Bernoulli's principle
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. Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in its potential energy and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same because in a reservoir the energy per unit volume is the same everywhere.
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.
File:Venturi5.svg|thumb|300x300px|The upstream static pressure is higher than in the constriction, 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 and non-adiabatic processes 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. 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:
- is the fluid flow speed at a point,
- is the acceleration due to gravity,
- is the elevation of the point above a reference plane, with the positive -direction pointing upward—so in the direction opposite to the gravitational acceleration,
- is the static pressure at the chosen point, and
- is the density of the fluid at all points in the fluid.
The following assumptions must be met for this Bernoulli equation to apply:
- the flow must be steady, that is, the flow parameters at any point cannot change with time,
- the flow must be incompressible—even though pressure varies, the density must remain constant along a streamline;
- friction by viscous forces must be negligible.
where is the force potential at the point considered. For example, for the Earth's gravity.
By multiplying with the fluid density, equation can be rewritten as:
or:
where
- is dynamic pressure,
- is the piezometric head or hydraulic head and
- is the stagnation pressure.
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."
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 and dynamic pressure. Their sum is defined to be the total pressure. The significance of Bernoulli's principle can now be summarized as "total pressure is constant in any region free of viscous forces". If the fluid flow is brought to rest at some point, this point is called a stagnation point, and at this point the static pressure is equal to the stagnation pressure.
If the fluid flow is irrotational, the total pressure is uniform 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, Bernoulli's principle importantly does not apply in the boundary layer such as in flow through long pipes.
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 of a velocity potential. In that case, and for a constant density, the momentum equations of the Euler equations can be integrated to:which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here denotes the partial derivative of the velocity potential with respect to time, and is the flow speed. The function depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case and are constants so equation can be applied in every point of the fluid domain. Further can be made equal to zero by incorporating it into the velocity potential using the transformation:
resulting in:
Note that the relation of the potential to the flow velocity is unaffected by this transformation:.
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 mechanics.
Compressible flow equation
Bernoulli developed his principle from observations on liquids, and Bernoulli's equation is valid for ideal fluids: those that are inviscid, incompressible and subjected only to conservative forces. It is sometimes valid for the flow of gases as well, 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. 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 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 along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.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.