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 DutchSwiss 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. 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.^{[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. A common form of Bernoulli's equation, valid at any arbitrary point along a streamline where gravity is constant, is:
where:
If gravity is constant, then Bernouli's equation can be generalized as:
where:
The following assumptions must be met for these equations to apply:
By multiplying with the mass density ρ, the above equation can be rewritten as:
or:
where:
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.
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 p_{0} is called total pressure, and q is dynamic pressure^{[8]}. 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."^{[9]}
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 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.^{[10]} 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.
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.
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
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).
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.
For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,
where:
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
where, in addition to the terms listed above:
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:
Another useful form of the equation, suitable for use in thermodynamics, is:
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 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.
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 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 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.^{[15]} In the form of the workenergy theorem, stating that^{[16]}
Therefore,
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: The work done by the forces consists of two parts:
And the total work done in this time interval Δt is The increase in kinetic energy is Putting these together, the workkinetic energy theorem W = ΔE_{kin} gives:^{[15]} or After dividing by the mass Δm = ρ A_{1} v_{1} Δt = ρ A_{2} v_{2} Δt the result is:^{[15]} or, as stated in the first paragraph:
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:
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
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
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.
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 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}:
Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by A_{1} and A_{2} is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero, 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: 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 ΔE_{2} may easily be constructed. So now setting 0 = ΔE_{1} − ΔE_{2}: which can be rewritten as: Now, using the previouslyobtained result from conservation of mass, this may be simplified to obtain which is the Bernoulli equation for compressible flow. 
In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle.
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 ^{[27]} ^{[28]}^{[29]}.
Several of these explanations use the Bernoulli 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 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^{[30]}^{[31]}^{[32]}^{[33]}.
