correctly is important.^[51] What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa.^[52] Thus, Bernoulli's principle concerns itself with changes in speed and changes in pressure within a flow field. It cannot be used to compare different flow fields.

A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve.^[53]^[54]^[55]^[56] Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed, i.e. that as the air passes over the paper it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration.^[57]^[58]^[59]

Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure".^[60]^[61]^[62]^[63]^[64]^[65]^[66]

References

↑ Clancy, L.J., Aerodynamics, Chapter 3.
1 2 3 Batchelor, G.K. (1967), Section 3.5, pp. 156–64.
↑ "Hydrodynamica". Britannica Online Encyclopedia. Retrieved 2008-10-30.
↑ Streeter, V.L., Fluid Mechanics, Example 3.5, McGraw–Hill Inc. (1966), New York.
↑ "If the particle is in a region of varying pressure (a non-vanishing pressure gradient in the $x$ -direction) and if the particle has a finite size $l$ , then the front of the particle will be ‘seeing’ a different pressure from the rear. More precisely, if the pressure drops in the $x$ -direction ( $dp / dx < 0$ ) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton’s second law, this force causes an acceleration and the particle’s velocity increases as it moves along the streamline... Bernoulli's equation describes this mathematically (see the complete derivation in the appendix)."Babinsky, Holger (November 2003), "How do wings work?" (PDF), Physics Education
↑ "Acceleration of air is caused by pressure gradients. Air is accelerated in direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity." Weltner, Klaus; Ingelman-Sundberg, Martin, Misinterpretations of Bernoulli's Law, archived from the original on April 29, 2009
↑ " The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely if the parcel is moving into a region of lower pressure, there will be an higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton’s laws of motion." See How It Flies John S. Denker http://www.av8n.com/how/htm/airfoils.html
↑ Resnick, R. and Halliday, D. (1960), section 18-4, Physics, John Wiley & Sons, Inc.
1 2 Batchelor, G.K. (1967), §5.1, p. 265.
↑ Mulley, Raymond (2004). Flow of Industrial Fluids: Theory and Equations. CRC Press. pp. 43–44. ISBN 0-8493-2767-9.
↑ Chanson, Hubert (2004). Hydraulics of Open Channel Flow: An Introduction. Butterworth-Heinemann. p. 22. ISBN 0-7506-5978-5.
↑ Oertel, Herbert; Prandtl, Ludwig; Böhle, M.; Mayes, Katherine (2004). Prandtl's Essentials of Fluid Mechanics. Springer. pp. 70–71. ISBN 0-387-40437-6.
↑ "Bernoulli's Equation". NASA Glenn Research Center. Retrieved 2009-03-04.
1 2 Clancy, L.J., Aerodynamics, Section 3.5.
↑ Clancy, L.J. Aerodynamics, Equation 3.12
1 2 Batchelor, G.K. (1967), p. 383
↑ White, Frank M. Fluid Mechanics, 6th ed. McGraw-Hill International Edition. p. 602.
↑ Clarke C. and Carswell B., Astrophysical Fluid Dynamics
↑ Clancy, L.J., Aerodynamics, Section 3.11
↑ Landau & Lifshitz (1987, §5)
↑ Van Wylen, G.J., and Sonntag, R.E., (1965), Fundamentals of Classical Thermodynamics, Section 5.9, John Wiley and Sons Inc., New York
1 2 3 Feynman, R.P.; Leighton, R.B.; Sands, M. (1963). The Feynman Lectures on Physics. ISBN 0-201-02116-1. , Vol. 2, §40–3, pp. 40–6 – 40–9.
↑ Tipler, Paul (1991). Physics for Scientists and Engineers: Mechanics (3rd extended ed.). W. H. Freeman. ISBN 0-87901-432-6. , p. 138.
↑ Feynman, R.P.; Leighton, R.B.; Sands, M. (1963). The Feynman Lectures on Physics. ISBN 0-201-02116-1. , Vol. 1, §14–3, p. 14–4.
↑ Physics Today, May 1010, "The Nearly Perfect Fermi Gas", by John E. Thomas, p 34.
↑ Clancy, L.J., Aerodynamics, Section 5.5 ("When a stream of air flows past an airfoil, there are local changes in flow speed round the airfoil, and consequently changes in static pressure, in accordance with Bernoulli's Theorem. The distribution of pressure determines the lift, pitching moment and form drag of the airfoil, and the position of its centre of pressure.")
↑ Resnick, R. and Halliday, D. (1960), Physics, Section 18–5, John Wiley & Sons, Inc., New York ("Streamlines are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.")
↑ Eastlake, Charles N. (March 2002). "An Aerodynamicist’s View of Lift, Bernoulli, and Newton" (PDF). The Physics Teacher. 40. "The resultant force is determined by integrating the surface-pressure distribution over the surface area of the airfoil."
↑ Clancy, L.J., Aerodynamics, Section 3.8
↑ Mechanical Engineering Reference Manual Ninth Edition
↑ Glenn Research Center (2006-03-15). "Incorrect Lift Theory". NASA. Retrieved 2010-08-12.
↑ Chanson, H. (2009). Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows. CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages. ISBN 978-0-415-49271-3.
↑ "Newton vs Bernoulli".
↑ Ison, David. Bernoulli Or Newton: Who's Right About Lift? Retrieved on 2009-11-26
↑ Phillips, O.M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 0-521-29801-6. Section 2.4.
↑ Batchelor, G.K. (1967). Sections 3.5 and 5.1
↑ Lamb, H. (1994) §17–§29
↑ Weltner, Klaus; Ingelman-Sundberg, Martin. "Physics of Flight – reviewed". "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause."
↑ "Bernoulli's law and experiments attributed to it are fascinating. Unfortunately some of these experiments are explained erroneously..." Weltner, Klaus; Ingelman-Sundberg, Martin. "Misinterpretations of Bernoulli's Law". Department of Physics, University Frankfurt. Archived from the original on June 21, 2012. Retrieved June 25, 2012.
↑ "This occurs because of Bernoulli’s principle — fast-moving air has lower pressure than non-moving air." Make Magazine http://makeprojects.com/Project/Origami-Flying-Disk/327/1
↑ " Faster-moving fluid, lower pressure. ... When the demonstrator holds the paper in front of his mouth and blows across the top, he is creating an area of faster-moving air." University of Minnesota School of Physics and Astronomy http://www.physics.umn.edu/outreach/pforce/circus/Bernoulli.html
↑ "Bernoulli's Principle states that faster moving air has lower pressure... You can demonstrate Bernoulli's Principle by blowing over a piece of paper held horizontally across your lips." "Educational Packet" (PDF). Tall Ships Festival – Channel Islands Harbor. Archived from the original (PDF) on December 3, 2013. Retrieved June 25, 2012.
↑ "If the lift in figure A were caused by "Bernoulli principle," then the paper in figure B should droop further when air is blown beneath it. However, as shown, it raises when the upward pressure gradient in downward-curving flow adds to atmospheric pressure at the paper lower surface." Craig, Gale M. "Physical Principles of Winged Flight". Retrieved March 31, 2016.
↑ "In fact, the pressure in the air blown out of the lungs is equal to that of the surrounding air..." Babinsky http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf
↑ "...air does not have a reduced lateral pressure (or static pressure...) simply because it is caused to move, the static pressure of free air does not decrease as the speed of the air increases, it misunderstanding Bernoulli's principle to suggest that this is what it tells us, and the behavior of the curved paper is explained by other reasoning than Bernoulli's principle." Peter Eastwell Bernoulli? Perhaps, but What About Viscosity? The Science Education Review, 6(1) 2007 http://www.scienceeducationreview.com/open_access/eastwell-bernoulli.pdf
↑ "Make a strip of writing paper about 5 cm × 25 cm. Hold it in front of your lips so that it hangs out and down making a convex upward surface. When you blow across the top of the paper, it rises. Many books attribute this to the lowering of the air pressure on top solely to the Bernoulli effect. Now use your fingers to form the paper into a curve that it is slightly concave upward along its whole length and again blow along the top of this strip. The paper now bends downward...an often-cited experiment, which is usually taken as demonstrating the common explanation of lift, does not do so..." Jef Raskin Coanda Effect: Understanding Why Wings Work http://karmak.org/archive/2003/02/coanda_effect.html
↑ "Blowing over a piece of paper does not demonstrate Bernoulli’s equation. While it is true that a curved paper lifts when flow is applied on one side, this is not because air is moving at different speeds on the two sides... It is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation." Holger Babinsky How Do Wings Work Physics Education 38(6) http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf
↑ "An explanation based on Bernoulli’s principle is not applicable to this situation, because this principle has nothing to say about the interaction of air masses having different speeds... Also, while Bernoulli’s principle allows us to compare fluid speeds and pressures along a single streamline and... along two different streamlines that originate under identical fluid conditions, using Bernoulli’s principle to compare the air above and below the curved paper in Figure 1 is nonsensical; in this case, there aren’t any streamlines at all below the paper!" Peter Eastwell Bernoulli? Perhaps, but What About Viscosity? The Science Education Review 6(1) 2007 http://www.scienceeducationreview.com/open_access/eastwell-bernoulli.pdf
↑ "The well-known demonstration of the phenomenon of lift by means of lifting a page cantilevered in one’s hand by blowing horizontally along it is probably more a demonstration of the forces inherent in the Coanda effect than a demonstration of Bernoulli’s law; for, here, an air jet issues from the mouth and attaches to a curved (and, in this case pliable) surface. The upper edge is a complicated vortex-laden mixing layer and the distant flow is quiescent, so that Bernoulli’s law is hardly applicable." David Auerbach Why Aircreft Fly European Journal of Physics Vol 21 p 295 http://iopscience.iop.org/0143-0807/21/4/302/pdf/0143-0807_21_4_302.pdf
↑ "Millions of children in science classes are being asked to blow over curved pieces of paper and observe that the paper "lifts"... They are then asked to believe that Bernoulli's theorem is responsible... Unfortunately, the "dynamic lift" involved...is not properly explained by Bernoulli's theorem." Norman F. Smith "Bernoulli and Newton in Fluid Mechanics" The Physics Teacher Nov 1972
↑ "Bernoulli’s principle is very easy to understand provided the principle is correctly stated. However, we must be careful, because seemingly-small changes in the wording can lead to completely wrong conclusions." See How It Flies John S. Denker http://www.av8n.com/how/htm/airfoils.html#sec-bernoulli
↑ "A complete statement of Bernoulli's Theorem is as follows: "In a flow where no energy is being added or taken away, the sum of its various energies is a constant: consequently where the velocity increasees the pressure decreases and vice versa."" Norman F Smith Bernoulli, Newton and Dynamic Lift Part I School Science and Mathematics Vol 73 Issue 3 http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb08998.x/pdf
↑ "...if a streamline is curved, there must be a pressure gradient across the streamline, with the pressure increasing in the direction away from the centre of curvature." Babinsky http://iopscience.iop.org/0031-9120/38/6/001/pdf/pe3_6_001.pdf
↑ "The curved paper turns the stream of air downward, and this action produces the lift reaction that lifts the paper." Norman F. Smith Bernoulli, Newton, and Dynamic Lift Part II School Science and Mathematics vol 73 Issue 4 pg 333 http://onlinelibrary.wiley.com/doi/10.1111/j.1949-8594.1973.tb09040.x/pdf
↑ "The curved surface of the tongue creates unequal air pressure and a lifting action. ... Lift is caused by air moving over a curved surface." AERONAUTICS An Educator’s Guide with Activities in Science, Mathematics, and Technology Education by NASA pg 26 http://www.nasa.gov/pdf/58152main_Aeronautics.Educator.pdf
↑ "Viscosity causes the breath to follow the curved surface, Newton's first law says there a force on the air and Newton’s third law says there is an equal and opposite force on the paper. Momentum transfer lifts the strip. The reduction in pressure acting on the top surface of the piece of paper causes the paper to rise." The Newtonian Description of Lift of a Wing David F. Anderson & Scott Eberhardt pg 12 http://www.integener.com/IE110522Anderson&EberhardtPaperOnLift0902.pdf
↑ '"Demonstrations" of Bernoulli's principle are often given as demonstrations of the physics of lift. They are truly demonstrations of lift, but certainly not of Bernoulli's principle.' David F Anderson & Scott Eberhardt Understanding Flight pg 229 https://books.google.com/books?id=52Hfn7uEGSoC&pg=PA229
↑ "As an example, take the misleading experiment most often used to "demonstrate" Bernoulli's principle. Hold a piece of paper so that it curves over your finger, then blow across the top. The paper will rise. However most people do not realize that the paper would not rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is ambient air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli." From The Aeronautics File By Max Feil https://www.mat.uc.pt/~pedro/ncientificos/artigos/aeronauticsfile1.ps Archived May 17, 2015, at the Wayback Machine.
↑ "Some people blow over a sheet of paper to demonstrate that the accelerated air over the sheet results in a lower pressure. They are wrong with their explanation. The sheet of paper goes up because it deflects the air, by the Coanda effect, and that deflection is the cause of the force lifting the sheet. To prove they are wrong I use the following experiment: If the sheet of paper is pre bend the other way by first rolling it, and if you blow over it than, it goes down. This is because the air is deflected the other way. Airspeed is still higher above the sheet, so that is not causing the lower pressure." Pim Geurts. sailtheory.com http://www.sailtheory.com/experiments.html
↑ "Finally, let’s go back to the initial example of a ball levitating in a jet of air. The naive explanation for the stability of the ball in the air stream, 'because pressure in the jet is lower than pressure in the surrounding atmosphere,' is clearly incorrect. The static pressure in the free air jet is the same as the pressure in the surrounding atmosphere..." Martin Kamela Thinking About Bernoulli The Physics Teacher Vol. 45, September 2007 http://tpt.aapt.org/resource/1/phteah/v45/i6/p379_s1
↑ "Aysmmetrical flow (not Bernoulli's theorem) also explains lift on the ping-pong ball or beach ball that floats so mysteriously in the tilted vacuum cleaner exhaust..." Norman F. Smith, Bernoulli and Newton in Fluid Mechanics" The Physics Teacher Nov 1972 p 455
↑ "Bernoulli’s theorem is often obscured by demonstrations involving non-Bernoulli forces. For example, a ball may be supported on an upward jet of air or water, because any fluid (the air and water) has viscosity, which retards the slippage of one part of the fluid moving past another part of the fluid." Bauman, Robert P. "The Bernoulli Conundrum" (PDF). Professor of Physics Emeritus, University of Alabama at Birmingham. Archived from the original (PDF) on February 25, 2012. Retrieved June 25, 2012.
↑ "In a demonstration sometimes wrongly described as showing lift due to pressure reduction in moving air or pressure reduction due to flow path restriction, a ball or balloon is suspended by a jet of air." Craig, Gale M. "Physical Principles of Winged Flight". Retrieved March 31, 2016.
↑ "A second example is the confinement of a ping-pong ball in the vertical exhaust from a hair dryer. We are told that this is a demonstration of Bernoulli's principle. But, we now know that the exhaust does not have a lower value of ps. Again, it is momentum transfer that keeps the ball in the airflow. When the ball gets near the edge of the exhaust there is an asymmetric flow around the ball, which pushes it away from the edge of the flow. The same is true when one blows between two ping-pong balls hanging on strings." Anderson & Eberhardt The Newtonian Description of Lift on a Wing http://lss.fnal.gov/archive/2001/pub/Pub-01-036-E.pdf
↑ "This demonstration is often incorrectly explained using the Bernoulli principle. According to the INCORRECT explanation, the air flow is faster in the region between the sheets, thus creating a lower pressure compared with the quiet air on the outside of the sheets." "Thin Metal Sheets – Coanda Effect". University of Maryland – Physics Lecture-Demonstration Facility. Archived from the original on June 23, 2012. Retrieved October 23, 2012.
↑ "Although the Bernoulli effect is often used to explain this demonstration, and one manufacturer sells the material for this demonstration as "Bernoulli bags," it cannot be explained by the Bernoulli effect, but rather by the process of entrainment." "Answer #256". University of Maryland – Physics Lecture-Demonstration Facility. Archived from the original on December 13, 2014. Retrieved December 9, 2014.

External links

Wikimedia Commons has media related to Bernoulli's principle.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Bernoulli's principle

Incompressible flow equation

Simplified form

Applicability of incompressible flow equation to flow of gases

Unsteady potential flow

Compressible flow equation

Compressible flow in fluid dynamics

Compressible flow in thermodynamics

Derivations of the Bernoulli equation

Applications

Misunderstandings about the generation of lift

Misapplications of Bernoulli's principle in common classroom demonstrations

See also

References

Further reading

External links