Bernoulli's principle

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Bernoulli's equation redirects here; see Bernoulli differential equation for an unrelated topic in ordinary differential equations.

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Bernoulli's Principle^[1] states that for an ideal fluid (water is a good approximation), with no work being performed on the fluid, an increase in velocity occurs simultaneously with decrease in pressure or a change in the fluid's gravitational potential energy. Bernoulli's Principle is named in honour of Daniel Bernoulli.

Fluid particles are subject only to pressure and their own weight. If the speed of a particle increases it can only be because it is 'falling' from a region of higher pressure into a region of lower pressure. If its speed decreases, it can only be because it is 'climbing' out of a region of lower pressure into a region of higher pressure. Consequently, within a flowing fluid, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest.

This principle is a simplification of Bernoulli's equation, which states that the sum of all forms of energy in a fluid flowing along an enclosed path (a streamline) is the same at any two points in that path.

Contents 1 Incompressible flow equation 1.1 Simplified form 2 Compressible flow equation 2.1 Compressible flow in fluid dynamics 2.2 Compressible flow in thermodynamics 3 Derivations of Bernoulli equation 3.1 Incompressible fluids 3.2 Compressible fluids 4 Real World Application 5 A common misconception about wings 6 References 7 See also 8 External links

[edit] Incompressible flow equation

An increase in velocity is associated with a decrease in pressure, and a decrease in velocity is associated with an increase in pressure. This is called Bernoulli's Principle. The magnitudes of the change in velocity and pressure are given by Bernoulli's equation.

In most circumstances liquids can be considered to be of constant density. For this reason liquids can be considered to be incompressible and the flow of liquids can be described as incompressible flow. Bernoulli performed his experiments on liquids and his equation is valid only for incompressible flow. Bernoulli's equation is also valid for the flow of gases provided the velocity of the gas is sufficiently low that the variation in density of the gas along each streamline can be ignored.

The original form of Bernoulli's equation^[2] is:

<math>{v^2 \over 2}+gh+{p\over\rho}=\mathrm{constant}</math>

where:

<math>v\,</math> is the fluid velocity at a point on a streamline

<math>g\,</math> is the acceleration due to gravity

<math>h\,</math> is the height of the point above a reference plane

<math>p\,</math> is the pressure at the point

<math>\rho\,</math> 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:

<math>{\rho v^2 \over 2}+\rho gh+p=q+\rho gh+p=\mathrm{constant}</math>

where:

<math>q = \frac{\rho v^2}{2}</math> is dynamic pressure

The above equations suggest there is a velocity at which pressure is zero and at higher velocities the pressure is negative. Gases and liquids are not capable of negative pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid long before zero pressure is reached. The above equations use a linear relationship between velocity squared and pressure. For real fluids such as water and air this relationship is only linear for low velocities. At higher velocities in liquids, non-linear processes such as cavitation occur. At higher velocities in gases the changes in pressure and density become significant so that the assumption of constant density is invalid.

[edit] Simplified form

In many applications of Bernoulli's equation, the change in the <math>\rho\,gh</math> term along streamlines is zero or so small it can be ignored. This allows the above equation to be presented in the following simplified form:

<math>p + q = p_0\,</math>

where <math>p_0\,</math> is called total pressure, and <math>q\,</math> is dynamic pressure^[3]. Many authors refer to the pressure <math>p\,</math> as static pressure to distinguish it from total pressure <math>p_0\,</math> and dynamic pressure <math>q\,</math>. 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."^[4]

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

static pressure + dynamic pressure = total pressure^[5]

Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure <math>p</math>, dynamic pressure <math>q</math>, and total pressure <math>p_0</math>.

The significance of Bernoulli's principle can now be summarised 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 summarised 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.

[edit] 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 velocity 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.

[edit] Compressible flow in fluid dynamics

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

<math>\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} + \frac {v^2}{2} + gh = constant</math>^[6] (constant along a streamline)

where:

<math>\gamma\,</math> is the ratio of the specific heats of the fluid

<math>p\,</math> is the pressure at a point

<math>\rho\,</math> is the density at the point

<math>v\,</math> is the speed of the fluid at the point

<math>g\,</math> is the acceleration due to gravity

<math>h\,</math> 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 <math>gh\,</math> can be omitted. A very useful form of the equation is then:

<math>\left(\frac {\gamma}{\gamma-1}\right)\frac {p}{\rho} + \frac {v^2}{2} = \left(\frac {\gamma}{\gamma-1}\right)\frac {p_0}{\rho_0}</math>

where:

<math>p_0\,</math> is the total pressure

<math>\rho_0\,</math> is the total density

[edit] Compressible flow in thermodynamics

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

<math>{v^2 \over 2}+ gh + w =\mathrm{constant}</math>^[7]

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

Note that <math>w = \epsilon + \frac{p}{\rho}</math> where <math>\epsilon \,</math> 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 <math>b\,</math>. For steady inviscid adiabatic flow with no additional sources or sinks of energy, <math>b\,</math> is constant along any given streamline. More generally, when <math>b\,</math> may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below).

When the change in <math>gh\,</math> can be ignored, a very useful form of this equation is:

<math>{v^2 \over 2}+ w = w_0</math>

where <math>w_0\,</math> 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.

[edit] Derivations of Bernoulli equation

[edit] 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

<math>m \frac{dv}{dt}= -F </math>

<math>\rho A dx \frac{dv}{dt}= -A dp </math>

<math>\rho \frac{dv}{dt}= -\frac{dp}{dx} </math>

In steady flow, <math>v=v(x)</math> so

<math>\frac{dv}{dt}= \frac{dv}{dx}\frac{dx}{dt} = \frac{dv}{dx}v=\frac{d}{dx} \frac{v^2}{2}</math>

With <math>\rho</math> constant, the equation of motion can be written as

<math>\frac{d}{dx} \left( \rho \frac{v^2}{2} + p \right) =0</math>

or

<math> \frac{v^2}{2} + \frac{p}{\rho}= C</math>

where <math>C</math> 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.

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;

<math>W=\Delta KE. \;</math>

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

<math>F_{1} s_{1}-F_{2} s_{2}=p_{1} A_{1} v_

{1}\Delta t-p_{2} A_{2} v_{2}\Delta t. \;</math>

The decrease of potential energy is

<math>m g h_{1}-m g h_{2}=\rho g A

_{1} v_{1}\Delta t h_{1}-\rho g A_{2} v_{2} \Delta t h_{2} \;</math>

The increase in kinetic energy is

<math>\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2}=\frac{1}{2}\rho A_{2} v_{2}\Delta t v_{2}

^{2}-\frac{1}{2}\rho A_{1} v_{1}\Delta t v_{1}^{2}.</math>

Putting these together,

<math>p_{1} A_{1} v_{1}\Delta t-p_{2} A_{2} v_{2}\Delta t+\rho g A_{1} v_{1}\Delta t h_{1}-\rho g A_{2} v_{2}\Delta t h_{2}=\frac{1}{2}\rho A_{2} v_{2}\Delta t v_{2}^{2}-\frac{1}{2}\rho A_{1} v_{1}\Delta t v_{1}^{2}</math>

or

<math>\frac{\rho A_{1} v_{1}\Delta t v_{1}^{

2}}{2}+\rho g A_{1} v_{1}\Delta t h_{1}+p_{1} A_{1 } v_{1}\Delta t=\frac{\rho A_{2} v_{2}\Delta t v_{ 2}^{2}}{2}+\rho g A_{2} v_{2}\Delta t h_{2}+p_{2} A_{2} v_{2}\Delta t.</math>

After dividing by <math>\Delta t</math>, <math>\rho</math> and <math>A_{1} v_{1}</math> (= rate of fluid flow = <math>A_{2} v_{2}</math> as the fluid is incompressible):

<math>\frac{v_{1}^{2}}{2}+g h_{1}+\frac{p_{1}}{\rho}=\frac{v_{2}^{2}}{2}+g h_{2}+\frac{p_{2}}{\rho}</math>

or, as stated in the first paragraph:

<math>\frac{v^{2}}{2}+g h+\frac{p}{\rho}=C</math> (Eqn. 1)

Further division by <math>g\,</math> 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:

<math>\frac{v^{2}}{2 g}+h+\frac{p}{\rho g}=C</math> (Eqn. 2a)

The middle term, <math>h\,</math>, can be called an elevation head, although height is used throughout this discussion. <math>h_{elevation}\,</math> represents the internal energy of the fluid due to its height above a reference plane.

A free falling mass from a height <math>h\,</math> (in a vacuum) will reach a velocity

<math>v=\sqrt{{2 g}{h}},</math> or when we rearrange it as a head: <math>h_{v}=\frac{v^{2}}{2 g}</math>

The term <math>\frac{v^2}{2 g}</math> 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

<math>p=\rho g h \,</math>, or when we rearrange it as a head: <math>\psi=\frac{p}{\rho g}</math>

The term <math>\frac{p}{\rho g}</math> 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 velocity and the head due to static pressure with the elevation above a reference plane, we obtain a simple relationship useful for incompressible fluids.

<math>h_{v} + h_{elevation} + \psi = C\,</math> (Eqn. 2b)

If we were to multiply Eqn. 1 by the density of the fluid, we would get an equation with three pressure terms:

<math>\frac{\rho v^{2}}{2}+ \rho g h + p=C</math> (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.

[edit] 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 <math>\Delta t\,</math>, the amount of mass passing through the boundary defined by the area <math>A_1\,</math> is equal to the amount of mass passing outwards through the boundary defined by the area <math>A_2\,</math>:

<math>0= \Delta M_1 - \Delta M_2 = \rho_1 A_1 v_1 \, \Delta t - \rho_2 A_2 v_2 \, \Delta t</math>.

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 <math>A_1\,</math> and <math>A_2\,</math> 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,

<math>0= \Delta E_1 - \Delta E_2 \,</math>

where <math>\Delta E_1</math> and <math>\Delta E_2\,</math> are the energy entering through <math>A_1\,</math> and leaving through <math>A_2\,</math>, respectively.

The energy entering through <math>A_1\,</math> 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 <math>p\,dV</math> work:

<math>\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</math>

where <math>\phi=gh\,</math>, <math>g\,</math> is acceleration due to gravity, and <math>h\,</math> is height above a reference plane

A similar expression for <math>\Delta E_2 </math> may easily be constructed. So now setting <math> 0 = \Delta E_1 - \Delta E_2</math>:

<math>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</math>

which can be rewritten as:

<math> 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 </math>

Now, using the previously-obtained result from conservation of mass, this may be simplified to obtain

<math> \frac{1}{2}v^2 + \phi + \epsilon + \frac{p}{\rho} = {\rm constant} \equiv b </math>

which is the Bernoulli equation for compressible flow.

[edit] Real World Application

• 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 velocity 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, conservation of mass shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid velocity. Subsequenty 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 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.

[edit] A common misconception about wings

While lift generated by an airfoil is often attributed to Bernoulli's Principle, it cannot be used to explain this. The popular and faulty explanation is generally written as this:

• A wing is flat on the underside and curved on the top (is cambered). Thus air that moves over the top of the wing must travel a longer path to get to the end of the wing and therefore gets a higher velocity. According to Bernoulli's Principle, the faster moving air has lower pressure and the difference therefore results in lift.

This explanation is flawed in a critical aspect: it assumes that two parcels of air separated at the leading edge of the wing, with one traveling over and one traveling below, must meet again at the trailing edge of the wing. This does not happen. In fact the Kutta-Joukowski Theorem shows that if this was to happen, the wing would generate no lift.

The actual mechanism generating lift on an airfoil is Newton's Third Law of Motion.^[8] An airfoil is always flown at an angle of attack against the air flow. As the wing deflects air downwards the opposing reaction force on the wing pushes it upwards. Note that nearly all of the lift arises from airflow over the top of the wing being deflected downwards, due to the Coanda Effect; the deflection due to the underside of the wing makes only a small contribution. (This is one reason why wing-mounted jet engines are suspended below the wing, rather than being placed on top of it - the disruptioin to flow over the bottom surface of the wing has much smaller effect of lift than mounting the engines above the wings.) Contrary to Bernoulli's Principle this explains why an aircraft with a thrust-to-weight ratio less than 1.0 can fly on a level flight path while being upside down (instead of being pulled dramatically towards the ground); why slim wings, such as those of the F-104 Starfighter or those of a paper plane, generate lift despite the camber being nonexistent; and why some lifting body aircraft can fly despite being very bulbous on the underside.

The airflow can be observed by solving the Navier-Stokes equations for the appropriate flow regime (turbulent or laminar) and the pressure along the wings edges directly calculated. As pressure is simply a measurement of force per unit area, integration of the pressure along the wings surface (both top and bottom) provides an overall force, which is the amount of lift provided by the wing.

Though Bernoulli's Principle cannot be used to explain the lifting mechanism, it can still be used to accurately analyze the airflow around an airfoil. If you know either the air pressure or the air velocity over a wing you can use Bernoulli's equation to calculate the value of the other property. The equation is very often used this way. This high frequency of use has been cited as the reason the misconception has arisen.^[8]

[edit] References

[edit] 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

[edit] External links

• Denver University - Bernoulli's equation and Pressure measurement

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