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Bernoulli's principle

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(Redirected fromTotal pressure (fluids))
Principle relating to fluid dynamics
This article is about Bernoulli's principle and Bernoulli's equation in fluid dynamics. For Bernoulli's theorem in probability, seelaw of large numbers. For an unrelated topic inordinary differential equations, seeBernoulli differential equation.
A flow of air through aVenturi meter. The kinetic energy increases at the expense of thefluid pressure, as shown by the difference in height of the two columns of water.
Video of aVenturi meter used in a lab experiment
Part of a series on
Continuum mechanics
J=−Ddφdx{\displaystyle J=-D{\frac {d\varphi }{dx}}}

Bernoulli's principle is a key concept influid 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 inpressure[1]: Ch.3 [2]: 156–164, § 3.5  The principle is named after the Swiss mathematician and physicistDaniel Bernoulli, who published it in his bookHydrodynamica in 1738.[3] Although Bernoulli deduced that pressure decreases when the flow speed increases, it wasLeonhard Euler in 1752 who derivedBernoulli's equation in its usual form.[4][5]

Bernoulli's principle can be derived from the principle ofconservation 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 ofkinetic energy, potential energy andinternal energy remains constant.[2]: § 3.5  Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) 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 (the sum of pressure andgravitational potentialρ g h) is the same everywhere.[6]: Example 3.5 and p.116 

Bernoulli's principle can also be derived directly fromIsaac Newton's secondlaw 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.[a][b][c]

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.[10]

Bernoulli's principle is only applicable forisentropic flows: when the effects ofirreversible processes (liketurbulence) 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 forincompressible flows (e.g. mostliquid flows andgases moving at lowMach number). More advanced forms may be applied tocompressible flows at higher Mach numbers.

Incompressible flow equation

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In most flows of liquids, and of gases at lowMach number, thedensity 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 calledincompressible 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:

v22+gz+pρ=constant{\displaystyle {\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}={\text{constant}}}A

where:

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 whereviscous forcesdominate and erode the energy per unit mass.[6]: Example 3.5 and p.116 

The following assumptions must be met for this Bernoulli equation to apply:[2]: 265 

  • the flow must besteady, that is, the flow parameters (velocity, density, etc.) 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 byviscous forces must be negligible.

Forconservative force fields (not limited to thegravitational field), Bernoulli's equation can be generalized as:[2]: 265 v22+Ψ+pρ=constant{\displaystyle {\frac {v^{2}}{2}}+\Psi +{\frac {p}{\rho }}={\text{constant}}}whereΨ is the force potential at the point considered. For example, for the Earth's gravityΨ =gz.

By multiplying with the fluid densityρ, equation (A) can be rewritten as:12ρv2+ρgz+p=constant{\displaystyle {\tfrac {1}{2}}\rho v^{2}+\rho gz+p={\text{constant}}}or:q+ρgh=p0+ρgz=constant{\displaystyle q+\rho gh=p_{0}+\rho gz={\text{constant}}}where

The constant in the Bernoulli equation can be normalized. A common approach is in terms oftotal head orenergy headH:H=z+pρg+v22g=h+v22g,{\displaystyle H=z+{\frac {p}{\rho g}}+{\frac {v^{2}}{2g}}=h+{\frac {v^{2}}{2g}},}

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 forsound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

Simplified form

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In many applications of Bernoulli's equation, the change in theρgz 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 heightz is so small theρgz term can be omitted. This allows the above equation to be presented in the following simplified form:p+q=p0{\displaystyle p+q=p_{0}}wherep0 is calledtotal pressure, andq isdynamic pressure.[14] Many authors refer to the pressurep as static pressure to distinguish it from total pressurep0 and dynamic pressureq. InAerodynamics, 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."[1]: § 3.5 

The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:[1]: § 3.5 

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 pressurep and dynamic pressureq. Their sump +q is defined to be the total pressurep0. 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 thestagnation pressure.

If the fluid flow isirrotational, the total pressure is uniform and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow".[1]: Equation 3.12  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 theboundary layer such as in flow through longpipes.

Unsteady potential flow

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The Bernoulli equation for unsteady potential flow is used in the theory ofocean surface waves andacoustics. For an irrotational flow, theflow velocity can be described as thegradient∇φ of avelocity potentialφ. In that case, and for a constant densityρ, themomentum equations of theEuler equations can be integrated to:[2]: 383 ∂φ∂t+12v2+pρ+gz=f(t),{\displaystyle {\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 thepartial derivative of the velocity potentialφ with respect to timet, andv = |∇φ| is the flow speed. The functionf(t) depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some momentt applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which casef and⁠∂φ/∂t⁠ are constants so equation (A) can be applied in every point of the fluid domain.[2]: 383  Furtherf(t) can be made equal to zero by incorporating it into the velocity potential using the transformation:Φ=φ−∫t0tf(τ)dτ,{\displaystyle \Phi =\varphi -\int _{t_{0}}^{t}f(\tau )\,\mathrm {d} \tau ,}resulting in:∂Φ∂t+12v2+pρ+gz=0.{\displaystyle {\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 inLuke's variational principle, a variational description of free-surface flows using theLagrangian mechanics.

Compressible flow equation

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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 entirelyisobaric, orisochoric, 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 absolutetemperature; 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 individualisentropic (frictionlessadiabatic) 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 thespeed 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 thanMach 0.3 is generally considered to be slow enough.[15]

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 thefirst law of thermodynamics.

Compressible flow in fluid dynamics

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For a compressible fluid, with abarotropicequation of state, and under the action of conservative forces,[16]v22+∫p1pdp~ρ(p~)+Ψ=constant (along a streamline){\displaystyle {\frac {v^{2}}{2}}+\int _{p_{1}}^{p}{\frac {\mathrm {d} {\tilde {p}}}{\rho \left({\tilde {p}}\right)}}+\Psi ={\text{constant (along a streamline)}}}where:

  • p is the pressure
  • ρ is the density andρ(p) indicates that it is a function of pressure
  • v is the flow speed
  • Ψ is the potential associated with the conservative force field, often thegravitational 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 anideal gas becomes:[1]: § 3.11 v22+gz+(γγ−1)pρ=constant (along a streamline){\displaystyle {\frac {v^{2}}{2}}+gz+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}={\text{constant (along a streamline)}}}where, in addition to the terms listed above:

  • γ is theratio 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 termgz can be omitted. A very useful form of the equation is then:v22+(γγ−1)pρ=(γγ−1)p0ρ0{\displaystyle {\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:

Compressible flow in thermodynamics

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The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is:[2]: § 3.5 [17]: § 5 [18]: § 5.9 

v22+Ψ+w=constant.{\displaystyle {\frac {v^{2}}{2}}+\Psi +w={\text{constant}}.}

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

Note thatw=e+pρ   (=γγ−1pρ){\displaystyle w=e+{\frac {p}{\rho }}~~~\left(={\frac {\gamma }{\gamma -1}}{\frac {p}{\rho }}\right)}wheree is thethermodynamic energy per unit mass, also known as thespecificinternal energy. So, for constant internal energye{\displaystyle e} the equation reduces to the incompressible-flow form.

The constant on the right-hand side is often called the Bernoulli constant and denotedb. For steady inviscid adiabatic flow with no additional sources or sinks of energy,b is constant along any given streamline. More generally, whenb may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When the change inΨ can be ignored, a very useful form of this equation is:v22+w=w0{\displaystyle {\frac {v^{2}}{2}}+w=w_{0}}wherew0 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.

Whenshock waves are present, in areference 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

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For a compressible fluid, with a barotropic equation of state, the unsteady momentum conservation equation∂v→∂t+(v→⋅∇)v→=−g→−∇pρ{\displaystyle {\frac {\partial {\vec {v}}}{\partial t}}+\left({\vec {v}}\cdot \nabla \right){\vec {v}}=-{\vec {g}}-{\frac {\nabla p}{\rho }}}

With the irrotational assumption, namely, the flow velocity can be described as the gradient∇φ of a velocity potentialφ. The unsteady momentum conservation equation becomes∂∇ϕ∂t+∇(∇ϕ⋅∇ϕ2)=−∇Ψ−∇∫p1pdp~ρ(p~){\displaystyle {\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}})}}}which leads to∂ϕ∂t+∇ϕ⋅∇ϕ2+Ψ+∫p1pdp~ρ(p~)=constant{\displaystyle {\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}}}

In this case, the above equation for isentropic flow becomes:∂ϕ∂t+∇ϕ⋅∇ϕ2+Ψ+γγ−1pρ=constant{\displaystyle {\frac {\partial \phi }{\partial t}}+{\frac {\nabla \phi \cdot \nabla \phi }{2}}+\Psi +{\frac {\gamma }{\gamma -1}}{\frac {p}{\rho }}={\text{constant}}}

Derivations

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Bernoulli equation for incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by eitherintegratingNewton's second law of motion or by applying the law ofconservation of energy, ignoringviscosity, compressibility, and thermal effects.

Derivation by integrating Newton's second law of motion

The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen inVenturi effect. Let thex axis be directed down the axis of the pipe.

Define a parcel of fluid moving through a pipe with cross-sectional areaA, the length of the parcel isdx, and the volume of the parcelA dx. Ifmass density isρ, the mass of the parcel is density multiplied by its volumem =ρA dx. The change in pressure over distancedx isdp andflow velocityv =⁠dx/dt⁠.

ApplyNewton's second law of motion (force = mass × acceleration) and recognizing that the effective force on theparcel 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 thex axis.

mdvdt=FρAdxdvdt=−Adpρdvdt=−dpdx{\displaystyle {\begin{aligned}m{\frac {\mathrm {d} v}{\mathrm {d} t}}&=F\\\rho A\mathrm {d} x{\frac {\mathrm {d} v}{\mathrm {d} t}}&=-A\mathrm {d} p\\\rho {\frac {\mathrm {d} v}{\mathrm {d} t}}&=-{\frac {\mathrm {d} p}{\mathrm {d} x}}\end{aligned}}}

In steady flow the velocity field is constant with respect to time,v =v(x) =v(x(t)), sov itself is not directly a function of timet. It is only when the parcel moves throughx that the cross sectional area changes:v depends ont only through the cross-sectional positionx(t).dvdt=dvdxdxdt=dvdxv=ddx(v22).{\displaystyle {\frac {\mathrm {d} v}{\mathrm {d} t}}={\frac {\mathrm {d} v}{\mathrm {d} x}}{\frac {\mathrm {d} x}{\mathrm {d} t}}={\frac {\mathrm {d} v}{\mathrm {d} x}}v={\frac {\mathrm {d} }{\mathrm {d} x}}\left({\frac {v^{2}}{2}}\right).}

With densityρ constant, the equation of motion can be written asddx(ρv22+p)=0{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\left(\rho {\frac {v^{2}}{2}}+p\right)=0}by integrating with respect toxv22+pρ=C{\displaystyle {\frac {v^{2}}{2}}+{\frac {p}{\rho }}=C}whereC is a constant, sometimes referred to as the Bernoulli constant. It is not auniversal 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.

A streamtube of fluid moving to the right. Indicated are pressure, elevation, flow speed, distance (s), and cross-sectional area. Note that in this figure elevation is denoted ash, contrary to the text where it is given byz.
Derivation by using conservation of energy

Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy.[19] In the form of thework-energy theorem, stating that[20]

the change in the kinetic energyEkin of the system equals the net workW done on the system;W=ΔEkin.{\displaystyle W=\Delta E_{\text{kin}}.}

Therefore,

thework done by theforces in the fluid equals increase inkinetic energy.

The system consists of the volume of fluid, initially between the cross-sectionsA1 andA2. In the time intervalΔt fluid elements initially at the inflow cross-sectionA1 move over a distances1 =v1 Δt, while at the outflow cross-section the fluid moves away from cross-sectionA2 over a distances2 =v2 Δt. The displaced fluid volumes at the inflow and outflow are respectivelyA1s1 andA2s2. The associated displaced fluid masses are – whenρ is the fluid'smass density – equal to density times volume, soρA1s1 andρA2s2. By mass conservation, these two masses displaced in the time intervalΔt have to be equal, and this displaced mass is denoted by Δm:ρA1s1=ρA1v1Δt=Δm,ρA2s2=ρA2v2Δt=Δm.{\displaystyle {\begin{aligned}\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{aligned}}}

The work done by the forces consists of two parts:

ΔEpot,gravity=Δmgz2−Δmgz1.{\displaystyle \Delta E_{\text{pot,gravity}}=\Delta m\,gz_{2}-\Delta m\,gz_{1}.}Now, thework by the force of gravity is opposite to the change in potential energy,Wgravity = −ΔEpot,gravity: while the force of gravity is in the negativez-direction, the work—gravity force times change in elevation—will be negative for a positive elevation changeΔz =z2 −z1, while the corresponding potential energy change is positive.[21]: 14–4, §14–3  So:Wgravity=−ΔEpot,gravity=Δmgz1−Δmgz2.{\displaystyle W_{\text{gravity}}=-\Delta E_{\text{pot,gravity}}=\Delta m\,gz_{1}-\Delta m\,gz_{2}.}And therefore the total work done in this time intervalΔt isW=Wpressure+Wgravity.{\displaystyle W=W_{\text{pressure}}+W_{\text{gravity}}.}Theincrease in kinetic energy isΔEkin=12Δmv22−12Δmv12.{\displaystyle \Delta E_{\text{kin}}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}-{\tfrac {1}{2}}\Delta m\,v_{1}^{2}.}Putting these together, the work-kinetic energy theoremW = ΔEkin gives:[19]Δmp1ρ−Δmp2ρ+Δmgz1−Δmgz2=12Δmv22−12Δmv12{\displaystyle \Delta m{\frac {p_{1}}{\rho }}-\Delta m{\frac {p_{2}}{\rho }}+\Delta m\,gz_{1}-\Delta m\,gz_{2}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}-{\tfrac {1}{2}}\Delta m\,v_{1}^{2}}or12Δmv12+Δmgz1+Δmp1ρ=12Δmv22+Δmgz2+Δmp2ρ.{\displaystyle {\tfrac {1}{2}}\Delta m\,v_{1}^{2}+\Delta m\,gz_{1}+\Delta m{\frac {p_{1}}{\rho }}={\tfrac {1}{2}}\Delta m\,v_{2}^{2}+\Delta m\,gz_{2}+\Delta m{\frac {p_{2}}{\rho }}.}After dividing by the massΔm =ρA1v1 Δt =ρA2v2 Δt the result is:[19]12v12+gz1+p1ρ=12v22+gz2+p2ρ{\displaystyle {\tfrac {1}{2}}v_{1}^{2}+gz_{1}+{\frac {p_{1}}{\rho }}={\tfrac {1}{2}}v_{2}^{2}+gz_{2}+{\frac {p_{2}}{\rho }}}or, as stated in the first paragraph:

v22+gz+pρ=C{\displaystyle {\frac {v^{2}}{2}}+gz+{\frac {p}{\rho }}=C}Eqn. 1, which is also Equation (A)

Further division byg produces the following equation. Note that each term can be described in thelength dimension (such as meters). This is the head equation derived from Bernoulli's principle:

v22g+z+pρg=C{\displaystyle {\frac {v^{2}}{2g}}+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 designationzelevation.

Afree falling mass from an elevationz > 0 (in avacuum) will reach aspeedv=2gz,{\displaystyle v={\sqrt {{2g}{z}}},}when arriving at elevationz = 0. Or when rearranged ashead:hv=v22g{\displaystyle h_{v}={\frac {v^{2}}{2g}}}The term⁠v2/2g⁠ is called thevelocityhead, expressed as a length measurement. It represents the internal energy of the fluid due to its motion.

Thehydrostatic pressurep is defined asp=p0−ρgz,{\displaystyle p=p_{0}-\rho gz,}withp0 some reference pressure, or when rearranged ashead:ψ=pρg.{\displaystyle \psi ={\frac {p}{\rho g}}.}The term⁠p/ρg⁠ is also called thepressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. The head due to the flow speed and the head due to static pressure combined with the elevation above a reference plane, a simple relationship useful for incompressible fluids using the velocity head, elevation head, and pressure head is obtained.

hv+zelevation+ψ=C{\displaystyle h_{v}+z_{\text{elevation}}+\psi =C}Eqn. 2b

If Eqn. 1 is multiplied by the density of the fluid, an equation with three pressure terms is obtained:

ρv22+ρgz+p=C{\displaystyle {\frac {\rho v^{2}}{2}}+\rho gz+p=C}Eqn. 3

Note that the pressure of the system is constant in this form of the Bernoulli equation. If the static pressure of the system (the third term) increases, and if the pressure due to elevation (the middle term) is constant, then the dynamic pressure (the first term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, it must be due to an increase in the static pressure that is resisting the flow.

All three equations are merely simplified versions of an energy balance on a system.

Bernoulli equation for compressible fluids

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of timeΔt, the amount of mass passing through the boundary defined by the areaA1 is equal to the amount of mass passing outwards through the boundary defined by the areaA2:0=ΔM1−ΔM2=ρ1A1v1Δt−ρ2A2v2Δt.{\displaystyle 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 volumeof the streamtube bounded byA1 andA2 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,ΔE1−ΔE2=0{\displaystyle \Delta E_{1}-\Delta E_{2}=0}whereΔE1 andΔE2 are the energy entering throughA1 and leaving throughA2, respectively. The energy entering throughA1 is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic internal energy per unit of mass (ε1) entering, and the energy entering in the form of mechanicalp dV work:ΔE1=(12ρ1v12+Ψ1ρ1+ε1ρ1+p1)A1v1Δt{\displaystyle \Delta E_{1}=\left({\tfrac {1}{2}}\rho _{1}v_{1}^{2}+\Psi _{1}\rho _{1}+\varepsilon _{1}\rho _{1}+p_{1}\right)A_{1}v_{1}\,\Delta t}whereΨ =gz is aforce potential due to theEarth's gravity,g is acceleration due to gravity, andz is elevation above a reference plane. A similar expression forΔE2 may easily be constructed.So now setting0 = ΔE1 − ΔE2:0=(12ρ1v12+Ψ1ρ1+ε1ρ1+p1)A1v1Δt−(12ρ2v22+Ψ2ρ2+ε2ρ2+p2)A2v2Δt{\displaystyle 0=\left({\tfrac {1}{2}}\rho _{1}v_{1}^{2}+\Psi _{1}\rho _{1}+\varepsilon _{1}\rho _{1}+p_{1}\right)A_{1}v_{1}\,\Delta t-\left({\tfrac {1}{2}}\rho _{2}v_{2}^{2}+\Psi _{2}\rho _{2}+\varepsilon _{2}\rho _{2}+p_{2}\right)A_{2}v_{2}\,\Delta t}which can be rewritten as:0=(12v12+Ψ1+ε1+p1ρ1)ρ1A1v1Δt−(12v22+Ψ2+ε2+p2ρ2)ρ2A2v2Δt{\displaystyle 0=\left({\tfrac {1}{2}}v_{1}^{2}+\Psi _{1}+\varepsilon _{1}+{\frac {p_{1}}{\rho _{1}}}\right)\rho _{1}A_{1}v_{1}\,\Delta t-\left({\tfrac {1}{2}}v_{2}^{2}+\Psi _{2}+\varepsilon _{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 obtain12v2+Ψ+ε+pρ=constant≡b{\displaystyle {\tfrac {1}{2}}v^{2}+\Psi +\varepsilon +{\frac {p}{\rho }}={\text{constant}}\equiv b}which is the Bernoulli equation for compressible flow.

An equivalent expression can be written in terms of fluid enthalpy (h):12v2+Ψ+h=constant≡b{\displaystyle {\tfrac {1}{2}}v^{2}+\Psi +h={\text{constant}}\equiv b}

Applications

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Condensation visible over the upper surface of anAirbus A340 wing caused by the increase inrelative humidityaccompanying the fall in pressure and temperature

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,[22] and a small viscosity often has a large effect on the flow.

  • Bernoulli's principle can be used to calculate the lift force on anairfoil, 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 upwardslifting force.[d][23] 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,[24] which were established by Bernoulli over a century before the first man-made wings were used for the purpose of flight.
  • The basis of acarburetor used in manyreciprocating engines is a throat in the air flow 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 can be explained by Bernoulli's principle, where air in the throat is moving at its fastest speed and therefore it is at its lowest pressure. The carburetor may or may not use the difference between the two static pressures which result from the Venturi effect on the air flow in order to force the fuel to flow, and as a basis a carburetor may use the difference in pressure between the throat and local air pressure in the float bowl, or between the throat and a Pitot tube at the air entry.
  • Aninjector on asteam locomotive or a staticboiler.
  • Thepitot tube andstatic port on an aircraft are used to determine theairspeed of the aircraft. These two devices are connected to theairspeed indicator, which determines the dynamic pressure of the airflow past the aircraft. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays theindicated airspeed appropriate to the dynamic pressure.[1]: § 3.8 
  • ADe Laval nozzle utilizes Bernoulli's principle to create a force by turning pressure energy generated by the combustion ofpropellants into velocity. This then generates thrust by way ofNewton's third law of motion.
  • The flow speed of a fluid can be measured using a device such as a Venturi meter or anorifice 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 theVenturi 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 isTorricelli's law, which is compatible with Bernoulli's principle. Increased viscosity lowers this drain rate; this is reflected in the discharge coefficient, which is a function of theReynolds number and the shape of the orifice.[25]
  • TheBernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper.
  • During acricket match,bowlers continually polish one side of the ball. After some time, one side is quite rough and the other is still smooth. Hence, when the ball is bowled and passes through air, the speed on one side of the ball is faster than on the other, and this results in a pressure difference between the sides; this leads to the ball rotating ("swinging") while travelling through the air, giving advantage to the bowlers.

Misconceptions

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Main article:Lift (force)

Airfoil lift

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An illustration of the incorrect equal transit-time explanation of airfoil lift

One of the most common erroneous explanations of aerodynamic lift asserts that the air must traverse the upper and lower surfaces of a wing in the same amount of time, implying that since the upper surface presents a longer path the air must be moving over the top of the wing faster than over the bottom. Bernoulli's principle is then cited to conclude that the pressure on top of the wing must be lower than on the bottom.[26][27]

Equal transit time applies to the flow around a body generating no lift, but there is no physical principle that requires equal transit time in cases of bodies generating lift. In fact, theory predicts – and experiments confirm – that the air traverses the top surface of a body experiencing lift in ashorter time than it traverses the bottom surface; the explanation based on equal transit time is false.[28][29][30] While the equal-time explanation is false, it is not the Bernoulli principle that is false, because this principle is well established; Bernoulli's equation is used correctly in common mathematical treatments of aerodynamic lift.[31][32]

Common classroom demonstrations

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There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle.[33] 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".[34][35][36]

One problem with this explanation can be seen by blowing along the bottom of the paper: if the deflection was caused by faster moving air, then the paper should deflect downward; but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom.[37] Another problem is that when the air leaves the demonstrator's mouth it has thesame pressure as the surrounding air;[38] 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 isequal to the pressure of the surrounding air.[39][40] 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 aredifferent flow fields and Bernoulli's principle only applies within a flow field.[41][42][43][44]

As the wording of the principle can change its implications, stating the principle correctly is important.[45] 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.[46] Thus, Bernoulli's principle concerns itself withchanges in speed andchanges in pressurewithin a flow field. It cannot be used to compare different flow fields.

A correct explanation of why the paper rises would observe that theplume 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.[47][48][49][50] Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed; in other words, 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.[51][52][53]

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".[54][55][56][57][58][59][60][61]

See also

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Notes

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  1. ^If the particle is in a region of varying pressure (a non-vanishing pressure gradient in thex-direction) and if the particle has a finite·sizel, then the front of the particle will be 'seeing' a different pressure from the rear. More precisely, if the pressure drops in thex-direction (⁠dp/dx⁠ < 0) the pressure at the rear is higher than at the front and the particle experiences a (positive) net force. According to Newton's second law, this force causes an acceleration and the particle's velocity increases as it moves along the streamline... Bernoulli's equation describes this mathematically (see the complete derivation in the appendix).[7]
  2. ^Acceleration of air is caused by pressure gradients. Air is accelerated in the direction of the velocity if the pressure goes down. Thus the decrease of pressure is the cause of a higher velocity.[8]
  3. ^The idea is that as the parcel moves along, following a streamline, as it moves into an area of higher pressure there will be higher pressure ahead (higher than the pressure behind) and this will exert a force on the parcel, slowing it down. Conversely, if the parcel is moving into a region of lower pressure, there will be a higher pressure behind it (higher than the pressure ahead), speeding it up. As always, any unbalanced force will cause a change in momentum (and velocity), as required by Newton's laws of motion.[9]
  4. ^"When a stream of air flows past an airfoil, there are local changes in velocity round the airfoil, and consequently changes in static pressure, in accordance with Bernoulli's Theorem. The distribution of pressure determines the lift, pitching moment and form drag of the airfoil, and the position of its centre of pressure."[1]: § 5.5 

References

[edit]
  1. ^abcdefgClancy, L.J. (1975).Aerodynamics. Wiley.ISBN 978-0-470-15837-1.
  2. ^abcdefgBatchelor, G.K. (2000).An Introduction to Fluid Dynamics. Cambridge:Cambridge University Press.ISBN 978-0-521-66396-0.
  3. ^"Hydrodynamica". Britannica Online Encyclopedia. Retrieved2008-10-30.
  4. ^Anderson, J.D. (2016),"Some reflections on the history of fluid dynamics", in Johnson, R.W. (ed.),Handbook of fluid dynamics (2nd ed.), CRC Press,ISBN 9781439849576
  5. ^Darrigol, O.; Frisch, U. (2008), "From Newton's mechanics to Euler's equations",Physica D: Nonlinear Phenomena,237 (14–17):1855–1869,Bibcode:2008PhyD..237.1855D,doi:10.1016/j.physd.2007.08.003
  6. ^abStreeter, Victor Lyle (1966).Fluid mechanics. New York: McGraw-Hill.
  7. ^Babinsky, Holger (November 2003), "How do wings work?",Physics Education,38 (6):497–503,Bibcode:2003PhyEd..38..497B,doi:10.1088/0031-9120/38/6/001,S2CID 1657792
  8. ^"Weltner, Klaus; Ingelman-Sundberg, Martin,Misinterpretations of Bernoulli's Law, archived fromthe original on April 29, 2009
  9. ^Denker, John S. (2005)."3 Airfoils and Airflow".See How It Flies. Retrieved2018-07-27.
  10. ^Resnick, R.; Halliday, D. (1960).Physics. John Wiley & Sons. section 18–4.
  11. ^Mulley, Raymond (2004).Flow of Industrial Fluids: Theory and Equations. CRC Press. pp. 43–44.ISBN 978-0-8493-2767-4.
  12. ^Chanson, Hubert (2004).Hydraulics of Open Channel Flow. Elsevier. p. 22.ISBN 978-0-08-047297-3.
  13. ^Oertel, Herbert; Prandtl, Ludwig; Böhle, M.; Mayes, Katherine (2004).Prandtl's Essentials of Fluid Mechanics. Springer. pp. 70–71.ISBN 978-0-387-40437-0.
  14. ^"Bernoulli's Equation". NASA Glenn Research Center. Archived fromthe original on 2012-07-31. Retrieved2009-03-04.
  15. ^White, Frank M.Fluid Mechanics (6th ed.). McGraw-Hill International Edition. p. 602.
  16. ^Clarke, Cathie;Carswell, Bob (2007).Principles of Astrophysical Fluid Dynamics. Cambridge University Press. p. 161.ISBN 978-1-139-46223-5.
  17. ^Landau, L.D.;Lifshitz, E.M. (1987).Fluid Mechanics.Course of Theoretical Physics (2nd ed.). Pergamon Press.ISBN 978-0-7506-2767-2.
  18. ^Van Wylen, Gordon J.; Sonntag, Richard E. (1965).Fundamentals of Classical Thermodynamics. New York: John Wiley and Sons.
  19. ^abcFeynman, R.P.;Leighton, R.B.; Sands, M. (1963).The Feynman Lectures on Physics. Vol. 2. Addison-Wesley.ISBN 978-0-201-02116-5.{{cite book}}:ISBN / Date incompatibility (help): 40–6 to 40–9, §40–3 
  20. ^Tipler, Paul (1991).Physics for Scientists and Engineers: Mechanics (3rd extended ed.). W. H. Freeman.ISBN 978-0-87901-432-2., p. 138.
  21. ^Feynman, R.P.;Leighton, R.B.; Sands, M. (1963).The Feynman Lectures on Physics. Vol. 1. Addison-Wesley.ISBN 978-0-201-02116-5.{{cite book}}:ISBN / Date incompatibility (help)
  22. ^Thomas, John E. (May 2010)."The Nearly Perfect Fermi Gas"(PDF).Physics Today.63 (5):34–37.Bibcode:2010PhT....63e..34T.doi:10.1063/1.3431329.
  23. ^Resnick, R.; Halliday, D. (1960).Physics. New York: John Wiley & Sons. section 18–5.Streamlines are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.
  24. ^Eastlake, Charles N. (March 2002)."An Aerodynamicist's View of Lift, Bernoulli, and Newton"(PDF).The Physics Teacher.40 (3):166–173.Bibcode:2002PhTea..40..166E.doi:10.1119/1.1466553. "The resultant force is determined by integrating the surface-pressuredistribution over the surface area of the airfoil."
  25. ^Mechanical Engineering Reference Manual (9th ed.).
  26. ^Technical education research center (2006).Physics That Works. Kendall Hunt.ISBN 0787291811.OCLC 61918633.One of the most widely circulated, but incorrect, explanations can be labeled the "Longer Path" theory, or the "Equal Transit Time" theory.
  27. ^Smith, Norman F. (November 1972)."Bernoulli and Newton in Fluid Mechanics".The Physics Teacher.10 (8): 451.Bibcode:1972PhTea..10..451S.doi:10.1119/1.2352317.The airfoil of the airplane wing, according to the textbook explanation that is more or less standard in the United States, has a special shape with more curvature on top than on the bottom; consequently, the air must travel over the top surface farther than over the bottom surface. Because the air must make the trip over the top and bottom surfaces in the same elapsed time ..., the velocity over the top surface will be greater than over the bottom. According to Bernoulli's theorem, this velocity difference produces a pressure difference which is lift.[permanent dead link]
  28. ^Babinsky, Holger (2003)."How do wings work?"(PDF).Physics Education.38 (6):497–503.Bibcode:2003PhyEd..38..497B.doi:10.1088/0031-9120/38/6/001.S2CID 1657792....it is often asked why fluid particles should meet up again at the trailing edge. Or, in other words, why should two particles on either side of the wing take the same time to travel from S to T? There is no obvious explanation and real-life observations prove that this is wrong.
  29. ^"The actual velocity over the top of an airfoil is much faster than that predicted by the "Longer Path" theory and particles moving over the top arrive at the trailing edge before particles moving under the airfoil."
    Glenn Research Center (Aug 16, 2000)."Incorrect Lift Theory #1". NASA. Archived fromthe original on April 27, 2014. RetrievedJune 27, 2021.
  30. ^Anderson, John (2005).Introduction to Flight. Boston: McGraw-Hill Higher Education. p. 355.ISBN 978-0072825695.It is then assumed that these two elements must meet up at the trailing edge, and because the running distance over the top surface of the airfoil is longer than that over the bottom surface, the element over the top surface must move faster. This is simply not true. Experimental results and computational fluid dynamic calculations clearly show that a fluid element moving over the top surface of an airfoil leaves the trailing edge long before its companion element moving over the bottom surface arrives at the trailing edge.
  31. ^Anderson, David; Eberhardt, Scott."How Airplanes Fly".How Airplanes Fly: A Physical Description of Lift.Archived from the original on January 26, 2016. Retrieved26 January 2016.There is nothing wrong with the Bernoulli principle, or with the statement that the air goes faster over the top of the wing. But, as the above discussion suggests, our understanding is not complete with this explanation. The problem is that we are missing a vital piece when we apply Bernoulli's principle. We can calculate the pressures around the wing if we know the speed of the air over and under the wing, but how do we determine the speed?
  32. ^Anderson, John D. (2016). "Chapter 4. Basic Aerodynamics".Introduction to Flight (8th ed.). McGraw-Hill Education.
  33. ^"Bernoulli's law and experiments attributed to it are fascinating. Unfortunately some of these experiments are explained erroneously..."Weltner, Klaus; Ingelman-Sundberg, Martin."Misinterpretations of Bernoulli's Law". Department of Physics, University Frankfurt. Archived fromthe original on June 21, 2012. RetrievedJune 25, 2012.
  34. ^Tymony, Cy."Origami Flying Disk".MAKE Magazine. Archived fromthe original on 2013-01-03.This occurs because of Bernoulli's principle — fast-moving air has lower pressure than non-moving air.
  35. ^"Bernoulli Effects". School of Physics and Astronomy,University of Minnesota. Archived fromthe original on 2012-03-10.Faster-moving fluid, lower pressure. ... When the demonstrator holds the paper in front of his mouth and blows across the top, he is creating an area of faster-moving air.
  36. ^"Educational Packet"(PDF). Tall Ships Festival – Channel Islands Harbor. Archived from the original on December 3, 2013. RetrievedJune 25, 2012.Bernoulli's Principle states that faster moving air has lower pressure... You can demonstrate Bernoulli's Principle by blowing over a piece of paper held horizontally across your lips.
  37. ^Craig, Gale M."Physical Principles of Winged Flight". RetrievedMarch 31, 2016 – via rcgroups.com.If the lift in figure A were caused by "Bernoulli's principle," then the paper in figure B should droop further when air is blown beneath it. However, as shown, it raises when the upward pressure gradient in downward-curving flow adds to atmospheric pressure at the paper lower surface.
  38. ^Babinsky, Holger (2003)."How Do Wings Work"(PDF).Physics Education.38 (6). IOP Publishing: 497.Bibcode:2003PhyEd..38..497B.doi:10.1088/0031-9120/38/6/001.S2CID 1657792. RetrievedApril 7, 2022 – via iopscience.iop.org.In fact, the pressure in the air blown out of the lungs is equal to that of the surrounding air...
  39. ^Eastwell, Peter (2007)."Bernoulli? Perhaps, but What About Viscosity?"(PDF).The Science Education Review.6 (1). Archived fromthe original(PDF) on 2018-03-18. Retrieved2018-03-18....air does not have a reduced lateral pressure (or static pressure...) simply because it is caused to move, the static pressure of free air does not decrease as the speed of the air increases, it misunderstanding Bernoulli's principle to suggest that this is what it tells us, and the behavior of the curved paper is explained by other reasoning than Bernoulli's principle.
  40. ^Raskin, Jef (February 2003)."Coanda Effect: Understanding Why Wings Work".karmak.org.Make a strip of writing paper about 5 cm × 25 cm. Hold it in front of your lips so that it hangs out and down making a convex upward surface. When you blow across the top of the paper, it rises. Many books attribute this to the lowering of the air pressure on top solely to the Bernoulli effect. Now use your fingers to form the paper into a curve that it is slightly concave upward along its whole length and again blow along the top of this strip. The paper now bends downward...an often-cited experiment, which is usually taken as demonstrating the common explanation of lift, does not do so...
  41. ^Babinsky, Holger (2003)."How Do Wings Work"(PDF).Physics Education.38 (6). IOP Publishing: 497.Bibcode:2003PhyEd..38..497B.doi:10.1088/0031-9120/38/6/001.S2CID 1657792. RetrievedApril 7, 2022 – via iopscience.iop.org.Blowing over a piece of paper does not demonstrate Bernoulli's equation. While it is true that a curved paper lifts when flow is applied on one side, this is not because air is moving at different speeds on the two sides...It is false to make a connection between the flow on the two sides of the paper using Bernoulli's equation.
  42. ^Eastwell, Peter (2007)."Bernoulli? Perhaps, but What About Viscosity?"(PDF).The Science Education Review.6 (1). Archived fromthe original(PDF) on 2018-03-18. Retrieved2018-03-18.An explanation based on Bernoulli's principle is not applicable to this situation, because this principle has nothing to say about the interaction of air masses having different speeds... Also, while Bernoulli's principle allows us to compare fluid speeds and pressures along a single streamline and... along two different streamlines that originate under identical fluid conditions, using Bernoulli's principle to compare the air above and below the curved paper in Figure 1 is nonsensical; in this case, there aren't any streamlines at all below the paper!
  43. ^Auerbach, David."Why Aircraft Fly"(PDF).European Journal of Physics.21: 295 – via iopscience.iop.org.The well-known demonstration of the phenomenon of lift by means of lifting a page cantilevered in one's hand by blowing horizontally along it is probably more a demonstration of the forces inherent in the Coanda effect than a demonstration of Bernoulli's law; for, here, an air jet issues from the mouth and attaches to a curved (and, in this case pliable) surface. The upper edge is a complicated vortex-laden mixing layer and the distant flow is quiescent, so that Bernoulli's law is hardly applicable.
  44. ^Smith, Norman F. (November 1972). "Bernoulli and Newton in Fluid Mechanics".The Physics Teacher.Millions of children in science classes are being asked to blow over curved pieces of paper and observe that the paper 'lifts'... They are then asked to believe that Bernoulli's theorem is responsible... Unfortunately, the 'dynamic lift' involved...is not properly explained by Bernoulli's theorem.
  45. ^Denker, John S."Bernoulli's Principle".See How It Flies – via av8n.com.Bernoulli's principle is very easy to understand provided the principle is correctly stated. However, we must be careful, because seemingly-small changes in the wording can lead to completely wrong conclusions.
  46. ^Smith, Norman F. (1973)."Bernoulli, Newton and Dynamic Lift Part I".School Science and Mathematics.73 (3):181–186.doi:10.1111/j.1949-8594.1973.tb08998.x – via wiley.com.A complete statement of Bernoulli's Theorem is as follows: 'In a flow where no energy is being added or taken away, the sum of its various energies is a constant: consequently where the velocity increases the pressure decreases and vice versa.'
  47. ^Babinsky, Holger (2003)."How Do Wings Work"(PDF).Physics Education.38 (6). IOP Publishing: 497.Bibcode:2003PhyEd..38..497B.doi:10.1088/0031-9120/38/6/001.S2CID 1657792. RetrievedApril 7, 2022 – via iopscience.iop.org....if a streamline is curved, there must be a pressure gradient across the streamline, with the pressure increasing in the direction away from the centre of curvature.
  48. ^Smith, Norman F. (1973)."Bernoulli, Newton and Dynamic Lift Part II".School Science and Mathematics.73 (4): 3333.doi:10.1111/j.1949-8594.1973.tb09040.x – via wiley.com.The curved paper turns the stream of air downward, and this action produces the lift reaction that lifts the paper.
  49. ^Aeronautics: An Educator's Guide with Activities in Science, Mathematics, and Technology Education(PDF). NASA. p. 26 – via nasa.gov.The curved surface of the tongue creates unequal air pressure and a lifting action. ... Lift is caused by air moving over a curved surface.
  50. ^Anderson, David F.; Eberhardt, Scott."The Newtonian Description of Lift of a Wing"(PDF). p. 12. Archived fromthe original(PDF) on 2016-03-11 – via integener.com.Viscosity causes the breath to follow the curved surface, Newton's first law says there a force on the air and Newton's third law says there is an equal and opposite force on the paper. Momentum transfer lifts the strip. The reduction in pressure acting on the top surface of the piece of paper causes the paper to rise.
  51. ^Anderson, David F.; Eberhardt, Scott.Understanding Flight. p. 229 – via Google Books.'Demonstrations' of Bernoulli's principle are often given as demonstrations of the physics of lift. They are truly demonstrations of lift, but certainly not of Bernoulli's principle.
  52. ^Feil, Max.The Aeronautics File. Archived fromthe original on May 17, 2015.As an example, take the misleading experiment most often used to "demonstrate" Bernoulli's principle. Hold a piece of paper so that it curves over your finger, then blow across the top. The paper will rise. However most people do not realize that the paper wouldnot rise if it were flat, even though you are blowing air across the top of it at a furious rate. Bernoulli's principle does not apply directly in this case. This is because the air on the two sides of the paper did not start out from the same source. The air on the bottom is ambient air from the room, but the air on the top came from your mouth where you actually increased its speed without decreasing its pressure by forcing it out of your mouth. As a result the air on both sides of the flat paper actually has the same pressure, even though the air on the top is moving faster. The reason that a curved piece of paper does rise is that the air from your mouth speeds up even more as it follows the curve of the paper, which in turn lowers the pressure according to Bernoulli.
  53. ^Geurts, Pim."Some simple Experiments".sailtheory.com. Archived fromthe original on 2016-03-03. RetrievedApril 7, 2022.Some people blow over a sheet of paper to demonstrate that the accelerated air over the sheet results in a lower pressure. They are wrong with their explanation. The sheet of paper goes up because it deflects the air, by the Coanda effect, and that deflection is the cause of the force lifting the sheet. To prove they are wrong I use the following experiment: If the sheet of paper is pre bend the other way by first rolling it, and if you blow over it than, it goes down. This is because the air is deflected the other way. Airspeed is still higher above the sheet, so that is not causing the lower pressure.
  54. ^Bobrowsky, Matt."Q: Is It Really Caused by the Bernoulli Effect?".Science 101. National Science Teaching Association.The Bernoulli effect is commonly—and incorrectly—invoked to explain: :why two suspended balloons or table tennis balls move toward each other when you blow air between them; :why paper rises when you blow air over it; :why a pitched baseball curves; :why a spoon is drawn toward a stream of water; :why a ball remains suspended in an air jet. Here's the news: None of these phenomena is the result of the Bernoulli effect.
  55. ^Kamela, Martin (September 2007)."Thinking About Bernoulli".The Physics Teacher.45 (6). American Association of Physics Teachers:379–381.Bibcode:2007PhTea..45..379K.doi:10.1119/1.2768700. Archived fromthe original on February 23, 2013.Finally, let's go back to the initial example of a ball levitating in a jet of air. The naive explanation for the stability of the ball in the air stream, 'because pressure in the jet is lower than pressure in the surrounding atmosphere,' is clearly incorrect. The static pressure in the free air jet is the same as the pressure in the surrounding atmosphere...
  56. ^Smith, Norman F. (November 1972). "Bernoulli and Newton in Fluid Mechanics".The Physics Teacher.10 (8): 455.Bibcode:1972PhTea..10..451S.doi:10.1119/1.2352317.Asymmetrical flow (not Bernoulli's theorem) also explains lift on theping-pong ball orbeach ball that floats so mysteriously in the tilted vacuum cleaner exhaust...
  57. ^Bauman, Robert P."The Bernoulli Conundrum"(PDF).introphysics.info. Department of Physics, University of Alabama at Birmingham. Archived fromthe original(PDF) on February 25, 2012. RetrievedJune 25, 2012.Bernoulli's theorem is often obscured by demonstrations involving non-Bernoulli forces. For example, a ball may be supported on an upward jet of air or water, because any fluid (the air and water) has viscosity, which retards the slippage of one part of the fluid moving past another part of the fluid.
  58. ^Craig, Gale M."Physical Principles of Winged Flight". RetrievedMarch 31, 2016.In a demonstration sometimes wrongly described as showing lift due to pressure reduction in moving air or pressure reduction due to flow path restriction, a ball or balloon is suspended by a jet of air.
  59. ^Anderson, David F.; Eberhardt, Scott."The Newtonian Description of Lift of a Wing"(PDF). p. 12. Archived fromthe original(PDF) on 2016-03-11 – via integener.com.A second example is the confinement of aping-pong ball in the vertical exhaust from ahair dryer. We are told that this is a demonstration of Bernoulli's principle. But, we now know that the exhaust does not have a lower value of ps. Again, it is momentum transfer that keeps the ball in the airflow. When the ball gets near the edge of the exhaust there is an asymmetric flow around the ball, which pushes it away from the edge of the flow. The same is true when one blows between two ping-pong balls hanging on strings.
  60. ^"Thin Metal Sheets – Coanda Effect".physics.umd.edu. Physics Lecture-Demonstration Facility, University of Maryland. Archived fromthe original on June 23, 2012. RetrievedOctober 23, 2012.This demonstration is often incorrectly explained using the Bernoulli principle. According to the INCORRECT explanation, the air flow is faster in the region between the sheets, thus creating a lower pressure compared with the quiet air on the outside of the sheets.
  61. ^"Answer #256".physics.umd.edu. Physics Lecture-Demonstration Facility, University of Maryland. Archived fromthe original on December 13, 2014. RetrievedDecember 9, 2014.Although the Bernoulli effect is often used to explain this demonstration, and one manufacturer sells the material for this demonstration as 'Bernoulli bags,' it cannot be explained by the Bernoulli effect, but rather by the process of entrainment.

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