User:Mr swordfish/Bernoulli principle



In analytical fluid dynamics, Bernoulli's principle states that for an ideal fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. The principle is named after Daniel Bernoulli who published it in his book Hydrodynamica in 1738. . Actually nowadays it is no more considered as a principle but rather as a theorem of conservation of energy for Euler equations, which hold for ideal fluids. 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. Thus an increase in the speed of the fluid – implying an increase in both its dynamic pressure and kinetic energy – occurs with a simultaneous decrease in (the sum of) its static pressure, potential energy 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.

Bernoulli's principle can be applied to various types of ideal fluids, 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 gases moving at low Mach number). 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 formally derived from the Euler momentum equation basing on a differential identity. Bernoulli's principle can also be linked to an expression of Newton's 2nd law. 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.

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. 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.

Formal derivation
The following derivation is mostly based on Feynman's physics, Vol. 2, §40-3 and on Childress 2008, p. 20-21. For an ideal fluid, the Euler equations hold: the momentum equation among them put in lagrangian form is:

$$\frac{D\boldsymbol u}{D t}+\frac {\nabla p} \rho -\boldsymbol{g}=\boldsymbol{0}$$

or explicitly:

$$\frac{\partial \boldsymbol u}{\partial t} + \mathbf u \cdot \nabla \mathbf u +\nabla \left( \frac p \rho \right) -\boldsymbol{g}=\boldsymbol{0}$$

where:
 * $$\frac{D}{D t}$$ is the material derivative,
 * ρ is the fluid mass density,
 * u is the flow velocity vector, with components in a 3D space u1, u2, and u3,
 * p is the pressure.
 * g is the external field, not necessarily given by Earth's gravity. Generally it can include nonconservative terms.

The following tensor calculus identity holds for the covariant derivative of a sufficiently regular vector field:

$$ \mathbf a \cdot \nabla \mathbf a = ( \nabla \times \mathbf a ) \times \mathbf a + \nabla \left( \frac 12 a^2 \right) $$

By considering this identity for the covariant derivative of the flow velocity the momentum equation becomes:

$$ \nabla \left(\frac 1 2 u^2 + \frac p \rho \right) - \boldsymbol{g} = - \frac{\partial \boldsymbol u}{\partial t} - \boldsymbol \omega \times \mathbf u $$

where the vorticity vector has been defined:

$$\boldsymbol \omega \equiv \nabla \times \mathbf u $$.

Now if the external field is conservative, by indicating with φ the associated scalar potential energy:

$$\mathbf g \equiv - \nabla \phi $$.

then the Euler momentum equation becomes:

$$ \nabla \psi =  - \frac{\partial \boldsymbol u}{\partial t} -  \boldsymbol \omega \times \mathbf u $$

where the specific mechanical energy (also called Bernoulli function) has been defined:

$$ \psi \equiv \frac 1 2 u^2 + \frac p \rho + \phi$$

For a steady motion the time derivative of the kinetic potential vanishes:

$$ \nabla \psi = -  \boldsymbol \omega \times \mathbf u $$

Now in every domain where the fluid is not rotating:

$$ \boldsymbol \omega = \mathbf 0 $$

the head is completely uniform:

$$ \nabla \psi = \mathbf 0 $$

This is the banal solution. In general, the fluid is rotating so the head is nonuniform. We can still make the vorticity term disappear by dotting this equation for the flow velocity, i.e. considering the flow along a streamline:

$$\mathbf u \cdot \nabla \psi = - \mathbf u \cdot \boldsymbol \omega \times \mathbf u $$

The RHS vanishes by an identity for the scalar triple product. So we finally arrive to Bernoulli's principle: for a steady ideal fluid the hydraulic head is constant along a streamline.

$$\mathbf u \cdot \nabla \psi = 0$$


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!Connections with Kelvin's theorem


 * Bernoulli equation has been obtained essentially by dotting Euler momentum equation in case of conservative external field for the flow velocity, i.e. considering its projection along a streamline.

In case of incompressible fluids the Euler mass equation become:

$$ \nabla \cdot \mathbf u =0 $$

by taking the curl of this equation:

$$ \nabla \cdot \boldsymbol\omega =0 $$

so also the vorticity field is conservative.

One can complete Helmholtz decomposition of the vorticity field by crossing the Euler momentum equation for the flow velocity, i.e. considering its projection orthogonal to the streamline. The equation:

$$ \nabla \psi =  - \frac{\partial \mathbf u}{\partial t} -  \boldsymbol \omega \times \mathbf u $$

become by crossing and cahanging the order of time and spatial derivatives:

$$ \nabla \times \nabla \psi =  - \frac{\partial}{\partial t} \nabla \times \mathbf u -   \nabla \times (\boldsymbol \omega \times \mathbf u) $$

and basing on definition of vorticity and on the fact that the curl of a gradient is identically zero:

$$ \frac{\partial \boldsymbol \omega}{\partial t} +  \nabla \times (\boldsymbol \omega \times \mathbf u) = 0 $$

This equation is the thesis of the Kelvin's circulation theorem. Note that if as initial condition the vorticity is zero:

$$ \boldsymbol \omega (\mathbf x, 0) = \mathbf 0  $$

then a solution for this equation is:

$$ \boldsymbol \omega (\mathbf x, t) = \mathbf 0  $$

and for Helmholtz theorem, this is the solution for the IVP. This property and others, in contrast with our experience (think to the emptying of asink), suggests that even fluids that we would consider as inviscid like water actually are viscous.


 * }

Statements
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 flow. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. The informal statement in which Bernoulli's principle is reported is: for a steady ideal flow ,


 * $$\left({u^2 \over 2}+\phi+{p\over\rho} \right) \text{ is constant along a streamline}$$

where:
 * $$u\,$$ is the flow velocity at a point of the streamline considered,
 * $$\phi$$ is the external potential at the point. E.g. for the Earth's gravity φ = gz.
 * $$g\,$$ 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 point, and
 * $$\rho\,$$ is the density at the point.

While the same statement is written in rigorous form as the equation:


 * $$\mathbf u \cdot \nabla \left({u^2 \over 2}+\phi+{p\over\rho} \right) =0 \qquad$$ ($$)

where:
 * $$\nabla$$ indicates the gradient at a point,
 * $$\mathbf u \cdot \nabla$$ indicates the streamline directional derivative.

For both simplicity and respect for usual practice, we will keep using the informal statement with words followed by the formal statement given by the equation:


 * $$(x \text{ is c.a.a.s.}) \quad \Leftrightarrow \quad (\mathbf u \cdot \nabla x = 0) \qquad \forall x$$

In case of incompressible flow, studied in hydraulics:


 * $$\mathbf u \cdot \nabla \rho =0$$

by multiplying with the density $$\rho$$, the equation ($$) can be rewritten in total pressure form:


 * $$p_0\, \equiv p\, +\, q\,$$

where:

$$q\, \equiv \frac12\, \rho\, u^2$$

is the dynamic pressure.

So the particular statement used in hydraulics is "for a steady ideal incompressible fluid the sum between total pressure and external field pressure is constant along a streamline":


 * $$\mathbf u \cdot \nabla (p_0 + \rho \phi) = 0 \qquad$$ ($$)

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 incompressibility assumption is invalid.

One the other hand in applied fluid-dynamics the external potential is only due to Earth's gravity, the total specific energy is:


 * $$\psi (z) =g z +\, \frac{p}{\rho}\, +\, \frac{u^2}{2}$$

in the technical system it is common practise to rather adopt the total head, that is dimensionally a length:


 * $$H\, \equiv \frac {\psi (z)} g = \, z\, +\, \frac{p}{\rho g}\, +\, \frac{u^2}{2\,g}\, =\, h\, +\, \frac{u^2}{2\,g},$$

where:


 * $$h\, =\, z\, +\, \frac{p}{\rho g}$$ is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head).

So the particular statement used in applied fluid-dynamics is "for a steady ideal fluid in Earth's gravity the total head is constant along a streamline":


 * $$\mathbf u \cdot \nabla H =0 \qquad$$ ($$)

Thermal fluids form
A generalised form of the equation, suitable for thermal fluids is:


 * $${u^2 \over 2} + \phi + w =\text{constant}$$

Here w is the specific enthalpy, 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.

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 φ can be ignored, a very useful form of this equation is:


 * $${u^2 \over 2}+ w = w_0$$

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.

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 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 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 p0 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 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."

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 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. 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.

Potential flow
The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics.

For an irrotational flow with constant density, the momentum equations of the Euler equations can be integrated to:


 * $$\frac{\partial \varphi}{\partial t} + \psi = f(t),$$

which is a Bernoulli equation. Here ∂φ/∂t denotes the partial derivative of the velocity potential φ with respect to time t, and ψ is the fluid specific energy. 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 (becomning a total velocity potential) using the transformation:


 * $$\phi_0=\varphi-\int_{t_0}^t f(\tau)\, \text{d}\tau,$$ resulting in  $$\frac{\partial \phi_0}{\partial t} + \psi =0.$$

Note that the relation of the potential to the flow velocity is unaffected by this transformation: ∇φ0 = ∇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 function.

Compressible fluids: 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 – 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.

Compressible flow equation
Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and compressible fluids up to approximately Mach number 0.3. 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,


 * $$\frac {u^2}{2}+ \int_{p_1}^p \frac {d\tilde{p}}{\rho(\tilde{p})}\ + \phi = \text{constant}$$ (constant along a streamline)

where:
 * p is the pressure
 * ρ is the density
 * u is the flow speed
 * φ is the scalar potential associated with the conservative force field.

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


 * $$\frac {u^2}{2}+ gz+\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho}  = \text{constant}$$  (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:


 * $$\frac {u^2}{2}+\left( \frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}$$

where:
 * p0 is the total pressure
 * ρ0 is the total density

Newtonian explanation

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!Bernoulli equation for incompressible fluids
 * The Bernoulli equation for incompressible fluids can be explaines 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. Note that this is not a formal derivation since it does not demostrate by clear and necessary steps but rather introduces in order the terms that are supposed known since the beginning ('heuristic explanation'). In fact with such explanations one cannot exclude the presence of other neglected force terms in the equation.
 * The Bernoulli equation for incompressible fluids can be explaines 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. Note that this is not a formal derivation since it does not demostrate by clear and necessary steps but rather introduces in order the terms that are supposed known since the beginning ('heuristic explanation'). In fact with such explanations one cannot exclude the presence of other neglected force terms in the equation.

The simplest explanation 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 = ρ A 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.
 * Integrating Newton's Second Law of Motion
 * $$m \frac{\operatorname{d}v}{\operatorname{d}t}= F $$
 * $$\rho A  \operatorname{d}x \frac{\operatorname{d}v}{\operatorname{d}t}= -A \operatorname{d}p $$
 * $$\rho \frac{\operatorname{d}v}{\operatorname{d}t}= -\frac{\operatorname{d}p}{\operatorname{d}x} $$

In steady flow 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).
 * $$\frac{\operatorname{d}v}{\operatorname{d}t}=  \frac{\operatorname{d}v}{\operatorname{d}x}\frac{\operatorname{d}x}{\operatorname{d}t} = \frac{\operatorname{d}v}{\operatorname{d}x}v=\frac{d}{\operatorname{d}x} \left( \frac{v^2}{2} \right).$$

With density ρ constant, the equation of motion can be written as
 * $$\frac{\operatorname{d}}{\operatorname{d}x} \left( \rho \frac{v^2}{2} + p \right) =0$$

by integrating with respect to x
 * $$ \frac{v^2}{2} + \frac{p}{\rho}= C$$

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 explanation, no external work-energy principle is invoked. Rather, Bernoulli's principle was explatined by a simple manipulation of Newton's second law. Another way to explain Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that
 * Conservation of energy
 * the change in the kinetic energy Ekin 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 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 and ρ 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:

\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} $$ The work done by the forces consists of two parts:
 * The work done by the pressure acting on the areas A1 and A2
 * $$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 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
 * $$\Delta E_\text{pot,gravity} = \Delta m\, g z_2 - \Delta m\, g z_1. \;$$
 * Now, the work by the force of gravity is opposite to the change in potential energy, 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. 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 work-kinetic energy theorem W = ΔEkin 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 A1 v1 Δt = ρ A2 v2 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), 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:
 * $$\frac{v^{2}}{2 g}+z+\frac{p}{\rho g}=C$$ (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 zelevation. A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed
 * $$v=\sqrt,$$ when arriving at elevation z = 0. Or when we rearrange it as a head: $$h_v =\frac{v^2}{2 g}$$

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
 * $$p=p_0-\rho g z \,$$, with p0 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.
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!Bernoulli equation for compressible fluids
 * The explanation for compressible fluids is similar. Again, the explanation is based 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:
 * $$0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t$$.
 * $$0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t$$.

Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the 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,
 * $$0= \Delta E_1 - \Delta E_2 \,$$

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:
 * $$\Delta E_1 = \left[\frac{1}{2} \rho_1 v_1^2 + \Psi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t$$

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 previously-obtained 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.
 * }

Applications
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 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. 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 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 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 airspeed appropriate to the dynamic 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 Reynolds number and the shape of the orifice.


 * The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.

Misunderstandings about the generation of lift
Many explanations for the generation of lift (on airfoils, propeller blades, etc.) can be found; 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 both Bernoulli's principle and Newton's laws are relevant and either can be used to correctly describe lift.

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 relying 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.

Misapplications of Bernoulli's principle in common classroom demonstrations
There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle. One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it. As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure".

One problem with this explanation can be seen by blowing along the bottom of the paper - were the deflection due simply to faster moving air one would expect the paper to deflect downward, but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom. Another problem is that when the air leaves the demonstrator's mouth it has the same pressure as the surrounding air; the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is equal to the pressure of the surrounding air. A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation since the air above and below are different flow fields and Bernoulli's principle only applies within a flow field.

As the wording of the principle can change its implications, stating the principle correctly is important. What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa. Thus, Bernoulli's principle concerns itself with changes in speed and changes in pressure within a flow field. It cannot be used to compare different flow fields.

A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve. Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed, i.e. that as the air passes over the paper it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration.

Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure".

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 flow. 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
 * $$\rho\,$$ is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be generalized as:


 * $${v^2 \over 2}+\Psi+{p\over\rho}=\text{constant}$$

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:
 * the flow must be incompressible – even though pressure varies, the density must remain constant along a streamline;
 * friction by viscous forces has to be negligible. In long lines mechanical energy dissipation as heat will occur. This loss can be estimated e.g. using Darcy–Weisbach equation.

By multiplying with the fluid density $$\rho$$, 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) 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:


 * $$H\, =\, z\, +\, \frac{p}{\rho g}\, +\, \frac{v^2}{2\,g}\, =\, h\, +\, \frac{v^2}{2\,g},$$

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 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 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 p0 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 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."

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 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. 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.

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 – 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:


 * $$\frac{\partial \varphi}{\partial t} + \tfrac{1}{2} v^2 + \frac{p}{\rho} + gz = f(t),$$

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


 * $$\Phi=\varphi-\int_{t_0}^t f(\tau)\, \text{d}\tau,$$ 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 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 compressible fluids up to approximately Mach number 0.3. 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


 * $$\frac {v^2}{2}+ gz+\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho}  = \text{constant}$$  (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:


 * $$\frac {v^2}{2}+\left( \frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}$$

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 and for (quasi) steady flow, 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.

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 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

 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!Bernoulli equation for incompressible fluids 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 = ρ A 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.
 * 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
 * Derivation through integrating Newton's Second Law of Motion
 * $$m \frac{\operatorname{d}v}{\operatorname{d}t}= F $$
 * $$\rho A  \operatorname{d}x \frac{\operatorname{d}v}{\operatorname{d}t}= -A \operatorname{d}p $$
 * $$\rho \frac{\operatorname{d}v}{\operatorname{d}t}= -\frac{\operatorname{d}p}{\operatorname{d}x} $$

In steady flow 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).
 * $$\frac{\operatorname{d}v}{\operatorname{d}t}=  \frac{\operatorname{d}v}{\operatorname{d}x}\frac{\operatorname{d}x}{\operatorname{d}t} = \frac{\operatorname{d}v}{\operatorname{d}x}v=\frac{d}{\operatorname{d}x} \left( \frac{v^2}{2} \right).$$

With density ρ constant, the equation of motion can be written as
 * $$\frac{\operatorname{d}}{\operatorname{d}x} \left( \rho \frac{v^2}{2} + p \right) =0$$

by integrating with respect to x
 * $$ \frac{v^2}{2} + \frac{p}{\rho}= C$$

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 work-energy principle is invoked. Rather, Bernoulli's principle was derived by a simple manipulation of Newton's second law. Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. In the form of the work-energy theorem, stating that
 * Derivation by using conservation of energy
 * the change in the kinetic energy Ekin 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 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 and ρ 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:

\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} $$ The work done by the forces consists of two parts:
 * The work done by the pressure acting on the areas A1 and A2
 * $$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 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
 * $$\Delta E_\text{pot,gravity} = \Delta m\, g z_2 - \Delta m\, g z_1. \;$$
 * Now, the work by the force of gravity is opposite to the change in potential energy, 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. 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 work-kinetic energy theorem W = ΔEkin 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 A1 v1 Δt = ρ A2 v2 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), 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:
 * $$\frac{v^{2}}{2 g}+z+\frac{p}{\rho g}=C$$ (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 zelevation. A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed
 * $$v=\sqrt,$$ when arriving at elevation z = 0. Or when we rearrange it as a head: $$h_v =\frac{v^2}{2 g}$$

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
 * $$p=p_0-\rho g z \,$$, with p0 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.
 * }
 * {| class="toccolours collapsible collapsed" width="60%" style="text-align:left"

!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:
 * $$0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t$$.
 * $$0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t$$.

Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the 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,
 * $$0= \Delta E_1 - \Delta E_2 \,$$

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:
 * $$\Delta E_1 = \left[\frac{1}{2} \rho_1 v_1^2 + \Psi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t$$

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 previously-obtained 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.
 * }

Applications
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 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. 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 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 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 airspeed appropriate to the dynamic 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 Reynolds number and the shape of the orifice.


 * The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.

Misunderstandings about the generation of lift
Many explanations for the generation of lift (on airfoils, propeller blades, etc.) can be found; 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 both Bernoulli's principle and Newton's laws are relevant and either can be used to correctly describe lift.

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 relying 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.

Misapplications of Bernoulli's principle in common classroom demonstrations
There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle. One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it. As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure".

One problem with this explanation can be seen by blowing along the bottom of the paper - were the deflection due simply to faster moving air one would expect the paper to deflect downward, but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom. Another problem is that when the air leaves the demonstrator's mouth it has the same pressure as the surrounding air; the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is equal to the pressure of the surrounding air. A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli’s equation since the air above and below are different flow fields and Bernoulli's principle only applies within a flow field.

As the wording of the principle can change its implications, stating the principle correctly is important. What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa. Thus, Bernoulli's principle concerns itself with changes in speed and changes in pressure within a flow field. It cannot be used to compare different flow fields.

A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve. Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed, i.e. that as the air passes over the paper it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration.

Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure".