Bernoulli's principle

This article is about Bernoulli's principle and Bernoulli's equation in fluid dynamics. For an unrelated topic in ordinary differential equations, see Bernoulli differential equation.

A flow of air into a venturi meter. The kinetic energy increases at the expense of the fluid pressure, as shown by the difference in height of the two columns of water.

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's gravitational potential energy. Bernoulli's principle is named after the inventor Daniel Bernoulli.

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. But in fact there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (e.g. most liquid flows) and also for compressible flows (e.g. gases) moving at low Mach numbers. More advanced forms may in some cases be applied to compressible flows at higher Mach numbers (see the derivations of the Bernoulli equation).

Bernoulli's principle is equivalent to the principle of conservation of energy. This states that in a steady flow the sum of all forms of mechanical energy in a fluid along a streamline is the same at all points on that streamline. This requires that the sum of kinetic energy and potential energy remain constant. 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 mass (the sum of pressure and gravitational potential $\rho gh$) is the same everywhere.

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, the mass density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. For this reason the fluid in such flows can be considered to be incompressible and these flows can be described as incompressible flow. Bernoulli performed his experiments on liquids and his equation in its original form is valid only for incompressible flow.

The original form of Bernoulli's equation is:

${v^{2} \over 2}+gh+{p \over \rho }=\mathrm {constant}$

where:

$v\,$ is the fluid flow speed at a point on a streamline

$g\,$ is the acceleration due to gravity

$h\,$ is the height of the point above a reference plane

$p\,$ is the pressure at the point

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

The following assumptions must be met for the equation to apply:

• The fluid must be incompressible - even though pressure varies, the density must remain constant.
• The streamline must not enter the boundary layer. (Bernoulli's equation is not applicable where there are viscous forces, such as in the boundary layer.)

The above equation can be rewritten as:

${\rho v^{2} \over 2}+\rho gh+p=q+\rho gh+p=\mathrm {constant}$

where:

$q={\frac {\rho v^{2}}{2}}$ is dynamic pressure

The above equations suggest there is a flow speed at which pressure is zero and at higher speeds the pressure is negative. 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. The above equations use a linear relationship between flow speed squared and pressure. At higher velocities in liquids, non-linear processes such as (viscous) turbulent flow and cavitation occur. At higher flow speeds in gases the changes in pressure become significant so that the assumption of constant density is invalid.

Simplified form

In several applications of Bernoulli's equation, the change in the $\rho \,gh$ term along streamlines is zero or so small it can be ignored: for instance in the case of airfoils at low Mach number. This allows the above equation to be presented in the following simplified form:

$p+q=p_{0}\,$

where $p_{0}\,$ is called total pressure, and $q\,$ is dynamic pressure. Many authors refer to the pressure $p\,$ as static pressure to distinguish it from total pressure $p_{0}\,$ and dynamic pressure $q\,$. In Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure."

The simplified form of Bernoulli's equation can be summarized in the following memorable word equation:

static pressure + dynamic pressure = total pressure

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure $p$, dynamic pressure $q$, and total pressure $p_{0}$.

The significance of Bernoulli's principle can now be summarized as "total pressure is constant along a streamline." Furthermore, if the fluid flow originated in a reservoir, 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." However, it is important to remember that Bernoulli's principle does not apply in the boundary layer.

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 can not be assumed to be valid. However if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas, (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature, however this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle, or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough.

Unsteady potential flow

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

For an irrotational flow, the flow velocity can be described as the gradient ∇φ of a velocity potential φ. In that case, and for a constant density ρ, the momentum equations of the Euler equations can be integrated to:

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

which is a Bernoulli equation valid also for unsteady — or time dependent — flows. Here ∂φ/∂t denotes the partial derivative of the velocity potential φ with respect to time t, and v = |∇φ| is the flow speed. The function f(t) depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment t does not only apply along a certain streamline, but in the whole fluid domain. This is also true for the special case of a steady irrotational flow, when f is a constant.

Further f(t) can be made equal to zero by incorporating it into the velocity potential using the transformation

$\Phi =\varphi -\int _{t_{0}}^{t}f(\tau )\,{\text{d}}\tau$   resulting in   ${\frac {\partial \Phi }{\partial t}}+{\frac {1}{2}}v^{2}+{\frac {p}{\rho }}+gz=0.$

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

The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian (not to be confused with Lagrangian coordinates).

Compressible flow equation

Bernoulli developed his principle from his observations on liquids, and his equation is applicable only to incompressible fluids, and compressible fluids at very low speeds (perhaps up to 1/3 of the sound speed in the fluid). It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics.

Compressible flow in fluid dynamics

A useful form of the equation, suitable for use in compressible fluid dynamics, is:

${\frac {v^{2}}{2}}+gh+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}=\mathrm {constant}$ (constant along a streamline)

where:

$\gamma \,$ is the ratio of the specific heats of the fluid

$p\,$ is the pressure at a point

$\rho \,$ is the density at the point

$v\,$ is the speed of the fluid at the point

$g\,$ is the acceleration due to gravity

$h\,$ is the height of the point above a reference plane

In many applications of compressible flow, changes in height above a reference plane are negligible so the term $gh\,$ can be omitted. A very useful form of the equation is then:

${\frac {v^{2}}{2}}+\left({\frac {\gamma }{\gamma -1}}\right){\frac {p}{\rho }}=\left({\frac {\gamma }{\gamma -1}}\right){\frac {p_{0}}{\rho _{0}}}$

where:

$p_{0}\,$ is the total pressure

$\rho _{0}\,$ is the total density

Compressible flow in thermodynamics

Another useful form of the equation, suitable for use in thermodynamics, is:

${v^{2} \over 2}+gh+w=\mathrm {constant}$

$w\,$ is the enthalpy per unit mass, which is also often written as $h\,$ (which would conflict with the use of $h\,$ for "height" in this article).

Note that $w=\epsilon +{\frac {p}{\rho }}$ where $\epsilon \,$ is the thermodynamic energy per unit mass, also known as the specific internal energy or "sie."

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

When the change in $gh\,$ can be ignored, a very useful form of this equation is:

${v^{2} \over 2}+w=w_{0}$

where $w_{0}\,$ is total enthalpy.

When shock waves are present, in a reference frame moving with a shock, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter itself, however, remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy.

Derivations of Bernoulli equation

Bernoulli equation for incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law of conservation of energy in two sections along a streamline, ignoring viscosity, compressibility, and thermal effects. The simplest derivation is to first ignore gravity and consider constrictions and expansions in pipes that are otherwise straight, as seen in Venturi effect. Let the x axis be directed down the axis of the pipe. The equation of motion for a parcel of fluid on the axis of the pipe is $m{\frac {dv}{dt}}=-F$ $\rho Adx{\frac {dv}{dt}}=-Adp$ $\rho {\frac {dv}{dt}}=-{\frac {dp}{dx}}$ In steady flow, $v=v(x)$ so ${\frac {dv}{dt}}={\frac {dv}{dx}}{\frac {dx}{dt}}={\frac {dv}{dx}}v={\frac {d}{dx}}{\frac {v^{2}}{2}}$ With $\rho$ constant, the equation of motion can be written as ${\frac {d}{dx}}\left(\rho {\frac {v^{2}}{2}}+p\right)=0$ or ${\frac {v^{2}}{2}}+{\frac {p}{\rho }}=C$ where $C$ is a constant, sometimes referred to as the Bernoulli constant. It is not a universal constant, but rather a constant of a particular fluid system. We deduce that where the speed is large, pressure is low and vice versa. In the above derivation, no external work-energy principle is invoked. Rather, the work-energy principle was inherently derived by a simple manipulation of the momentum equation. A streamtube of fluid moving to the right. Indicated are pressure, height, flow speed, distance (s), and cross-sectional area. Applying conservation of energy in form of the work-kinetic energy theorem we find that: the change in KE of the system equals the net work done on the system; $W=\Delta KE.\;$ Therefore, the work done by the forces in the fluid + decrease in potential energy = increase in kinetic energy. The work done by the forces is $F_{1}s_{1}-F_{2}s_{2}=p_{1}A_{1}v_{1}\Delta t-p_{2}A_{2}v_{2}\Delta t.\;$ The decrease of potential energy is $mgh_{1}-mgh_{2}=\rho gA_{1}v_{1}\Delta th_{1}-\rho gA_{2}v_{2}\Delta th_{2}\;$ The increase in kinetic energy is ${\frac {1}{2}}mv_{2}^{2}-{\frac {1}{2}}mv_{1}^{2}={\frac {1}{2}}\rho A_{2}v_{2}\Delta tv_{2}^{2}-{\frac {1}{2}}\rho A_{1}v_{1}\Delta tv_{1}^{2}.$ Putting these together, $p_{1}A_{1}v_{1}\Delta t-p_{2}A_{2}v_{2}\Delta t+\rho gA_{1}v_{1}\Delta th_{1}-\rho gA_{2}v_{2}\Delta th_{2}={\frac {1}{2}}\rho A_{2}v_{2}\Delta tv_{2}^{2}-{\frac {1}{2}}\rho A_{1}v_{1}\Delta tv_{1}^{2}$ or ${\frac {\rho A_{1}v_{1}\Delta tv_{1}^{2}}{2}}+\rho gA_{1}v_{1}\Delta th_{1}+p_{1}A_{1}v_{1}\Delta t={\frac {\rho A_{2}v_{2}\Delta tv_{2}^{2}}{2}}+\rho gA_{2}v_{2}\Delta th_{2}+p_{2}A_{2}v_{2}\Delta t.$ After dividing by $\Delta t$, $\rho$ and $A_{1}v_{1}$ (= rate of fluid flow = $A_{2}v_{2}$ as the fluid is incompressible): ${\frac {v_{1}^{2}}{2}}+gh_{1}+{\frac {p_{1}}{\rho }}={\frac {v_{2}^{2}}{2}}+gh_{2}+{\frac {p_{2}}{\rho }}$ or, as stated in the first paragraph: ${\frac {v^{2}}{2}}+gh+{\frac {p}{\rho }}=C$ (Eqn. 1) Further division by $g\,$ produces the following equation. Note that each term can be described in the length dimension (such as meters). This is the head equation derived from Bernoulli's principle: ${\frac {v^{2}}{2g}}+h+{\frac {p}{\rho g}}=C$ (Eqn. 2a) The middle term, $h\,$, can be called head, although height is used throughout this discussion. $h_{\text{elevation}}\,$ represents the internal energy of the fluid due to its height above a reference plane. A free falling mass from a height $h\,$ (in a vacuum) will reach a speed $v={\sqrt {{2g}{h}}},$ or when we rearrange it as a head: $h_{v}={\frac {v^{2}}{2g}}$ The term ${\frac {v^{2}}{2g}}$ is called the velocity head, expressed as a length measurement. It represents the internal energy of the fluid due to its motion. The hydrostatic pressure p is defined as $p=\rho gh\,$, or when we rearrange it as a head: $\psi ={\frac {p}{\rho g}}$ The term ${\frac {p}{\rho g}}$ is also called the pressure head, expressed as a length measurement. It represents the internal energy of the fluid due to the pressure exerted on the container. When we combine the head due to the flow speed and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids. $h_{v}+h_{\text{elevation}}+\psi =C\,$ (Eqn. 2b) If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms: ${\frac {\rho v^{2}}{2}}+\rho gh+p=C$ (Eqn. 3) We note that the pressure of the system is constant in this form of the Bernoulli Equation. If the static pressure of the system (the far right term) increases, and if the pressure due to elevation (the middle term) is constant, then we know that the dynamic pressure (the left term) must have decreased. In other words, if the speed of a fluid decreases and it is not due to an elevation difference, we know it must be due to an increase in the static pressure that is resisting the flow. All three equations are merely simplified versions of an energy balance on a system.

Bernoulli equation for compressible fluids

The derivation for compressible fluids is similar. Again, the derivation depends upon (1) conservation of mass, and (2) conservation of energy. Conservation of mass implies that in the above figure, in the interval of time $\Delta t\,$, the amount of mass passing through the boundary defined by the area $A_{1}\,$ is equal to the amount of mass passing outwards through the boundary defined by the area $A_{2}\,$: $0=\Delta M_{1}-\Delta M_{2}=\rho _{1}A_{1}v_{1}\,\Delta t-\rho _{2}A_{2}v_{2}\,\Delta t$. Conservation of energy is applied in a similar manner: It is assumed that the change in energy of the volume of the streamtube bounded by $A_{1}\,$ and $A_{2}\,$ is due entirely to energy entering or leaving through one or the other of these two boundaries. Clearly, in a more complicated situation such as a fluid flow coupled with radiation, such conditions are not met. Nevertheless, assuming this to be the case and assuming the flow is steady so that the net change in the energy is zero, $0=\Delta E_{1}-\Delta E_{2}\,$ where $\Delta E_{1}$ and $\Delta E_{2}\,$ are the energy entering through $A_{1}\,$ and leaving through $A_{2}\,$, respectively. The energy entering through $A_{1}\,$ is the sum of the kinetic energy entering, the energy entering in the form of potential gravitational energy of the fluid, the fluid thermodynamic energy entering, and the energy entering in the form of mechanical $p\,dV$ work: $\Delta E_{1}=\left[{\frac {1}{2}}\rho _{1}v_{1}^{2}+\phi _{1}\rho _{1}+\epsilon _{1}\rho _{1}+p_{1}\right]A_{1}v_{1}\,\Delta t$ where $\phi =gh\,$, $g\,$ is acceleration due to gravity, and $h\,$ is height above a reference plane A similar expression for $\Delta E_{2}$ may easily be constructed. So now setting $0=\Delta E_{1}-\Delta E_{2}$: $0=\left[{\frac {1}{2}}\rho _{1}v_{1}^{2}+\phi _{1}\rho _{1}+\epsilon _{1}\rho _{1}+p_{1}\right]A_{1}v_{1}\,\Delta t-\left[{\frac {1}{2}}\rho _{2}v_{2}^{2}+\phi _{2}\rho _{2}+\epsilon _{2}\rho _{2}+p_{2}\right]A_{2}v_{2}\,\Delta t$ which can be rewritten as: $0=\left[{\frac {1}{2}}v_{1}^{2}+\phi _{1}+\epsilon _{1}+{\frac {p_{1}}{\rho _{1}}}\right]\rho _{1}A_{1}v_{1}\,\Delta t-\left[{\frac {1}{2}}v_{2}^{2}+\phi _{2}+\epsilon _{2}+{\frac {p_{2}}{\rho _{2}}}\right]\rho _{2}A_{2}v_{2}\,\Delta t$ Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain ${\frac {1}{2}}v^{2}+\phi +\epsilon +{\frac {p}{\rho }}={\rm {constant}}\equiv b$ which is the Bernoulli equation for compressible flow.

Real world application

In every-day life there are many observations that can be successfully explained by application of Bernoulli's principle.

• The air flowing past the top of the wing of an airplane, or the rotor blades of a helicopter, is moving much faster than the air flowing past the under-side of the wing or rotor blade. The air pressure on the top of the wing or rotor blade is much lower than the air pressure on the under-side, and this explains the origin of the lift force generated by a wing or rotor blade to keep the airplane or helicopter in the air. The fact that the air is moving very fast over the top of the wing or rotor blade and the air pressure is very low on the top of the wing or rotor blade is an example of Bernoulli's principle in action, even though Bernoulli established his famous principle over a century before the first man-made wings were used for the purpose of flight. (Bernoulli's principle does not explain why the air flows faster past the top of the wing and slower past the under-side. To understand why, it is helpful to understand circulation, the Kutta condition and the Kutta–Joukowski theorem.)

• The carburetor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburetor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli's principle - in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure.

• The pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to the airspeed indicator which determines the dynamic pressure of the airflow past the aircraft. Dynamic pressure is the difference between stagnation pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure.

• The flow speed of a fluid can be measured using a devices such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect.

• The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation, and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli's law, showing that Torricelli's law is compatible with Bernoulli's principle. Viscosity lowers this drain rate. This is reflected in the discharge coefficient which is a function of the Reynold's number and the shape of the orifice.

Misunderstandings about the generation of lift

Main article: Lift (force)

Many explanations for the generation of lift can be found; but some of these explanations can be misleading, and some are false. This has been a source of heated discussion over the years. In particular, there has been debate about whether lift is best explained by Bernoulli's principle or Newton's Laws. Modern writings agree that Bernoulli's principle and Newton's Laws are both relevant and correct.

Several of these explanations use Bernoulli's principle to connect the flow kinematics to the flow-induced pressures. In case of incorrect (or partially correct) explanations of lift, also relying at some stage on Bernoulli's principle, the errors generally occur in the assumptions on the flow kinematics, and how these are produced. It is not Bernoulli's principle itself that is questioned because this principle is well established.

References

• Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0521663962.
• Clancy, L.J. (1975). Aerodynamics. Pitman Publishing, London. ISBN 0273011200.
• Lamb, H. (1994). Hydrodynamics (6^th edition ed.). Cambridge University Press. ISBN 9780521458689. {{cite book}}: |edition= has extra text (help) Originally published in 1879, the 6^th extended edition appeared first in 1932.

See also

• Terminology in fluid dynamics
• Navier-Stokes equations – The Navier-Stokes equations for fluid flow
• Euler equations – A special case of the Navier-Stokes equation for an inviscid fluid
• Hydraulics – Applied fluid mechanics for liquids

External links

• Denver University - Bernoulli's equation and Pressure measurement
• Millersville University - Applications of Euler's Equation
• Nasa - Beginner's Guide to Aerodynamics