Bernoulli's Equation

Fluid Mechanics

Presenter: Ola Kadhim Jalil

Mechanical Engineering Department
Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid
occurs simultaneously with a decrease in static pressure or a decrease in
the fluid's potential energy.[1]: Ch.3 [2]: 156–164, § 3.5  The principle
is named after Daniel Bernoulli who published it in his
book Hydrodynamica in 1738.[3] Although Bernoulli deduced
that pressure decreases when the flow speed increases, it
was Leonhard Euler in 1752 who derived Bernoulli's
equation in its usual form.[4][5] The principle is only applicable
for isentropic flows: when the effects of irreversible
processes (like turbulence) and non-adiabatic
processes (e.g. heat radiation) are small and can be neglected.

Bernoulli's principle can be applied to various types of

fluid flow, resulting in various forms of Bernoulli's A flow of air through a venturi meter.
equation. The simple form of Bernoulli's equation is The kinetic energy increases at the
valid for incompressible flows (e.g. most liquid flows expense of the fluid pressure, as shown
and gases moving at low Mach number). More advanced by the difference in height of the two
forms may be applied to compressible flows at columns of water.
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]: § 3.5  Thus an increase in the speed of the fluid – implying an
increase in its kinetic energy (dynamic pressure) – occurs with a simultaneous decrease
in (the sum of) its potential energy (including the static pressure) 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.[6]: Example 3.5 
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. [a][b][c]
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
 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:

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:[2]: 265 
• the flow must be steady, i.e., the flow parameters (velocity, density, etc...) at any 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 must be negligible.

For conservative force fields (not limited to the gravitational field), Bernoulli's equation
can be generalized as:[2]: 265 

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:

or:
 where

• q = 1/2 ρv2 is dynamic pressure,

• h = z + p/ρg is the piezometric head or hydraulic head (the sum of the elevation z and
the pressure head)[11][12] and
• p0 = p + q is the stagnation pressure (the sum of the static pressure p and dynamic
pressure q).[13]

The constant in the Bernoulli equation can be normalized. A common approach is in

terms of total head or energy head H:

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:

where p0 is called "total pressure", and q is "dynamic pressure".[14] Many authors refer to
the pressure p as static pressure to distinguish it from total pressure p0 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."[1]: § 3.5 
The simplified form of Bernoulli's equation can be summarized in the following
memorable word equation:[1]: § 3.5 
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 p0. 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".[1]: Equation 3.12  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, Bernoulli's principle
importantly 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 valid for ideal fluids: those that are incompressible, irrotational,
inviscid, and subjected to conservative forces. It 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 ∇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

which is a Bernoulli equation valid also for unsteady—or time dependent—flows.

Here ∂φ/∂t denotes the partial derivative of the velocity potential φ with respect to
time t, and v = |∇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 and ∂φ/∂t are constants so
equation (A) can be applied in every point of the fluid domain.
 Further f(t) can be made equal to zero by incorporating it into the velocity potential
using the transformation

Note that the relation of the potential to the flow velocity is unaffected by this
transformation: ∇Φ = ∇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 steady compressible fluids up to
approximately Mach number 0.3.[15] 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

where:

• p is the pressure
• ρ is the density and indicates that it is a function of pressure

• 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 for an ideal gas becomes:[1]: § 3.11 

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:

where:

• p0 is the total pressure

• ρ0 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]: § 3.5 [17]: § 5 [18]: § 5.9 

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

Note that where is the thermodynamic energy per unit mass, also known as

the specific internal energy. So, for constant internal energy 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:

where w0 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.

Unsteady potential flow

For a compressible fluid, with a barotropic equation of state, the unsteady momentum
conservation equation

With the irrotational assumption, namely, the flow velocity can be described as
the gradient ∇φ of a velocity potential φ. The unsteady momentum conservation
equation becomes
which leads to

In this case, the above equation for isentropic flow becomes:

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 dx, and the volume of the parcel A dx. If mass
density is ρ, the mass of the parcel is density multiplied by its
volume m = ρA dx. The change in pressure over distance dx is dp and flow
velocity v = dx/dt.
Apply Newton's second law of motion (force = mass × acceleration) and
recognizing that the effective force on the parcel of fluid is −A dp. If the
pressure decreases along the length of the pipe, dp is negative but the force
resulting in flow is positive along the x axis.

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).

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

by integrating with respect to x

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.[19] In the form of the work-energy
theorem, stating that[20]
the change in the kinetic energy Ekin of the system equals the net
work W done on the system;

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 A1 and A2. In the time interval Δt fluid elements initially at the
inflow cross-section A1 move over a distance s1 = v1 Δt, while at the outflow
cross-section the fluid moves away from cross-section A2 over a
distance s2 = v2 Δt. The displaced fluid volumes at the inflow and outflow
are respectively A1s1 and A2s2. The associated displaced fluid masses are –
when ρ is the fluid's mass density – equal to density times volume,
so ρA1s1 and ρA2s2. By mass conservation, these two masses displaced in the
time interval Δt have to be equal, and this displaced mass is denoted
by Δm:

The work done by the forces consists of two parts:

• The work done by the pressure acting on the areas A1 and A2

• The work done by gravity: the gravitational potential energy in the
volume A1s1 is lost, and at the outflow in the volume A2s2 is gained. So, the
change in gravitational potential energy ΔEpot,gravity in the time interval Δt is

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

potential energy, Wgravity = −ΔEpot,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 Δz = z2 − z1, while the corresponding potential energy
change is positive.[21]: 14–4, §14–3  So:

And therefore the total work done in this time interval Δt is

The increase in kinetic energy is

Putting these together, the work-kinetic energy theorem W = ΔEkin gives:[19]

or

After dividing by the mass Δm = ρA1v1 Δt = ρA2v2 Δt the result is:[19]

or, as stated in the first paragraph:

(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:

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

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

The term v2/2g 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

with p0 some reference pressure, or when we rearrange it as a head:

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.

(Eqn. 2b)
If we were to multiply Eqn. 1 by the density of the fluid, we would get an
equation with three pressure terms:

(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 third term)
increases, and if the pressure due to elevation (the middle term) is
constant, then we know that the dynamic pressure (the first 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 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:
.
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 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,

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 gravitational energy of the fluid, the fluid
thermodynamic internal energy per unit of mass (ε1)
entering, and the energy entering in the form of
mechanical p dV work:

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 ΔE2 may easily be constructed. So now
setting 0 = ΔE1 − ΔE2:

which can be rewritten as:

Now, using the previously-obtained result from

conservation of mass, this may be simplified to
obtain

which is the Bernoulli equation for compressible

flow.

An equivalent expression can be written in terms of

fluid enthalpy (h):
 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[22] 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.[d][23] 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[24] – 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.[1]: § 3.8 
• A De Laval nozzle utilizes Bernoulli's principle to create a force by turning pressure
energy generated by the combustion of propellants into velocity. This then generates
thrust by way of Newton's third law of motion.
• 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.[25]

• The Bernoulli grip relies on this principle to create a non-contact adhesive force between
a surface and the gripper.
• Bernoulli's principle is also applicable in the swinging of a cricket ball. During a cricket
match, bowlers continually polish one side of the ball. After some time, one side is quite
rough and the other is still smooth. Hence, when the ball is bowled and passes through
air, the speed on one side of the ball is faster than on the other, due to this difference in
smoothness, and this results in a pressure difference between the sides; this leads to the
ball rotating ("swinging") while travelling through the air, giving advantage to the
bowlers.
 References:

1. ^ a b c d e f g Clancy, L.J. (1975). Aerodynamics. Wiley. ISBN 978-0-470-15837-1.

2. ^ a b c d e f g h Batchelor, G.K. (2000). An Introduction to Fluid Dynamics. Cambridge: Cambridge
University Press. ISBN 978-0-521-66396-0.
3. ^ "Hydrodynamica". Britannica Online Encyclopedia. Retrieved 2008-10-30.
4. ^ Anderson, J.D. (2016), "Some reflections on the history of fluid dynamics", in Johnson, R.W.
(ed.), Handbook of fluid dynamics(2nd ed.), CRC Press, ISBN 9781439849576
5. ^ Darrigol, O.; Frisch, U. (2008), "From Newton's mechanics to Euler's equations", Physica D:
Nonlinear Phenomena, 237 (14–17): 1855–
1869, Bibcode:2008PhyD..237.1855D, doi:10.1016/j.physd.2007.08.003
6. ^ Streeter, Victor Lyle (1966). Fluid mechanics. New York: McGraw-Hill.
7. ^ Babinsky, Holger (November 2003), "How do wings work?", Physics Education, 38 (6): 497–
503, Bibcode:2003PhyEd..38..497B, doi:10.1088/0031-9120/38/6/001
8. ^ "Weltner, Klaus; Ingelman-Sundberg, Martin, Misinterpretations of Bernoulli's Law, archived
from the original on April 29, 2009
9. ^ Denker, John S. (2005). "3 Airfoils and Airflow". See How It Flies. Retrieved 2018-07-27.
10. ^ Resnick, R. and Halliday, D. (1960), section 18-4, Physics, John Wiley & Sons, Inc.
11. ^ Mulley, Raymond (2004). Flow of Industrial Fluids: Theory and Equations. CRC Press. pp. 43–
44. ISBN 978-0-8493-2767-4.
12. ^ a b Chanson, Hubert (2004). Hydraulics of Open Channel Flow. Elsevier. p. 22. ISBN 978-0-08-
047297-3.
13. ^ Oertel, Herbert; Prandtl, Ludwig; Böhle, M.; Mayes, Katherine (2004). Prandtl's Essentials of
Fluid Mechanics. Springer. pp. 70–71. ISBN 978-0-387-40437-0.
14. ^ "Bernoulli's Equation". NASA Glenn Research Center. Archived from the original on 2012-07-31.
Retrieved 2009-03-04.
15. ^ White, Frank M. Fluid Mechanics, 6th ed. McGraw-Hill International Edition. p. 602.
16. ^ Clarke, Cathie; Carswell, Bob (2007). Principles of Astrophysical Fluid Dynamics. Cambridge
University Press. p. 161. ISBN 978-1-139-46223-5.
17. ^ Landau, L.D.; Lifshitz, E.M. (1987). Fluid Mechanics. Course of Theoretical Physics (2nd ed.).
Pergamon Press. ISBN 978-0-7506-2767-2.
18. ^ Van Wylen, Gordon J.; Sonntag, Richard E. (1965). Fundamentals of Classical Thermodynamics.
New York: John Wiley and Sons.
19. ^ a b c Feynman, R.P.; Leighton, R.B.; Sands, M. (1963). The Feynman Lectures on Physics. Vol.
2. ISBN 978-0-201-02116-5. : 40–6 to 40–9, §40–3 
20. ^ Tipler, Paul (1991). Physics for Scientists and Engineers: Mechanics(3rd extended ed.). W. H.
Freeman. ISBN 978-0-87901-432-2., p. 138.
21. ^ Feynman, R.P.; Leighton, R.B.; Sands, M. (1963). The Feynman Lectures on Physics. Vol.
1. ISBN 978-0-201-02116-5.
22. ^ Thomas, John E. (May 2010). "The Nearly Perfect Fermi Gas" (PDF). Physics Today. 63 (5): 34–
37. Bibcode:2010PhT....63e..34T. doi:10.1063/1.3431329.
23. ^ 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.")
24. ^ Eastlake, Charles N. (March 2002). "An Aerodynamicist's View of Lift, Bernoulli, and
Newton" (PDF). The Physics Teacher. 40 (3): 166–
173. Bibcode:2002PhTea..40..166E. doi:10.1119/1.1466553. "The resultant force is determined by
integrating the surface-pressure distribution over the surface area of the airfoil."
25. ^ Mechanical Engineering Reference Manual Ninth Edition
26. ^ Glenn Research Center (2006-03-15). "Incorrect Lift Theory". NASA. Retrieved 2010-08-12.