Bernoulli's principle

right|thumb|A flow of air through a [[Venturi meter. The kinetic energy increases at the expense of the fluid pressure, as shown by the difference in height of the two columns of water.]]

thumb|Video of a [[Venturi effect|Venturi meter used in a lab experiment]]

Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. For example, for a fluid flowing horizontally, Bernoulli's principle states that an increase in the speed occurs simultaneously with a decrease in pressure. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form.

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 energy in a fluid is the same at all points that are free of viscous forces. This requires that the sum of kinetic energy, potential energy and internal energy remains constant.

Bernoulli's principle can also be derived directly from Isaac Newton's second law of motion. When a fluid is flowing horizontally from a region of high pressure to a region of low pressure, there is more pressure from 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.

thumb|300x300px|The upstream [[static pressure (1) is higher than in the constriction (2), and the fluid speed at "1" is slower than at "2", because the cross-sectional area at "1" is greater than at "2".]]

Bernoulli's principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. thermal radiation) are small and can be neglected. However, the principle can be applied to various types of flow within these bounds, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers.

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 flows. 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 is:

where:

<math>v</math> is the fluid flow speed at a point,
<math>g</math> is the acceleration due to gravity,
<math>z</math> is the elevation of the point above a reference plane, with the positive <math>z</math>-direction pointing upward—so in the direction opposite to the gravitational acceleration,
<math>p</math> is the static pressure at the chosen point, and
<math>\rho</math> is the density of the fluid at all points in the fluid.

Bernoulli's equation and the Bernoulli constant are applicable throughout any region of flow where the energy per unit mass is uniform. Because the energy per unit mass of liquid in a well-mixed reservoir is uniform throughout, Bernoulli's equation can be used to analyze the fluid flow everywhere in that reservoir (including pipes or flow fields that the reservoir feeds) except where viscous forces dominate and erode the energy per unit mass. and

is the stagnation pressure (the sum of the static pressure and dynamic pressure ).

The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head :

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 term 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 is so small the term can be omitted. This allows the above equation to be presented in the following simplified form:

where is called total pressure, and is dynamic pressure. 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."

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,

<math display="block">\frac {v^2}{2}+ \int_{p_1}^p \frac {\mathrm{d}\tilde{p{\rho\left(\tilde{p}\right)} + \Psi = \text{constant (along a streamline)}</math>

where:

is the pressure
is the density and indicates that it is a function of pressure
is the flow speed
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 for an ideal gas becomes:

<math display="block">\frac{v^2}{2} + \Psi + w = \text{constant}.</math>

Here is the enthalpy per unit mass (also known as specific enthalpy), which is also often written as (not to be confused with "head" or "height").

Note that

<math display="block">w =e + \frac{p}{\rho} ~~~\left(= \frac{\gamma}{\gamma-1} \frac{p}{\rho}\right)</math>

where is the thermodynamic energy per unit mass, also known as the specific internal energy. So, for constant internal energy <math>e</math> the equation reduces to the incompressible-flow form.

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. 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 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.

Unsteady potential flow

For a compressible fluid, with a barotropic equation of state, the unsteady momentum conservation equation

<math display="block">\frac{\partial \vec{v{\partial t} + \left(\vec{v}\cdot \nabla\right)\vec{v} = -\vec{g} - \frac{\nabla p}{\rho}</math>

With the irrotational assumption, namely, the flow velocity can be described as the gradient of a velocity potential . The unsteady momentum conservation equation becomes

<math display="block">\frac{\partial \nabla \phi}{\partial t} + \nabla \left(\frac{\nabla \phi \cdot \nabla \phi}{2}\right) = -\nabla \Psi - \nabla \int_{p_1}^{p}\frac{d \tilde{p{\rho(\tilde{p})}</math>

which leads to

<math display="block">\frac{\partial \phi}{\partial t} + \frac{\nabla \phi \cdot \nabla \phi}{2} + \Psi + \int_{p_1}^{p}\frac{d \tilde{p{\rho(\tilde{p})} = \text{constant}</math>

In this case, the above equation for isentropic flow becomes:

<math display="block">\frac{\partial \phi}{\partial t} + \frac{\nabla \phi \cdot \nabla \phi}{2} + \Psi + \frac{\gamma}{\gamma-1}\frac{p}{\rho} = \text{constant}</math>

Derivations

{\mathrm{d}x} \left( \frac{v^2}{2} \right).</math>

With density constant, the equation of motion can be written as

<math display="block">\frac{\mathrm{d{\mathrm{d}x} \left( \rho \frac{v^2}{2} + p \right) =0</math>

by integrating with respect to

where 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.

thumb|center|600px|A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (), and cross-sectional area. Note that in this figure elevation is denoted as , contrary to the text where it is given by .

; Derivation by using conservation of energy

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

Therefore,

The system consists of the volume of fluid, initially between the cross-sections and . In the time interval fluid elements initially at the inflow cross-section move over a distance , while at the outflow cross-section the fluid moves away from cross-section over a distance . The displaced fluid volumes at the inflow and outflow are respectively and . The associated displaced fluid masses are – when is the fluid's mass density – equal to density times volume, so and . By mass conservation, these two masses displaced in the time interval have to be equal, and this displaced mass is denoted by :

<math display="block">\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}</math>

The work done by the forces consists of two parts:

The work done by the pressure acting on the areas and <math display="block">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}.</math>
The work done by gravity: the gravitational potential energy in the volume is lost, and at the outflow in the volume is gained. So, the change in gravitational potential energy in the time interval is

<math display="block">\Delta E_\text{pot,gravity} = \Delta m\, g z_2 - \Delta m\, g z_1. </math>

Now, the work by the force of gravity is opposite to the change in potential energy, : while the force of gravity is in the negative -direction, the work—gravity force times change in elevation—will be negative for a positive elevation change , while the corresponding potential energy change is positive. So: <math display="block">W_\text{gravity} = -\Delta E_\text{pot,gravity} = \Delta m\, g z_1 - \Delta m\, g z_2.</math>

And therefore the total work done in this time interval is

<math display="block">W = W_\text{pressure} + W_\text{gravity}.</math>

The increase in kinetic energy is

<math display="block">\Delta E_\text{kin} = \tfrac12 \Delta m\, v_2^2-\tfrac12 \Delta m\, v_1^2.</math>

Putting these together, the work-kinetic energy theorem gives:

. By redirecting air flows, it is possible to eliminate vibration of materials during their grasping and manipulation by robots.]]

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.