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# Bernoulli's principle: Map

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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 potential energy. Bernoulli's principle is named after the Dutch-Swiss mathematician Daniel Bernoulli who published his principle in his book Hydrodynamica in 1738.

Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation. In fact, there are different forms of the Bernoulli equation for different types of flow. The simple form of Bernoulli's principle is valid for incompressible flows (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 can be derived from 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 ρ g h) 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 -- valid at any point along a streamline -- is:

{v^2 \over 2}+\Psi+{p\over\rho}=\text{constant}

where:
v is the fluid flow speed at a point on a streamline,
Ψ is the gravitational potential,
p is the pressure at the point, and
ρ 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 a boundary layer. (Bernoulli's equation is not applicable where there are viscous forces, such as in a boundary layer.)

If the acceleration due to gravity (g) does not change over the length scale of the problem (eg the situation occurs at an elevation which is low compared with the radius of the Earth) then Ψ = g z, where z is the elevation of the point above some reference plane (positive z-direction points upward, in the opposite direction to the gravitational acceleration). This will be assumed for the rest of this section.

By multiplying with the mass density ρ, the above equation can be rewritten as:

\tfrac12\, \rho\, v^2\, +\, \rho\, g\, z\, +\, p\, =\, \text{constant}\,

or:

q\, +\, \rho\, g\, h\,
 =\, p_0\, +\, \rho\, g\, z\,
=\, \text{constant}\,

where:
q\, =\, \tfrac12\, \rho\, v^2 is dynamic pressure,

h\, =\, z\, +\, \frac{p}{\rho g} is the piezometric head or hydraulic head (the sum of the elevation z and the pressure head) , 410 pages. See pp. 43-44. , 650 pages. See p. 22. and

p_0\, =\, p\, +\, q\, is the total pressure (the sum of the static pressure p and dynamic pressure q).

The constant in the Bernoulli equation can be normalised. A common approach is in terms of total head or energy head H:

H\, =\, z\, +\, \frac{p}{\rho g}\, +\, \frac{v^2}{2\,g}\, =\, h\, +\, \frac{v^2}{2\,g}, so divide the above constant by ρ and g to get the total head H in terms of metres of fluid column.

The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids -- when the pressure becomes too low -- cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid.

### Simplified form

In many applications of Bernoulli's equation, the change in the ρ g z term along the streamline is so small compared with the other terms it can be ignored. For example, in the case of aircraft in flight, the change in height z along a streamline is so small the ρ g z term can be omitted. This allows the above equation to be presented in the following simplified form:

p + q = p_0\,

where p0 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 p0 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 and dynamic pressure q. Their sum p + q is defined to be the total pressure p0. The significance of Bernoulli's principle can now be summarized as total pressure is constant along a streamline.

If the fluid flow is irrotational, 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. It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight, and ships moving in open bodies of water. However, it is important to remember that Bernoulli's principle does not apply in the boundary layer or in fluid flow through long pipes.

### Applicability of incompressible flow equation to flow of gases

Bernoulli's equation is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation -- in its incompressible flow form -- can not be assumed to be valid. However if the gas process is entirely isobaric, 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} + \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 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, in which case 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)\, \operatorname{d}\tau   resulting in   \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 in Luke's variational principle, a variational description of free-surface flows using the Lagrangian (not to be confused with Lagrangian coordinates).

Lift and Drag curves for a typical airfoil

## 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

For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces,

\frac {v^2}{2}+ \int_{p_1}^p \frac {d\tilde{p}}{\rho(\tilde{p})}\ + \Psi = \text{constant} (constant along a streamline)

where:
p is the pressure
ρ is the density
v is the flow speed
Ψ is the potential associated with the conservative force field, often the gravitational 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 becomes

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

where, in addition to the terms listed above:
γ is the ratio 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 term gz 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:
p0 is the total pressure
ρ0 is the total density

### Compressible flow in thermodynamics

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

{v^2 \over 2} + \Psi + w =\text{constant}.

Here w is the enthalpy per unit mass, which is also often written as h (not to be confused with "head" or "height").

Note that w = \epsilon + \frac{p}{\rho} where ε 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 Ψ can be ignored, a very useful form of this equation is:

{v^2 \over 2}+ w = w_0

where w0 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

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## Real world application

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

• Bernoulli's Principle can be used to calculate the lift force on an airfoil if you know the behavior of the fluid flow in the vicinity of the foil. 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 upwards lift force. 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 -- established by Bernoulli 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 flow speed of a fluid can be measured using a device 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.

• The principle also makes it possible for sail-powered craft to travel faster than the wind that propels them (if friction can be sufficiently reduced). If the wind passing in front of the sail is fast enough to experience a significant reduction in pressure, the sail is pulled forward, in addition to being pushed from behind. Although boats in water must contend with the friction of the water along the hull, ice sailing and land sailing vehicles can travel faster than the wind.

## Misunderstandings about the generation of lift

Many explanations for the generation of lift(on airfoils, propellerblades, etc.) 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 of motion. 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

1. Clancy, L.J., Aerodynamics, Chapter 3.
2. Batchelor, G.K. (1967), Section 3.5, pp. 156-64.
3. Streeter, V.L., Fluid Mechanics, Example 3.5, McGraw–Hill Inc. (1966), New York.
4. Clancy, L.J., Aerodynamics, Section 3.4.
5. Clancy, L.J., Aerodynamics, Section 3.5.
6. Clancy, L.J. Aerodynamics, Equation 3.12
7. Batchelor, G.K. (1967), p. 383
8. Clarke C. and Carswell B., Astrophysical Fluid Dynamics
9. Clancy, L.J., Aerodynamics, Section 3.11
10. Van Wylen, G.J., and Sonntag, R.E., (1965), Fundamentals of Classical Thermodynamics, Section 5.9, John Wiley and Sons Inc., New York
11. , p. 138.
12. , Vol. 1, §14-3, p. 14-4.
13. Clancy, L.J., Aerodynamics, Section 5.5 ("When a stream of air flows past an airfoil, there are local changes in flow speed 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.")
14. Resnick, R. and Halliday, D. (1960), Physics, Section 18-5, John Wiley & Sons, Inc., New York ("[streamlines] are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.")
15. "The resultant force is determined by integrating the surface-pressure distribution over the surface area of the airfoil."
16. Clancy, L.J., Aerodynamics, Section 3.8
17. Mechanical Engineering Reference Manual Ninth Edition
18. Ice Sailing Manual, p. 2
19. Wind Sports – Ice sailing hand held sails
20. Ison, David. Bernoulli Or Newton: Who's Right About Lift? Retrieved on 2009-11-26
21. Section 2.4.
22. Batchelor, G.K. (1967). Sections 3.5 and 5.1
23. Lamb, H. (1994). §17-§29
24. "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause."

## Notes

1. Clancy, L.J., Aerodynamics, Chapter 3.
2. Batchelor, G.K. (1967), Section 3.5, pp. 156-64.
3. Streeter, V.L., Fluid Mechanics, Example 3.5, McGraw–Hill Inc. (1966), New York.
4. Clancy, L.J., Aerodynamics, Section 3.4.
5. Clancy, L.J., Aerodynamics, Section 3.5.
6. Clancy, L.J. Aerodynamics, Equation 3.12
7. Batchelor, G.K. (1967), p. 383
8. Clarke C. and Carswell B., Astrophysical Fluid Dynamics
9. Clancy, L.J., Aerodynamics, Section 3.11
10. Van Wylen, G.J., and Sonntag, R.E., (1965), Fundamentals of Classical Thermodynamics, Section 5.9, John Wiley and Sons Inc., New York
11. , p. 138.
12. , Vol. 1, §14-3, p. 14-4.
13. Clancy, L.J., Aerodynamics, Section 5.5 ("When a stream of air flows past an airfoil, there are local changes in flow speed 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.")
14. Resnick, R. and Halliday, D. (1960), Physics, Section 18-5, John Wiley & Sons, Inc., New York ("[streamlines] are closer together above the wing than they are below so that Bernoulli's principle predicts the observed upward dynamic lift.")
15. "The resultant force is determined by integrating the surface-pressure distribution over the surface area of the airfoil."
16. Clancy, L.J., Aerodynamics, Section 3.8
17. Mechanical Engineering Reference Manual Ninth Edition
18. Ice Sailing Manual, p. 2
19. Wind Sports – Ice sailing hand held sails
20. Ison, David. Bernoulli Or Newton: Who's Right About Lift? Retrieved on 2009-11-26
21. Section 2.4.
22. Batchelor, G.K. (1967). Sections 3.5 and 5.1
23. Lamb, H. (1994). §17-§29
24. "The conventional explanation of aerodynamical lift based on Bernoulli’s law and velocity differences mixes up cause and effect. The faster flow at the upper side of the wing is the consequence of low pressure and not its cause."

• Originally published in 1879; the 6th extended edition appeared first in 1932.

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, having a length dx, mass density ρ, mass m = ρ A dx and flow velocity v = dx / dt, moving along the axis of the horizontal pipe, with cross-sectional area A is

m \frac{\operatorname{d}v}{\operatorname{d}t}= F
\rho A \operatorname{d}x \frac{\operatorname{d}v}{\operatorname{d}t}= -A \operatorname{d}p
\rho \frac{\operatorname{d}v}{\operatorname{d}t}= -\frac{\operatorname{d}p}{\operatorname{d}x}

In steady flow, v = v(x) so

\frac{\operatorname{d}v}{\operatorname{d}t}= \frac{\operatorname{d}v}{\operatorname{d}x}\frac{\operatorname{d}x}{\operatorname{d}t} = \frac{\operatorname{d}v}{\operatorname{d}x}v=\frac{d}{\operatorname{d}x} \left( \frac{v^2}{2} \right).

With density ρ constant, the equation of motion can be written as

\frac{\operatorname{d}}{\operatorname{d}x} \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. 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 inherently derived by a simple manipulation of the momentum equation.

Another way to derive Bernoulli's principle for an incompressible flow is by applying conservation of energy. , Vol. 2, §40-3, p. 40-6 -- 40-9. In the form of the work-energy theorem, stating that

the change in the kinetic energy Ekin of the system equals the net work W done on the system;

W = \Delta E_\text{kin}. \;

Therefore,
the work done by the forces in the fluid = increase in kinetic energy.

The system consists of the volume of fluid, initially between the cross-sections A1 and A2. In the time interval Δt fluid elements initially at the inflow cross-section A1 move over a distance s1 = v1 Δt, while at the outflow cross-section the fluid moves away from cross-section A2 over a distance s2 = v2 Δt. The displaced fluid volumes at the inflow and outflow are respectively A1 s1 and A2 s2. The associated displaced fluid masses are -- when ρ is the fluid's mass density -- equal to density times volume, so ρ A1 s1 and ρ A2 s2. By mass conservation, these two masses displaced in the time interval Δt have to be equal, and this displaced mass is denoted by Δm:

\begin{align}
 \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{align}

The work done by the forces consists of two parts:
• The work done by the pressure acting on the area's A1 and A2

: W_\text{pressure}=F_{1,\text{pressure}}\; s_{1}\, -\, F_{2,\text{pressure}}\; s_{2}=p_{1} A_{1} s_1 - p_{2} A_{2} s_{2} = \Delta m\, \frac{p_1}{\rho} - \Delta m\, \frac{p_2}{\rho}. \;

• The work done by gravity: the gravitational potential energy in the volume A1 s1 is lost, and at the outflow in the volume A2 s2 is gained. So, the change in gravitational potential energy ΔEpot,gravity in the time interval Δt is

: \Delta E_\text{pot,gravity} = \Delta m\, g z_{2} - \Delta m\, g z_{1}. \;

Now, the work by the force of gravity is opposite to the change in potential energy, Wgravity = −ΔEpot,gravity: while the force of gravity is in the negative z-direction, the work--gravity force times change in elevation--will be negative for a positive elevation change Δz = z2 − z1, while the corresponding potential energy change is positive. So:

: W_\text{gravity} = -\Delta E_\text{pot,gravity} = \Delta m\, g z_{1} - \Delta m\, g z_{2}. \;

And the total work done in this time interval \Delta t is

W = W_\text{pressure} + W_\text{gravity}. \,

The increase in kinetic energy is

\Delta E_\text{kin} = \frac{1}{2} \Delta m\, v_{2}^{2}-\frac{1}{2} \Delta m\, v_{1}^{2}.

Putting these together, the work-kinetic energy theorem W = ΔEkin gives:

\Delta m\, \frac{p_{1}}{\rho} - \Delta m\, \frac{p_{2}}{\rho} + \Delta m\, g z_{1} - \Delta m\, g z_{2} = \frac{1}{2} \Delta m\, v_{2}^{2} - \frac{1}{2} \Delta m\, v_{1}^{2}

or

\frac12 \Delta m\, v_{1}^{2} + \Delta m\, g z_{1} + \Delta m\, \frac{p_{1}}{\rho} = \frac12 \Delta m\, v_{2}^{2} + \Delta m\, g z_{2} + \Delta m\, \frac{p_{2}}{\rho}.

After dividing by the mass Δm = ρ A1 v1 Δt = ρ A2 v2 Δt the result is:

\frac12 v_{1}^{2}+g z_{1}+\frac{p_{1}}{\rho}=\frac12 v_{2}^{2}+g z_{2}+\frac{p_{2}}{\rho}

or, as stated in the first paragraph:

\frac{v^{2}}{2}+g z+\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}}{2 g}+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 designation zelevation.

A free falling mass from an elevation z > 0 (in a vacuum) will reach a speed

v=\sqrt{{2 g}{z}}, when arriving at elevation z=0. Or when we rearrange it as a head: h_{v}=\frac{v^{2}}{2 g}

The term v2 / (2 g) 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=p_0-\rho g z \,, with p0 some reference pressure, or when we rearrange it as a head: \psi=\frac{p}{\rho g}

The term p / (ρg) 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 using the velocity head, elevation head, and pressure head.

h_{v} + z_\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 g z + 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 Δt, the amount of mass passing through the boundary defined by the area A1 is equal to the amount of mass passing outwards through the boundary defined by the area A2:

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 by A1 and A2 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 ΔE1 and ΔE2 are the energy entering through A1 and leaving through A2, respectively.

The energy entering through A1 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 + \Psi_1 \rho_1 + \epsilon_1 \rho_1 + p_1 \right] A_1 v_1 \, \Delta t

where Ψ = gz is a force potential due to the Earth's gravity, g is acceleration due to gravity, and z is elevation 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+ \Psi_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 + \Psi_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 + \Psi_1 + \epsilon_1 + \frac{p_1}{\rho_1} \right] \rho_1 A_1 v_1 \, \Delta t - \left[ \frac{1}{2} v_2^2 + \Psi_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 + \Psi + \epsilon + \frac{p}{\rho} = {\rm constant} \equiv b

which is the Bernoulli equation for compressible flow.

 Embed code: at The Full Wiki'>


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# 1.6 Mechanisms of heat transfer  (Page 12/27)

 Page 12 / 27

## Key equations

 Linear thermal expansion $\text{Δ}L=\alpha L\text{Δ}T$ Thermal expansion in two dimensions $\text{Δ}A=2\alpha A\text{Δ}T$ Thermal expansion in three dimensions $\text{Δ}V=\beta V\text{Δ}T$ Heat transfer $Q=mc\text{Δ}T$ Transfer of heat in a calorimeter ${Q}_{\text{cold}}+{Q}_{\text{hot}}=0$ Heat due to phase change (melting and freezing) $Q=m{L}_{\text{f}}$ Heat due to phase change (evaporation and condensation) $Q=m{L}_{\text{v}}$ Rate of conductive heat transfer $P=\frac{kA\left({T}_{h}-{T}_{c}\right)}{d}$ Net rate of heat transfer by radiation ${P}_{\text{net}}=\sigma eA\left({T}_{2}{}^{4}-{T}_{1}{}^{4}\right)$

## Conceptual questions

What are the main methods of heat transfer from the hot core of Earth to its surface? From Earth’s surface to outer space?

When our bodies get too warm, they respond by sweating and increasing blood circulation to the surface to transfer thermal energy away from the core. What effect will those processes have on a person in a $40.0\text{-}\text{°}\text{C}$ hot tub?

Increasing circulation to the surface will warm the person, as the temperature of the water is warmer than human body temperature. Sweating will cause no evaporative cooling under water or in the humid air immediately above the tub.

Shown below is a cut-away drawing of a thermos bottle (also known as a Dewar flask), which is a device designed specifically to slow down all forms of heat transfer. Explain the functions of the various parts, such as the vacuum, the silvering of the walls, the thin-walled long glass neck, the rubber support, the air layer, and the stopper.

Some electric stoves have a flat ceramic surface with heating elements hidden beneath. A pot placed over a heating element will be heated, while the surface only a few centimeters away is safe to touch. Why is ceramic, with a conductivity less than that of a metal but greater than that of a good insulator, an ideal choice for the stove top?

It spread the heat over the area above the heating elements, evening the temperature there, but does not spread the heat much beyond the heating elements.

Loose-fitting white clothing covering most of the body, shown below, is ideal for desert dwellers, both in the hot Sun and during cold evenings. Explain how such clothing is advantageous during both day and night.

One way to make a fireplace more energy-efficient is to have room air circulate around the outside of the fire box and back into the room. Detail the methods of heat transfer involved.

Heat is conducted from the fire through the fire box to the circulating air and then convected by the air into the room (forced convection).

On cold, clear nights horses will sleep under the cover of large trees. How does this help them keep warm?

When watching a circus during the day in a large, dark-colored tent, you sense significant heat transfer from the tent. Explain why this occurs.

The tent is heated by the Sun and transfers heat to you by all three processes, especially radiation.

Satellites designed to observe the radiation from cold (3 K) dark space have sensors that are shaded from the Sun, Earth, and the Moon and are cooled to very low temperatures. Why must the sensors be at low temperature?

Why are thermometers that are used in weather stations shielded from the sunshine? What does a thermometer measure if it is shielded from the sunshine? What does it measure if it is not?

If shielded, it measures the air temperature. If not, it measures the combined effect of air temperature and net radiative heat gain from the Sun.

Maxwell's stress tensor is
if 6.0×10^13 electrons are placed on a metal sphere of charge 9.0micro Coulombs, what is the net charge on the sphere
18.51micro Coulombs
ASHOK
Is it possible to find the magnetic field of a circular loop at the centre by using ampere's law?
Is it possible to find the magnetic field of a circular loop at it's centre?
yes
Brother
The density of a gas of relative molecular mass 28 at a certain temperature is 0.90 K kgmcube.The root mean square speed of the gas molecules at that temperature is 602ms.Assuming that the rate of diffusion of a gas in inversely proportional to the square root of its density,calculate the density of
A hot liquid at 80degree Celsius is added to 600g of the same liquid originally at 10 degree Celsius. when the mixture reaches 30 degree Celsius, what will be the total mass of the liquid?
what is electrostatics
Study of charges which are at rest
himanshu
Explain Kinematics
Two equal positive charges are repelling each other. The force on the charge on the left is 3.0 Newtons. Using your notes on Coulomb's law, and the forces acting on each of the charges, what is the force on the charge on the right?
Using the same two positive charges, the left positive charge is increased so that its charge is 4 times LARGER than the charge on the right. Using your notes on Coulomb's law and changes to the charge, once the charge is increased, what is the new force of repulsion between the two positive charges?
Nya
A mass 'm' is attached to a spring oscillates every 5 second. If the mass is increased by a 5 kg, the period increases by 3 second. Find its initial mass 'm'
a hot water tank containing 50,000g of water is heated by an electric immersion heater rated at 3kilowatt,240volt, calculate the current
what is charge
product of current and time
Jaffar
Why always amber gain electrons and fur loose electrons? Why the opposite doesn't happen?
A closely wound search coil has an area of 4cm^2,1000 turns and a resistance of 40ohm. It is connected to a ballistic galvanometer whose resistance is 24 ohm. When coil is rotated from a position parallel to uniform magnetic field to one perpendicular to field,the galvanometer indicates a charge
Using Kirchhoff's rules, when choosing your loops, can you choose a loop that doesn't have a voltage?
how was the check your understand 12.7 solved?

format: plain


text:
# I^i means WHAT?

1. Aug 10, 2010

### trevdna

i^i means WHAT?!?!?!?!?!

My friend gave me the brain-teaser "i^i = what?", and with a little bit of coaching I finally discovered that

i^i = e^(-pi/2)

Which is cool, I suppose. But the more I think about it, the more I wonder:

what the heck does it mean???

For that matter, what does anything raised to an imaginary power mean?
For instance:
e^(ix) = cos(x)+i*sin(x)

It's so counter-intuitive, is there some way to make sense of it by analogy with real numbers? Or will it just remain abstract until the end of time?

2. Aug 10, 2010

### g_edgar

Re: i^i means WHAT?!?!?!?!?!

For e^(ix) one approach is to note that for real constants a, y=e^(ax) is the solution of y'(x) = ay(x), y(0)=1. So for complex number i, to evaluate e^(ix) we want complex-valued function y such that y'(x) = iy(x), y(0)=1. The solution is y(x) = cos(x) + i sin(x).

3. Aug 10, 2010

### mathman

Re: i^i means WHAT?!?!?!?!?!

e^(ix) = cos(x)+i*sin(x) may not be intuitive, but you can prove it by expanding both sides into power series and see they are the same.

4. Aug 12, 2010

### chaoseverlasting

Re: i^i means WHAT?!?!?!?!?!

Why are real numbers any more 'real' that complex numbers? They are just convenient representations of things that we observe around us.

The function $$e^x$$ gives us an easy/convenient way to represent other quantities. I assume you're taking some form of coaching for competetive examinations from your post. At this level, there arent a lot of examples that I can think of. It's a very useful function when dealing with geometrical rotation of a quantity or geometrical figures in general (eg. a circle of unit radius or other regular polygons).

You can also define any possible mathematical function in terms of $$e^x$$ with something called Fourier Analysis, which is one of the corner stones of signal analysis and digital electronics.

In Electrical engineering, I doubt you can go through a single topic without encountering the exponential function in some form. Its also lends itself to very very convenient representations of AC waveforms and provides a graphical analogy to the same (Phaser Diagrams).

There are countless examples of the use of the exponential function in 'real' life. Hope that helps.

5. Aug 16, 2010

### Jamma

Re: i^i means WHAT?!?!?!?!?!

It doesn't mean that, $$i^{i}$$ means an infinite number of different values.

Think about square roots, or things raised to the power of 1/2. Then this means a set of two different numbers (except for 0). For irrational numbers, you can get an infinite number of (complex) values.

For complex numbers, we can define a general power $$z^{v}$$ as:

$$z^{v}=e^{v.log(z)}$$

and log(z) can have an infinite number of values. This definition of exponentiation is basically just a rule which is consistent with all the older laws for use with real numbers.

So in your case:

$$i^{i}=e^{i.log(i)}=e^{i((-1/2+2n).\pi .i)}$$

for n any natural number. So $$e^{-\pi /2}$$ is an answer, but there are many other values also.

6. Sep 7, 2010

### sEsposito

Re: i^i means WHAT?!?!?!?!?!

I like to think of imaginary numbers to be a convenient way to "extend the number-line" so to speak... If you look back in history to when we first conceived the "real" numbers, to the time when we formalized them into the set $$R$$ and then to the point where imaginary numbers came into play -- none of it is intuitive. It's only intuitive to us because we're comfortable with it.

There's a nice documentary on this subject, I think it was put out by BBC... It's on YouTube somewhere. You'll find it if you search "Mathematics Documentary".

7. Sep 10, 2010

### Tac-Tics

Re: i^i means WHAT?!?!?!?!?!

What does it mean for a number to be raised to a fractional power? if 2^2 = 2 x 2, what the heck does 2^(1/2) mean?

Generalizations don't always make sense. The power of generalizing in math is that you throw away some of your assumptions (exponentiation has a literal interpretation in terms of multiplication) while maintaining handy invariants (x^(a+b) = x^a * x^b) in order to describe a larger (and often more abstract) class of mathematical objects (real numbers, as opposed to the integers).

Exponentiation is interesting, too, in that it loses one interpretation (iterated multiplication) and gains a new one (converting rectangular form to polar form).


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# HUP and Photoelectric effect

1. Dec 9, 2006

### lightarrow

A sinusoidal EM wave which lenght l is very great, hits a photoelectric device.

According to HUP, a sinusoidal EM wave with finite lenght l has a spectrum of frequencies which broadness is inversely proportional to l. So, if l is very great, the energy uncertainty is very low, and this, still for HUP, should mean that the time required to make that energy measurement is high. Or not?

If this were true, why, instead,the photoelectrons are emitted almost instantly?

2. Dec 9, 2006

### DrChinese

The HUP does *not* say that there is a connection between a photon's frequency "spectrum" and its wavelength. The HUP says there is a relationship between the frequency (actually energy) spectrum and the time spectrum.

$$\Delta E \Delta t & \ge \frac{\hbar}{2}$$

If you work through it, you will see that the wavelength itself is not a factor, even though wavelength and frequency themselves are inversely proportional. I.e. it is the deltas that are related by the HUP.

3. Dec 9, 2006

### lightarrow

In my post I talked about the lenght l of the wave train, not the wavelenght. Probably I didn't explain well.

4. Dec 12, 2006

### RandallB

And since as you put it the electron is produce "almost instantly" you must have some uncertainty about when it is, and I assume you have some uncertainty about where it or its energy is. Sounds consistent with the uncertainty defined with the originating photon, enough so as to not be able to question “why”.

5. Dec 12, 2006

### CarlB

Yes, the length of the EM wave train is very long so delta E is very small. This means that the time at which the photon is absorbed (and the electron is emitted) is very imprecise. The HUP implication is not that the electron is spread out in time, but instead that the arrival time of the electron cannot be predicted in advance. And the energy of the electron is not ill defined, just the time that it arrives relative to the duration of the EM wave. If you run the experiment many times you will get many different electron arrival times. Let me explain this in more detail.

For example, suppose the wave is 500 feet long, and consists of a single photon. The wave takes about 500ns to pass a point in space, so you cannot expect the electron to appear except within a period of time of 500ns. If you run the experiment, you will find that the electron is emitted at various times, depending on how lucky you are.

Now any one of those electrons will be measured to arrive at a very specific time. For exmple, one run might detect the electron at 139ns +- 1ns after the beginning of the wave train. But the +- 1ns is not the delta t of the HUP. The +- 1ns is the inaccuracy in your measurement apparatus. The delta t of the HUP is the fact that the photon is annihilated and the energetic electron appears at any time in that 500 ns.

Is that the explanation you were looking for?

6. Dec 12, 2006

### lightarrow

Yes, but not with a 500 feet long photon, with a 500 light seconds long one.

7. Dec 12, 2006

### CarlB

500 feet is already very very long compared to the wavelength of light, and so the delta E of light is very accurately determined. (I.e. something like 1 part in 10^8).

If you make the light wavetrain instead be 500 seconds long, it just makes delta E that much smaller, and correspondingly increases the length of time over which the electron may arrive (i.e. it increases the delta t).

8. Dec 12, 2006

### reilly

The standard way to do photoelectric assumes pure plane waves for the incident photon and the emittted electron. When you restrict the photon to a finite spatial range you get a term like sinkr1/kr1 - sinkr2/kr2 rather than the usual momentum delta function. r1-r2 is the length of the incident wave packet.(Don't worry, momentum is still conserved.Recall that the incident photon is in a state that encompasses all frequencies from 0 to infinity.So the term above needs to be integrated over the photon spectra) So the finite wavetrain will kick out an electron, with a probability dependent upon the momentum and the spatial characteristics of the incident wave train.

I'll leave it at that.

Re the HUP, first it is built into the computation of the matrix element. And, in fact, with a bit of effort, you can trace a Gaussian (or any other packet) to see how the uncertainties propagate through the system.

Regards,
Reilly Atkinson

9. Dec 13, 2006

### lightarrow

So $$\Delta t$$ should be infinite in that case?

10. Dec 13, 2006

### CarlB

Yes, delta t is infinite.

11. Dec 14, 2006


text:
## Multistable randomly switching oscillators: The odds of meeting a ghost

We consider oscillators whose parameters randomly switch between two values at equal time intervals. If random switching is fast compared to the oscillator's intrinsic time scale, one expects the switching system to follow the averaged system, obtained by replacing the random variables with their mean. The averaged system is multistable and one of its attractors is not shared by the switching system and acts as a ghost attractor for the switching system. Starting from the attraction basin of the averaged system's ghost attractor, the trajectory of the switching system can converge near the ghost attractor with high probability or may escape to another attractor with low probability. Applying our recent general results on convergent properties of randomly switching dynamical systems [1,2], we derive explicit bounds that connect these probabilities, the switching frequency, and the chosen initial conditions.

Published in:
European Physical Journal-Special Topics, 222, 10, 2497-2507
Year:
2013
Publisher:
Heidelberg, Springer Verlag
ISSN:
1951-6355
Laboratories:


text:
# Dynamic Programming Help

Sometimes I can see the recurrence relationship and other times I cannot. This is one of those times. I've gone through a few iterations of trying to solve it but I also find a set of values where my idea fails. If I go exponenetial I can figure it out... but that's a specific no-no.

I'm good with nudges in the right direction. Please make it a little strong though.

Problem

• Welcome to CS.SE! What did you try? We're happy to help you understand the concepts but just solving exercises for you is unlikely to achieve that. You might want to review the tag guidance: cs.stackexchange.com/tags/dynamic-programming/info, and try applying the techniques there, then edit your question to show what progress you've made and which you've tried and what you're stuck on. Also, take a look at meta.cs.stackexchange.com/q/1284/755. – D.W. Nov 3 '16 at 4:46

When you encounter these types of problems, the typical thing to do is to find some "decision scheme" that can decompose your big problem into a series of smaller problems. The gist of this problem is finding a set $(h_1, \dots, h_n)$ that maximizes your mean grade $\frac{\sum_{i \le n} f_i(h_i)}{n}$ subjected to the constraint that $\sum\limits_{i \le n} h_i \le H$. Notice that maximizing $\frac{\sum_{i \le n} f_i(h_i)}{n}$ is equivalent to just maximizing $\sum\limits_{i \le n} f_i(h_i)$ since the denominator $n$ is independent of the choices of $(h_1, \dots, h_n)$ that we make.

Now, there are various dynamic programming schemes that we can choose to solve this problem, each of which depends on some special interpretation of an "optimal functional" that we define. For me, the intuitive one is based on the idea of variable elimination, in which we eliminate one of the variables $h_i$ at each turn.

Here, let $\textrm{Best}(k, h)$ denote the maximum cumulative grade you can gain for the classes $C_k, C_{k+1}, \dots, C_n$ so that you spend no more than $h$ hours on all of their final projects. Given this interpretation, it is clear that our problem is to find $\textrm{Best}(n, H)$.

Now comes the fun/interesting part. First, we need to convince ourselves that this functional displays what we would call an "optimal substructure." In layman's words, this claims that if we know the best distribution of no more than 5 hours to Networking and Programming Languages' final projects, then we also know the best distribution of 8 hours to the Networking, PL, and OS' final projects if we decide to allocate 3 hours to the OS project. It's simple, we just use the same optimal budget of 5 hours for Networking and Programming languages that we already have.

In other words, this optimal substructure property guides us on the decision we have to make at each turn (because we don't have to worry about the subproblems). Here, $\textrm{Best}(k, h)$, according to the substructural property we've introduced above, must make a decision on what $h_k$ should be assigned. In addition, it must

1. Ensure that $h_k$ does not exceed its allocated hours of $h$, and
2. Maximize the grades contingent on the choices that it can make.

Therefore, when we consider the problem of computing $\textrm{Best}(k, h)$, there's a range of valid decisions that we can make, each corresponds with an assignment of a single value to $h_k$. For example, say I'm considering $\textrm{Best}(\textbf{Networking}, 5)$, which corresponds to the maximum cumulative grade that I can get in classes Networking, PL, and OS with only 5 hours to distribute between them. Now, I have to make a decision about how I will budge for Networking. Say I budget exactly 0 hours to networking, then I can spend 5 hours on PL and OS; alternative, I can budget 1 hour to networking and spend 4 hours between PL and OS, 2 hours to networking and 3 between PL and OS, etc. Each choice of time budget for Networking also determines a subproblem that I have to solve between PL and OS with the remaining time. Once we consider all of these choices (and subproblems), we must then commit to the problem that will give us the best solution.

This thought process should yield a recurrence for Best that can be computed in some increasing order of its parameter. This computation then takes the form of a tabulation over an $n \times H$ grid, each time spending approximately $O(k)$ (where $k \le n$ is the column number) time between the $O(k)$ choices, which gives an $O(n^2 H)$ algorithm.

• Thank you very much. I gave myself barely enough time to finish (midnight after my post). What I ended up doing was precomputing all grades from 1->H hrs for every class, then changing the table such that every cell noted the class and grade/hrs. Each column was then sorted descending. From z = H to H/2 determined all factors of H-z. This translated into hours spent in n classes per column. For example H=10 if z = 8 there is 8x1, 2x1, and 8x1, 1x2. n is the second number. From high to low summed the top n entries of the columns skipping any class that already was accounted for in higher col. – Nolan Robidoux Nov 8 '16 at 7:00
• With each set of factors determined if the sum was higher than prev. If so became new budget for class projects. I know the running time is #classes*#hours... not sure exactly what the number of factors are for a given number but I believe I met the runtime requirement. – Nolan Robidoux Nov 8 '16 at 7:03


text: The Classical Limit of Dynamics for Spaces Quantized by an Action of ${\Bbb R}^{\lowercase{d}}$ We have previously shown how to construct a deformation quantization of any locally compact space on which a vector group acts. Within this framework we show here that, for a natural class of Hamiltonians, the quantum evolutions will have the classical evolution as their classical limit. Categories:46L60, 46l55, 81S30