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 book Hydrodynamica 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. But 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 is equivalent to 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 ρgh) 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.
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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[5] is:
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 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.The following assumptions must be met for the equation to apply:
The above equation can be rewritten, in two different ways, as:
where:
is dynamic pressure,
is the piezometric head or hydraulic head, the sum of the elevation z and the pressure head (the static pressure contribution) p/(ρg),[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:
so divide the above constant by ρ and g to get the total head H in terms of metres of fluid column.[7][6]The above equations suggest there is a flow speed at which pressure is zero and at higher speeds the pressure is negative. 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. The above equations use a linear relationship between flow speed squared and pressure. At higher velocities in liquids, non-linear processes such as (viscous) turbulent flow and cavitation occur. At higher flow speeds in gases the changes in pressure become significant so that the assumption of constant density is invalid.
In several applications of Bernoulli's equation, the change in the 
term along streamlines is zero or so small it can be ignored: for instance in the case of airfoils at low Mach number. This allows the above equation to be presented in the following simplified form:
where 
is called total pressure, and 
is dynamic pressure[9]. Many authors refer to the pressure as static pressure to distinguish it from total pressure and dynamic pressure . 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:
Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure p, dynamic pressure q, and total pressure p0.
The significance of Bernoulli's principle can now be summarized as "total pressure is constant along a streamline." Furthermore, if the fluid flow originated in a reservoir, 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." However, it is important to remember that Bernoulli's principle does not apply in the boundary layer.
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 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 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.
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:[11]
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.[11]
Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation
resulting in 
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).
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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.
A useful form of the equation, suitable for use in compressible fluid dynamics, is:
[12] (constant along a streamline)where:
is the ratio of the specific heats of the fluid is the pressure at a point is the density at the point is the speed of the fluid at the point is the acceleration due to gravity is the elevation of the point above a reference planeIn many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term 
can be omitted. A very useful form of the equation is then:
where:
is the total pressure
is the total densityAnother useful form of the equation, suitable for use in thermodynamics, is:
[13]
is the enthalpy per unit mass, which is also often written as 
(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 or "sie."
The constant on the right hand side is often called the Bernoulli constant and denoted 
. For steady inviscid adiabatic flow with no additional sources or sinks of energy, is constant along any given streamline. More generally, when may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).
When the change in can be ignored, a very useful form of this equation is:
where 
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.
| 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 on the axis of the pipe is In steady flow, v = v(x) so With 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. We deduce that where the speed is large, pressure is low and vice versa. In the above derivation, no external work-energy principle is invoked. Rather, the work-energy principle was inherently derived by a simple manipulation of the momentum equation. 
Applying conservation of energy in form of the work-kinetic energy theorem we find that:
Therefore,
The work done by the forces is The decrease of potential energy is The increase in kinetic energy is Putting these together, or After dividing by Δt, ρ and A1v1 (= rate of fluid flow = A2v2 as the fluid is incompressible): or, as stated in the first paragraph:
Further division by
The middle term, A free falling mass from an elevation
The term The hydrostatic pressure p is defined as
The term 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.
If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms:
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  , the amount of mass passing through the boundary defined by the area  is equal to the amount of mass passing outwards through the boundary defined by the area  :
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 where ΔE1 and The energy entering through where 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. |
In every-day life there are many observations that can be successfully explained by application of Bernoulli's principle.
Many explanations for the generation of lift 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. Modern writings agree that Bernoulli's principle and Newton's Laws are both relevant and correct. [19][20]
Several of these explanations use Bernoulli's principle to connect the flow kinematics to the flow-induced 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[21][22][23].
(Eqn. 1)
(Eqn. 2a)
(in a 
when arriving at elevation z=0, or when we rearrange it as a head: 
is called the velocity 
, with p0 some reference pressure, or when we rearrange it as a head: 
is also called the 
(Eqn. 2b)
(Eqn. 3)
.
are the energy entering through 
work:![\Delta E_1 = \left[\frac{1}{2} \rho_1 v_1^2 + \phi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t](http://upload.wikimedia.org/math/9/3/6/9364fdec271c653b66acf1071156bb5c.png)
is a ![0 = \left[\frac{1}{2} \rho_1 v_1^2+ \phi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t - \left[ \frac{1}{2} \rho_2 v_2^2 + \phi_2\rho_2 + \epsilon_2 \rho_2 + p_2 \right] A_2 v_2 \, \Delta t](http://upload.wikimedia.org/math/4/2/c/42ca460091570d3e1e4411e0f23f431c.png)
![0 = \left[ \frac{1}{2} v_1^2 + \phi_1 + \epsilon_1 + \frac{p_1}{\rho_1} \right] \rho_1 A_1 v_1 \, \Delta t - \left[ \frac{1}{2} v_2^2 + \phi_2 + \epsilon_2 + \frac{p_2}{\rho_2} \right] \rho_2 A_2 v_2 \, \Delta t](http://upload.wikimedia.org/math/c/a/3/ca33bead05e8d1d09da0924e542c7923.png)
