BumpDumper
03-03-2013, 11:48 PM
In fluid dynamics (http://en.wikipedia.org/wiki/Fluid_dynamics), Bernoulli's principle states that for an inviscid flow (http://en.wikipedia.org/wiki/Inviscid_flow), an increase in the speed of the fluid occurs simultaneously with a decrease in pressure (http://en.wikipedia.org/wiki/Pressure) or a decrease in the fluid (http://en.wikipedia.org/wiki/Fluid)'s potential energy (http://en.wikipedia.org/wiki/Potential_energy).[1] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-1)[2] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Batchelor_3.5-2) Bernoulli's principle is named after the Swiss scientist Daniel Bernoulli (http://en.wikipedia.org/wiki/Daniel_Bernoulli) who published his principle in his book Hydrodynamica in 1738.[3] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-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 (http://en.wikipedia.org/wiki/Incompressible_flow) (e.g. most liquid (http://en.wikipedia.org/wiki/Liquid) flows) and also for compressible flows (http://en.wikipedia.org/wiki/Compressible_flow) (e.g. gases (http://en.wikipedia.org/wiki/Gas)) moving at low Mach numbers (http://en.wikipedia.org/wiki/Mach_number). More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (http://en.wikipedia.org/wiki/Mach_number) (see the derivations of the Bernoulli equation (http://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_Bernoulli_equ ation)).
Bernoulli's principle can be derived from the principle of conservation of energy (http://en.wikipedia.org/wiki/Conservation_of_energy). This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline (http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines) is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both itsdynamic pressure (http://en.wikipedia.org/wiki/Dynamic_pressure) and kinetic energy (http://en.wikipedia.org/wiki/Kinetic_energy), and a decrease in its static pressure (http://en.wikipedia.org/wiki/Static_pressure) and potential energy (http://en.wikipedia.org/wiki/Potential_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.[4] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-4)
Bernoulli's principle can also be derived directly from 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.[5] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Babinsky-5)[6] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Weltner-6)[7] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-7)
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.
Incompressible flow equation
In most flows of liquids, and of gases at low Mach number (http://en.wikipedia.org/wiki/Mach_number), the density (http://en.wikipedia.org/wiki/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 (http://en.wikipedia.org/wiki/Arbitrary) point along a streamline (http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines), is:
http://upload.wikimedia.org/math/6/1/a/61a840e7e6b25040825c61fd519756ae.png
(A (http://en.wikipedia.org/wiki/Bernoulli's_principle#equation_A))
where:
http://upload.wikimedia.org/math/2/d/3/2d3fdc651d296cf7a5bde9d58fa58c47.png is the fluid flow speed (http://en.wikipedia.org/wiki/Speed) at a point on a streamline,http://upload.wikimedia.org/math/f/3/1/f31f123f5b510e1c58b2be1990dcada8.png is the acceleration due to gravity (http://en.wikipedia.org/wiki/Earth%27s_gravity),http://upload.wikimedia.org/math/7/7/6/77698ae92ac0435f8da1e266eeb528e3.png is the elevation (http://en.wikipedia.org/wiki/Elevation) of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,http://upload.wikimedia.org/math/5/a/3/5a34bb082daf037b3c4b14c13af6855b.png is the pressure (http://en.wikipedia.org/wiki/Pressure) at the chosen point, andhttp://upload.wikimedia.org/math/a/b/4/ab4c699d5daae16f90abf620d960811a.png is the density (http://en.wikipedia.org/wiki/Density) of the fluid at all points in the fluid.For conservative force (http://en.wikipedia.org/wiki/Conservative_force) fields, Bernoulli's equation can be generalized as:[8] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Batchelor_265-8)
http://upload.wikimedia.org/math/a/5/8/a58b68f3bb891d015f4b62ad34e4a014.pngwhere Ψ is the force potential (http://en.wikipedia.org/wiki/Conservative_force) 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:[8] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Batchelor_265-8)
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.
By multiplying with the fluid density http://upload.wikimedia.org/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png, equation (A (http://en.wikipedia.org/wiki/Bernoulli's_principle#math_A)) can be rewritten as:
http://upload.wikimedia.org/math/8/f/e/8fe8d95e8b31880e495d141a3256e3db.pngor:
http://upload.wikimedia.org/math/9/c/0/9c025a603b3fd6f9fb8acec39d82d1f9.pngwhere:
http://upload.wikimedia.org/math/f/c/f/fcfbd59ede279e644c8caea318f219df.png is dynamic pressure (http://en.wikipedia.org/wiki/Dynamic_pressure),http://upload.wikimedia.org/math/2/c/0/2c0b8fbdd96903c200687176146b7d30.png is the piezometric head (http://en.wikipedia.org/wiki/Piezometric_head) or hydraulic head (http://en.wikipedia.org/wiki/Hydraulic_head) (the sum of the elevation z and the pressure head (http://en.wikipedia.org/wiki/Pressure_head))[9] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Mulley_43_44-9)[10] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Chanson_22-10) andhttp://upload.wikimedia.org/math/b/f/e/bfe314b14ae96c0130ca00cc6fe66f2e.png is the total pressure (http://en.wikipedia.org/wiki/Total_pressure) (the sum of the static pressure p and dynamic pressure q).[11] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-11)The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head H:
http://upload.wikimedia.org/math/1/8/7/187d6853e6b3183e324fadb92b51735a.pngThe 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 (http://en.wikipedia.org/wiki/Cavitation) occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound (http://en.wikipedia.org/wiki/Sound) waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.
[edit (http://en.wikipedia.org/w/index.php?title=Bernoulli%27s_principle&action=edit§ion=2)]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 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:
http://upload.wikimedia.org/math/e/2/1/e21481196d18d888f0079c4398a7b481.pngwhere p0 is called 'total pressure', and q is 'dynamic pressure (http://en.wikipedia.org/wiki/Dynamic_pressure)'.[12] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-12) Many authors refer to the pressure (http://en.wikipedia.org/wiki/Pressure) p as static pressure (http://en.wikipedia.org/wiki/Static_pressure) to distinguish it from total pressure p0 and dynamic pressure (http://en.wikipedia.org/wiki/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."[13] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Clancy3.5-13)
The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:
static pressure + dynamic pressure = total pressure[13] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Clancy3.5-13)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 (http://en.wikipedia.org/wiki/Irrotational_flow), 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.[14] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-14) 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 (http://en.wikipedia.org/wiki/Boundary_layer) or in fluid flow through long pipes (http://en.wikipedia.org/wiki/Pipe_flow).
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 (http://en.wikipedia.org/wiki/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 – can not be assumed to be valid. However if the gas process is entirely isobaric (http://www.spurstalk.com/wiki/Isobaric_process), or isochoric (http://www.spurstalk.com/wiki/Isochoric_process), 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 (http://www.spurstalk.com/wiki/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 (http://www.spurstalk.com/wiki/Isentropic) (frictionless (http://www.spurstalk.com/wiki/Friction) adiabatic (http://www.spurstalk.com/wiki/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 (http://www.spurstalk.com/wiki/Speed_of_sound), such that the variation in density of the gas (due to this effect) along each streamline (http://www.spurstalk.com/wiki/Streamlines,_streaklines_and_pathlines) can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.
[edit (http://www.spurstalk.com/w/index.php?title=Bernoulli%27s_principle&action=edit§ion=4)] Unsteady potential flow
The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves (http://www.spurstalk.com/wiki/Ocean_surface_wave) and acoustics (http://www.spurstalk.com/wiki/Acoustics).
For an irrotational flow (http://www.spurstalk.com/wiki/Irrotational_flow), the flow velocity (http://www.spurstalk.com/wiki/Flow_velocity) can be described as the gradient (http://www.spurstalk.com/wiki/Gradient) ∇φ of a velocity potential (http://www.spurstalk.com/wiki/Velocity_potential) φ. In that case, and for a constant density (http://www.spurstalk.com/wiki/Density) ρ, the momentum (http://www.spurstalk.com/wiki/Momentum) equations of the Euler equations (http://www.spurstalk.com/wiki/Euler_equations_(fluid_dynamics)) can be integrated to:[15] (http://www.spurstalk.com/forums/#cite_note-Batch383-15)
http://upload.wikimedia.org/math/1/4/7/147add5aeb10b133f57a503afd75afe3.pngwhich is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here ∂φ/∂t denotes the partial derivative (http://www.spurstalk.com/wiki/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.[15] (http://www.spurstalk.com/forums/#cite_note-Batch383-15)
Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation
http://upload.wikimedia.org/math/4/0/9/4090c5e33799eaf7fbd572e67539073a.pngNote 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 (http://www.spurstalk.com/wiki/Luke%27s_variational_principle), a variational description of free-surface flows using the Lagrangian (http://www.spurstalk.com/wiki/Lagrangian) (not to be confused with Lagrangian coordinates (http://www.spurstalk.com/wiki/Lagrangian_coordinates)).
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 (http://en.wikipedia.org/wiki/Incompressible_flow) (e.g. most liquid (http://en.wikipedia.org/wiki/Liquid) flows) and also for compressible flows (http://en.wikipedia.org/wiki/Compressible_flow) (e.g. gases (http://en.wikipedia.org/wiki/Gas)) moving at low Mach numbers (http://en.wikipedia.org/wiki/Mach_number). More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (http://en.wikipedia.org/wiki/Mach_number) (see the derivations of the Bernoulli equation (http://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_Bernoulli_equ ation)).
Bernoulli's principle can be derived from the principle of conservation of energy (http://en.wikipedia.org/wiki/Conservation_of_energy). This states that, in a steady flow, the sum of all forms of mechanical energy in a fluid along a streamline (http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines) is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. Thus an increase in the speed of the fluid occurs proportionately with an increase in both itsdynamic pressure (http://en.wikipedia.org/wiki/Dynamic_pressure) and kinetic energy (http://en.wikipedia.org/wiki/Kinetic_energy), and a decrease in its static pressure (http://en.wikipedia.org/wiki/Static_pressure) and potential energy (http://en.wikipedia.org/wiki/Potential_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.[4] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-4)
Bernoulli's principle can also be derived directly from 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.[5] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Babinsky-5)[6] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Weltner-6)[7] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-7)
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.
Incompressible flow equation
In most flows of liquids, and of gases at low Mach number (http://en.wikipedia.org/wiki/Mach_number), the density (http://en.wikipedia.org/wiki/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 (http://en.wikipedia.org/wiki/Arbitrary) point along a streamline (http://en.wikipedia.org/wiki/Streamlines,_streaklines,_and_pathlines), is:
http://upload.wikimedia.org/math/6/1/a/61a840e7e6b25040825c61fd519756ae.png
(A (http://en.wikipedia.org/wiki/Bernoulli's_principle#equation_A))
where:
http://upload.wikimedia.org/math/2/d/3/2d3fdc651d296cf7a5bde9d58fa58c47.png is the fluid flow speed (http://en.wikipedia.org/wiki/Speed) at a point on a streamline,http://upload.wikimedia.org/math/f/3/1/f31f123f5b510e1c58b2be1990dcada8.png is the acceleration due to gravity (http://en.wikipedia.org/wiki/Earth%27s_gravity),http://upload.wikimedia.org/math/7/7/6/77698ae92ac0435f8da1e266eeb528e3.png is the elevation (http://en.wikipedia.org/wiki/Elevation) of the point above a reference plane, with the positive z-direction pointing upward – so in the direction opposite to the gravitational acceleration,http://upload.wikimedia.org/math/5/a/3/5a34bb082daf037b3c4b14c13af6855b.png is the pressure (http://en.wikipedia.org/wiki/Pressure) at the chosen point, andhttp://upload.wikimedia.org/math/a/b/4/ab4c699d5daae16f90abf620d960811a.png is the density (http://en.wikipedia.org/wiki/Density) of the fluid at all points in the fluid.For conservative force (http://en.wikipedia.org/wiki/Conservative_force) fields, Bernoulli's equation can be generalized as:[8] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Batchelor_265-8)
http://upload.wikimedia.org/math/a/5/8/a58b68f3bb891d015f4b62ad34e4a014.pngwhere Ψ is the force potential (http://en.wikipedia.org/wiki/Conservative_force) 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:[8] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Batchelor_265-8)
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.
By multiplying with the fluid density http://upload.wikimedia.org/math/f/7/f/f7f177957cf064a93e9811df8fe65ed1.png, equation (A (http://en.wikipedia.org/wiki/Bernoulli's_principle#math_A)) can be rewritten as:
http://upload.wikimedia.org/math/8/f/e/8fe8d95e8b31880e495d141a3256e3db.pngor:
http://upload.wikimedia.org/math/9/c/0/9c025a603b3fd6f9fb8acec39d82d1f9.pngwhere:
http://upload.wikimedia.org/math/f/c/f/fcfbd59ede279e644c8caea318f219df.png is dynamic pressure (http://en.wikipedia.org/wiki/Dynamic_pressure),http://upload.wikimedia.org/math/2/c/0/2c0b8fbdd96903c200687176146b7d30.png is the piezometric head (http://en.wikipedia.org/wiki/Piezometric_head) or hydraulic head (http://en.wikipedia.org/wiki/Hydraulic_head) (the sum of the elevation z and the pressure head (http://en.wikipedia.org/wiki/Pressure_head))[9] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Mulley_43_44-9)[10] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Chanson_22-10) andhttp://upload.wikimedia.org/math/b/f/e/bfe314b14ae96c0130ca00cc6fe66f2e.png is the total pressure (http://en.wikipedia.org/wiki/Total_pressure) (the sum of the static pressure p and dynamic pressure q).[11] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-11)The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head H:
http://upload.wikimedia.org/math/1/8/7/187d6853e6b3183e324fadb92b51735a.pngThe 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 (http://en.wikipedia.org/wiki/Cavitation) occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound (http://en.wikipedia.org/wiki/Sound) waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.
[edit (http://en.wikipedia.org/w/index.php?title=Bernoulli%27s_principle&action=edit§ion=2)]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 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:
http://upload.wikimedia.org/math/e/2/1/e21481196d18d888f0079c4398a7b481.pngwhere p0 is called 'total pressure', and q is 'dynamic pressure (http://en.wikipedia.org/wiki/Dynamic_pressure)'.[12] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-12) Many authors refer to the pressure (http://en.wikipedia.org/wiki/Pressure) p as static pressure (http://en.wikipedia.org/wiki/Static_pressure) to distinguish it from total pressure p0 and dynamic pressure (http://en.wikipedia.org/wiki/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."[13] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Clancy3.5-13)
The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:
static pressure + dynamic pressure = total pressure[13] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-Clancy3.5-13)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 (http://en.wikipedia.org/wiki/Irrotational_flow), 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.[14] (http://en.wikipedia.org/wiki/Bernoulli's_principle#cite_note-14) 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 (http://en.wikipedia.org/wiki/Boundary_layer) or in fluid flow through long pipes (http://en.wikipedia.org/wiki/Pipe_flow).
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 (http://en.wikipedia.org/wiki/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 – can not be assumed to be valid. However if the gas process is entirely isobaric (http://www.spurstalk.com/wiki/Isobaric_process), or isochoric (http://www.spurstalk.com/wiki/Isochoric_process), 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 (http://www.spurstalk.com/wiki/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 (http://www.spurstalk.com/wiki/Isentropic) (frictionless (http://www.spurstalk.com/wiki/Friction) adiabatic (http://www.spurstalk.com/wiki/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 (http://www.spurstalk.com/wiki/Speed_of_sound), such that the variation in density of the gas (due to this effect) along each streamline (http://www.spurstalk.com/wiki/Streamlines,_streaklines_and_pathlines) can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.
[edit (http://www.spurstalk.com/w/index.php?title=Bernoulli%27s_principle&action=edit§ion=4)] Unsteady potential flow
The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves (http://www.spurstalk.com/wiki/Ocean_surface_wave) and acoustics (http://www.spurstalk.com/wiki/Acoustics).
For an irrotational flow (http://www.spurstalk.com/wiki/Irrotational_flow), the flow velocity (http://www.spurstalk.com/wiki/Flow_velocity) can be described as the gradient (http://www.spurstalk.com/wiki/Gradient) ∇φ of a velocity potential (http://www.spurstalk.com/wiki/Velocity_potential) φ. In that case, and for a constant density (http://www.spurstalk.com/wiki/Density) ρ, the momentum (http://www.spurstalk.com/wiki/Momentum) equations of the Euler equations (http://www.spurstalk.com/wiki/Euler_equations_(fluid_dynamics)) can be integrated to:[15] (http://www.spurstalk.com/forums/#cite_note-Batch383-15)
http://upload.wikimedia.org/math/1/4/7/147add5aeb10b133f57a503afd75afe3.pngwhich is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here ∂φ/∂t denotes the partial derivative (http://www.spurstalk.com/wiki/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.[15] (http://www.spurstalk.com/forums/#cite_note-Batch383-15)
Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation
http://upload.wikimedia.org/math/4/0/9/4090c5e33799eaf7fbd572e67539073a.pngNote 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 (http://www.spurstalk.com/wiki/Luke%27s_variational_principle), a variational description of free-surface flows using the Lagrangian (http://www.spurstalk.com/wiki/Lagrangian) (not to be confused with Lagrangian coordinates (http://www.spurstalk.com/wiki/Lagrangian_coordinates)).