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.
Bernoulli's principle is named after the
DutchSwiss mathematician Daniel Bernoulli who published his
principle in his book Hydrodynamica in 1738.
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 a
streamline 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 potential
ρ g h) is the same everywhere.
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.
The original form of Bernoulli's equation  valid at any point
along a
streamline 
is:
 {v^2 \over 2}+\Psi+{p\over\rho}=\text{constant}
where:
 v is the fluid flow speed at a
point on a streamline,
 Ψ is the gravitational potential,
 p is the pressure at the
point, and
 ρ is the density of the fluid
at all points in the fluid.
The following assumptions must be met for the equation to apply:
 The fluid must be incompressible  even though pressure varies,
the density must remain constant.
 The streamline must not enter a boundary layer. (Bernoulli's equation is not
applicable where there are viscous forces, such as in a boundary layer.)
If the acceleration due to gravity (
g) does not change
over the length scale of the problem (eg the situation occurs at an
elevation which is low compared with the
radius of the Earth) then
Ψ =
g z, where
z
is the elevation of the point above some reference plane (positive
zdirection points upward, in the opposite direction to
the gravitational acceleration). This will be assumed for the rest
of this section.
By multiplying with the mass density
ρ, the above equation
can be rewritten as:
\tfrac12\, \rho\, v^2\, +\, \rho\, g\, z\, +\, p\, =\,
\text{constant}\,
or:
q\, +\, \rho\, g\, h\,
=\, p_0\, +\, \rho\, g\, z\,
=\, \text{constant}\,
where:
 q\, =\, \tfrac12\, \rho\, v^2 is dynamic pressure,
 h\, =\, z\, +\, \frac{p}{\rho g} is the piezometric head or hydraulic head (the sum of the elevation
z and the pressure head) ,
410 pages. See pp. 4344. , 650 pages. See p. 22. and
 p_0\, =\, p\, +\, q\, is the total pressure
(the sum of the static pressure p and dynamic pressure
q).
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}{2\,g}\, =\,
h\, +\, \frac{v^2}{2\,g}, so divide the above constant by
ρ and g to get the total head H in terms
of metres of fluid column.
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
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:
 p + q = p_0\,
where
p_{0} is called total pressure, and
q is
dynamic pressure.
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."
The simplified form of Bernoulli's equation can be summarized in
the following memorable word equation:
 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. 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.
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 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 nonzero 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:
 \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 ∂
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 is a
constant.
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)\, \operatorname{d}\tau
resulting in \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:
∇
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 freesurface flows using the
Lagrangian (not to be confused with
Lagrangian coordinates).
Lift and Drag curves for a typical airfoil
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.
Compressible flow in fluid dynamics
For a compressible fluid, with a
barotropic equation
of state, and under the action of
conservative forces,
 \frac {v^2}{2}+ \int_{p_1}^p \frac
{d\tilde{p}}{\rho(\tilde{p})}\ + \Psi = \text{constant} (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
 \frac {v^2}{2}+ gz+\left(\frac {\gamma}{\gamma1}\right)\frac
{p}{\rho} = \text{constant} (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}{\gamma1}\right)\frac
{p}{\rho} = \left(\frac {\gamma}{\gamma1}\right)\frac
{p_0}{\rho_0}
where:
 p_{0} is the total pressure
 ρ_{0} is the total density
Compressible flow in thermodynamics
Another useful form of the equation, suitable for use in
thermodynamics, is:
 {v^2 \over 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 = \epsilon + \frac{p}{\rho} 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:
 {v^2 \over 2}+ w = w_0
where
w_{0} is total enthalpy.
When
shock waves are present, in a
reference frame moving with a shock,
many of the parameters in the Bernoulli equation suffer abrupt
changes in passing through the shock. The Bernoulli parameter
itself, however, remains unaffected. An exception to this rule is
radiative shocks, which violate the assumptions leading to the
Bernoulli equation, namely the lack of additional sinks or sources
of energy.
Derivations of Bernoulli equation
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Real world application
In modern everyday life there are many observations that can be
successfully explained by application of Bernoulli's
principle.
 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. 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  established by Bernoulli over a
century before the first manmade 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 KuttaJoukowski 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 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.
 The principle also makes it possible for sailpowered 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
sailing vehicles can travel faster than the wind.
Misunderstandings about the generation of lift
Many explanations for the generation of
lift(on
airfoils,
propellerblades, 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.
Several of these explanations use Bernoulli's principle to connect
the flow kinematics to the flowinduced pressures. In case of
incorrect (or partially correct) explanations of lift, also relying
at some stage on Bernoulli's principle, the errors generally occur
in the assumptions on the flow kinematics, and how these are
produced. It is not Bernoulli's principle itself that is questioned
because this principle is well established.
References
 Clancy, L.J., Aerodynamics, Chapter 3.
 Batchelor, G.K. (1967), Section 3.5, pp. 15664.
 Streeter, V.L., Fluid Mechanics, Example 3.5,
McGraw–Hill Inc. (1966), New York.
 Clancy, L.J., Aerodynamics, Section 3.4.
 Clancy, L.J., Aerodynamics, Section 3.5.
 Clancy, L.J. Aerodynamics, Equation 3.12
 Batchelor, G.K. (1967), p. 383
 Clarke C. and Carswell B., Astrophysical Fluid
Dynamics
 Clancy, L.J., Aerodynamics, Section 3.11
 Van Wylen, G.J., and Sonntag, R.E., (1965), Fundamentals of
Classical Thermodynamics, Section 5.9, John Wiley and
Sons Inc., New York
 , p. 138.
 , Vol. 1, §143, p. 144.
 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 185, 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.")
 "The resultant force is determined by integrating the
surfacepressure distribution over the surface area of the
airfoil."
 Clancy, L.J., Aerodynamics, Section 3.8
 Mechanical Engineering Reference Manual Ninth Edition
 Ice Sailing Manual, p. 2
 Wind Sports – Ice sailing hand held sails
 Ison, David. Bernoulli Or Newton: Who's Right About Lift?
Retrieved on 20091126
 Section 2.4.
 Batchelor, G.K. (1967). Sections 3.5 and 5.1
 Lamb, H. (1994). §17§29
 "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."
Notes
 Clancy, L.J., Aerodynamics, Chapter 3.
 Batchelor, G.K. (1967), Section 3.5, pp. 15664.
 Streeter, V.L., Fluid Mechanics, Example 3.5,
McGraw–Hill Inc. (1966), New York.
 Clancy, L.J., Aerodynamics, Section 3.4.
 Clancy, L.J., Aerodynamics, Section 3.5.
 Clancy, L.J. Aerodynamics, Equation 3.12
 Batchelor, G.K. (1967), p. 383
 Clarke C. and Carswell B., Astrophysical Fluid
Dynamics
 Clancy, L.J., Aerodynamics, Section 3.11
 Van Wylen, G.J., and Sonntag, R.E., (1965), Fundamentals of
Classical Thermodynamics, Section 5.9, John Wiley and
Sons Inc., New York
 , p. 138.
 , Vol. 1, §143, p. 144.
 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 185, 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.")
 "The resultant force is determined by integrating the
surfacepressure distribution over the surface area of the
airfoil."
 Clancy, L.J., Aerodynamics, Section 3.8
 Mechanical Engineering Reference Manual Ninth Edition
 Ice Sailing Manual, p. 2
 Wind Sports – Ice sailing hand held sails
 Ison, David. Bernoulli Or Newton: Who's Right About Lift?
Retrieved on 20091126
 Section 2.4.
 Batchelor, G.K. (1967). Sections 3.5 and 5.1
 Lamb, H. (1994). §17§29
 "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."
Further reading
 Originally published in 1879; the 6^{th} extended
edition appeared first in 1932.
See also
External links
Bernoulli equation for incompressible fluids 

The Bernoulli equation for incompressible fluids can be derived
by integrating the Euler equations, or applying the law of
conservation of energy in two
sections along a streamline, ignoring viscosity, compressibility, and thermal
effects.
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 crosssectional area
A is
 m \frac{\operatorname{d}v}{\operatorname{d}t}= F
 \rho A \operatorname{d}x
\frac{\operatorname{d}v}{\operatorname{d}t}= A
\operatorname{d}p
 \rho \frac{\operatorname{d}v}{\operatorname{d}t}=
\frac{\operatorname{d}p}{\operatorname{d}x}
In steady flow, v = v(x)
so
 \frac{\operatorname{d}v}{\operatorname{d}t}=
\frac{\operatorname{d}v}{\operatorname{d}x}\frac{\operatorname{d}x}{\operatorname{d}t}
=
\frac{\operatorname{d}v}{\operatorname{d}x}v=\frac{d}{\operatorname{d}x}
\left( \frac{v^2}{2} \right).
With density ρ constant, the equation of motion can be
written as
 \frac{\operatorname{d}}{\operatorname{d}x} \left( \rho
\frac{v^2}{2} + p \right) =0
or
 \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 workenergy principle is
invoked. Rather, Bernoulli's principle was inherently derived by a
simple manipulation of the momentum equation.
Another way to derive Bernoulli's principle for an incompressible
flow is by applying conservation of energy. , Vol. 2, §403, p.
406  409. In the form of the workenergy theorem, stating that
 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 = increase in kinetic energy.
The system consists of the volume of fluid, initially between the
crosssections A_{1} and A_{2}.
In the time interval Δt fluid elements initially at the
inflow crosssection A_{1} move over a distance
s_{1} = v_{1} Δt,
while at the outflow crosssection the fluid moves away from
crosssection A_{2} over a distance
s_{2} = v_{2} Δt.
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
ρ A_{1} s_{1}
and
ρ A_{2} s_{2}.
By mass conservation, these two masses displaced in the time
interval Δt have to be equal, and this displaced mass is
denoted by Δm:
\begin{align}
\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{align}
The work done by the forces consists of two parts:
 The work done by the pressure acting on the area's
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
ΔE_{pot,gravity} in the time interval Δt
is
 : \Delta E_\text{pot,gravity} = \Delta m\, g z_{2}  \Delta m\,
g z_{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 zdirection,
the workgravity force times change in elevationwill be negative
for a positive elevation change
Δz = z_{2} − z_{1},
while the corresponding potential energy change is positive.
So:
 : 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 \Delta t is
 W = W_\text{pressure} + W_\text{gravity}. \,
The increase in kinetic energy is
 \Delta E_\text{kin} = \frac{1}{2} \Delta m\,
v_{2}^{2}\frac{1}{2} \Delta m\, v_{1}^{2}.
Putting these together, the workkinetic energy theorem
W = ΔE_{kin} gives:
 \Delta m\, \frac{p_{1}}{\rho}  \Delta m\, \frac{p_{2}}{\rho} +
\Delta m\, g z_{1}  \Delta m\, g z_{2} = \frac{1}{2} \Delta m\,
v_{2}^{2}  \frac{1}{2} \Delta m\, v_{1}^{2}
or
 \frac12 \Delta m\, v_{1}^{2} + \Delta m\, g z_{1} + \Delta m\,
\frac{p_{1}}{\rho} = \frac12 \Delta m\, v_{2}^{2} + \Delta m\, g
z_{2} + \Delta m\, \frac{p_{2}}{\rho}.
After dividing by the mass
Δm = ρ A_{1} v_{1} Δt = ρ A_{2} v_{2} Δt
the result is:
 \frac12 v_{1}^{2}+g z_{1}+\frac{p_{1}}{\rho}=\frac12
v_{2}^{2}+g z_{2}+\frac{p_{2}}{\rho}
or, as stated in the first paragraph:
 \frac{v^{2}}{2}+g z+\frac{p}{\rho}=C (Eqn.
1)
Further division by g produces the following equation.
Note that each term can be described in the length dimension (such as meters). This is the head
equation derived from Bernoulli's principle:
 \frac{v^{2}}{2 g}+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{{2 g}{z}}, when arriving at elevation z=0. Or
when we rearrange it as a head: h_{v}=\frac{v^{2}}{2
g}
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 g z \,, 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 g z + 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 Δt, 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 volumeof 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 ΔE_{1} and ΔE_{2} are the
energy entering through A_{1} and leaving through
A_{2}, respectively.
The energy entering through A_{1} is the sum of
the kinetic energy entering, the energy entering in the form of
potential gravitational energy of the fluid, the fluid
thermodynamic energy entering, and the energy entering in the form
of mechanical p dV work:
 \Delta E_1 = \left[\frac{1}{2} \rho_1 v_1^2 + \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 \Delta E_2 may easily be constructed.So
now setting 0 = \Delta E_1  \Delta E_2:
 0 = \left[\frac{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[ \frac{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[ \frac{1}{2} v_1^2 + \Psi_1 + \epsilon_1 +
\frac{p_1}{\rho_1} \right] \rho_1 A_1 v_1 \, \Delta t  \left[
\frac{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 previouslyobtained result from conservation of
mass, this may be simplified to obtain
 \frac{1}{2}v^2 + \Psi + \epsilon + \frac{p}{\rho} = {\rm
constant} \equiv b
which is the Bernoulli equation for compressible flow. 