Bernoulli's principle

From Wikipedia, the free encyclopedia

This article is about Bernoulli's principle and Bernoulli's equation in fluid dynamics. For
Bernoulli's Theorem (probability), see Law of large numbers. For an unrelated topic
in ordinary differential equations, seeBernoulli differential equation.

A flow of air 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.

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in

the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease
in the fluid's potential energy.[1][2] Bernoulli's principle is named after the Dutch-
Swiss mathematician Daniel Bernoulli who published his principle in his
bookHydrodynamica in 1738.[3]

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is
loosely denoted as Bernoulli's equation. In fact, there are different forms of the
Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is
valid for incompressible flows (e.g. most liquid flows) and also for compressible
flows(e.g. gases) moving at low Mach numbers. More advanced forms may in some
cases 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 mechanical energy in a fluid along
astreamline is the same at all points on that streamline. This requires that the sum of
kinetic energy and potential energy remain constant. 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 mass (the sum of pressure and gravitational
potentialρ g h) is the same everywhere.[4]

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 mass density of a fluid
parcel can be considered to be constant, regardless of pressure variations in the flow.
For this reason the fluid in such flows can be considered to be incompressible and
these flows can be described as incompressible flow. Bernoulli performed his
experiments on liquids and 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 where gravity is constant, is:

(
A
 

)
where:
 is the fluid flow speed at a point on a streamline,
 is the acceleration due to gravity,
 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,
 is the pressure at the point, and
 is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be

generalized as:[5]
 where Ψ is the force potential at the point considered on
the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this

Bernoulli equation to apply:[5]

 the fluid must be incompressible — even though pressure

varies, the density must remain constant along a
streamline;
 friction by viscous forces has to be negligible.
By multiplying with the fluid density ρ, equation (A) can
be rewritten as:

or:

where:

 is dynamic pressure,

 is the piezometric head or hydraulic head (the sum of the

elevation z and the pressure head)[6][7] and
 is the total pressure (the sum of the static pressure p and dynamic
pressure q).[8]

The constant in the Bernoulli equation can be normalised. 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.
[edit]Simplified form
In many applications of Bernoulli's equation, the change in
the ρ g z term along the streamline is so small compared
with the other terms 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 ρ g z 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[9]. 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."[10]

The simplified form of Bernoulli's equation can be

summarized in the following memorable word equation:

static pressure + dynamic pressure = total pressure[10]

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.[11] 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 stream line is

brought to rest, this point is called a stagnation
point, and at this point the total pressure is equal to
the stagnation pressure.
[edit]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—can not be assumed to be valid.
However if the gas process is entirelyisobaric,
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.
[edit]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:[12]

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 functionf(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.[12]

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 withLagrangian 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 at very low speeds
(perhaps up to 1/3 of the sound speed in the fluid). 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.
[edit]Compressible flow in fluid dynamics
For a compressible fluid, with a barotropic equation of state, and under the action
of conservative forces,

[13]
   (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

[14]
   (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:

where:

p0 is the total pressure

ρ0 is the total density

Compressible flow in thermodynamics

Another useful form of the equation, suitable for use in thermodynamics, is:

[15]

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   where ε is the thermodynamic energy per unit mass, also

known as the specific internal energy or "sie."

The constant on the right hand side is often called the Bernoulli constant and
denoted b. For steady inviscid adiabatic flow with no additional sources or sinks of
energy, b is constant along any given streamline. More generally, when b may vary
along streamlines, it still proves a useful parameter, related to the "head" of the
fluid (see below).

When 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.

Derivations of Bernoulli equation

[hide]Bernoulli equation for incompressible fluids

The Bernoulli equation for incompressible fluids can be

derived by integrating the Euler equations, or applying the
law ofconservation of energy in two sections along a
streamline, ignoring viscosity, compressibility, and thermal
effects.

The simplest derivation is to first ignore gravity and consider

constrictions and expansions in pipes that are otherwise
straight, as seen in Venturi effect. Let the x axis be directed
down the axis of the pipe.

The equation of motion for a parcel of fluid, having a length

dx, mass density ρ, mass m = ρ A dx and flow
velocity v = dx / dt, moving along the axis of the horizontal
pipe, with cross-sectional area A is

In steady flow, v = v(x) so

With density ρ constant, the equation

of motion can be written as

or

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
inherently derived by a
simple manipulation of the
momentum equation.
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.

Another way to derive Bernoulli's principle for an incompressible flow is by applying

conservation of energy.[16] In the form of thework-energy theorem, stating that[17]

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 = 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 A1 s1 and A2 s2. The associated displaced fluid masses are—
when ρ is the fluid's mass density – equal to density times volume,
so ρ A1 s1 andρ A2 s2. 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 area's A1 and A2

 The work done by gravity: the gravitational potential energy in the

volume A1 s1 is lost, and at the outflow in the volume A2 s2 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.[18] So:
Failed to parse (lexing error): W_\text{gravity} = -\Delta E_\text{pot,\,gravity}
= \Delta m\, g z_1 - \Delta m\, g z_2. \;

And 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:[16]

or

After dividing by the mass Δm = ρ A1 v1 Δt = ρ A2 v2 Δt the result is:[16]

 or, as stated in the first paragraph:

   (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:

   (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 / (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

, 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 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 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 energy 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.

Real-world application

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 [19] 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 you know the
behavior of the fluid flow in the vicinity of the foil. 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 lift force.[nb 1][20] 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[21]—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. To understand why, it is helpful to understand circulation,
the Kutta condition, and the Kutta–Joukowski theorem.

 The carburetor used in many reciprocating engines contains a venturi to create a region

of low pressure to draw fuel into the carburetor 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.

 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 airspeedappropriate to the dynamic
pressure.[22]

 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 Reynold's number and the shape of the
orifice.[23]

 In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and its

extension were recently developed[24]. It was proved that the depth-averaged specific energy
reaches a minimum in converging accelerating free-surface flow over weirs and flumes
(also [25][26]). Further, in general, a channel control with minimum specific energy in curvilinear
flow is not isolated from water waves, as customary state in open-channel hydraulics.

 The principle also makes it possible for sail-powered craft to travel faster than the wind
that propels them (if friction can be sufficiently reduced). If the wind passing in front of the
sail is fast enough to experience a significant reduction in pressure, the sail is pulled
forward, in addition to being pushed from behind. Although boats in water must contend with
the friction of the water along the hull, ice sailing and land sailingvehicles can travel faster
than the wind.[27][28]
Misunderstandings about the generation of lift

Main article: Lift (force)

Many explanations for the generation of lift (on airfoils, propeller blades, etc.) can be

found; but some of these explanations can be misleading, and some are false. 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 Bernoulli's principle and Newton's laws are both relevant
and correct [29][30][31].

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 of
lift, also relying at some stage 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. [32][33]
[34][35]