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Figure is the schematics of a fluid flowing in an “S”-shaped pipeline with the cross-section area reducing from A1 (left bottom part) to A2 (right top part). The left bottom part is at the height y1 above the ground; right top part is at the height h2 above the ground. Fluid moves with the velocity v1 at the bottom part and v2 at the top part. Volume of fluid dv takes dx1 in the part of pipeline and dx2 in the top part of pipeline.
The geometry used for the derivation of Bernoulli’s equation.

We also assume that there are no viscous forces in the fluid, so the energy of any part of the fluid will be conserved. To derive Bernoulli’s equation, we first calculate the work that was done on the fluid:

d W = F 1 d x 1 F 2 d x 2
d W = p 1 A 1 d x 1 p 2 A 2 d x 2 = p 1 d V p 2 d V = ( p 1 p 2 ) d V .

The work done was due to the conservative force of gravity and the change in the kinetic energy of the fluid. The change in the kinetic energy of the fluid is equal to

d K = 1 2 m 2 v 2 2 1 2 m 1 v 1 2 = 1 2 ρ d V ( v 2 2 v 1 2 ) .

The change in potential energy is

d U = m g y 2 m g y 1 = ρ d V g ( y 2 y 1 ) .

The energy equation then becomes

d W = d K + d U ( p 1 p 2 ) d V = 1 2 ρ d V ( v 2 2 v 1 2 ) + ρ d V g ( y 2 y 1 ) ( p 1 p 2 ) = 1 2 ρ ( v 2 2 v 1 2 ) + ρ g ( y 2 y 1 ) .

Rearranging the equation gives Bernoulli’s equation:

p 1 + 1 2 ρ v 1 2 + ρ g y 1 = p 2 + 1 2 ρ v 2 2 + ρ g y 2 .

This relation states that the mechanical energy of any part of the fluid changes as a result of the work done by the fluid external to that part, due to varying pressure along the way. Since the two points were chosen arbitrarily, we can write Bernoulli’s equation more generally as a conservation principle along the flow.

Bernoulli’s equation

For an incompressible, frictionless fluid, the combination of pressure and the sum of kinetic and potential energy densities is constant not only over time, but also along a streamline:

p + 1 2 ρ v 2 + ρ g y = constant

A special note must be made here of the fact that in a dynamic situation, the pressures at the same height in different parts of the fluid may be different if they have different speeds of flow.

Analyzing bernoulli’s equation

According to Bernoulli’s equation, if we follow a small volume of fluid along its path, various quantities in the sum may change, but the total remains constant. Bernoulli’s equation is, in fact, just a convenient statement of conservation of energy for an incompressible fluid in the absence of friction.

The general form of Bernoulli’s equation has three terms in it, and it is broadly applicable. To understand it better, let us consider some specific situations that simplify and illustrate its use and meaning.

Bernoulli’s equation for static fluids

First consider the very simple situation where the fluid is static—that is, v 1 = v 2 = 0 . Bernoulli’s equation in that case is

p 1 + ρ g h 1 = p 2 + ρ g h 2 .

We can further simplify the equation by setting h 2 = 0 . (Any height can be chosen for a reference height of zero, as is often done for other situations involving gravitational force, making all other heights relative.) In this case, we get

p 2 = p 1 + ρ g h 1 .

This equation tells us that, in static fluids, pressure increases with depth. As we go from point 1 to point 2 in the fluid, the depth increases by h 1 , and consequently, p 2 is greater than p 1 by an amount ρ g h 1 . In the very simplest case, p 1 is zero at the top of the fluid, and we get the familiar relationship p = ρ g h . ( Recall that p = ρ g h and Δ U g = m g h . ) Thus, Bernoulli’s equation confirms the fact that the pressure change due to the weight of a fluid is ρ g h . Although we introduce Bernoulli’s equation for fluid motion, it includes much of what we studied for static fluids earlier.

Questions & Answers

explain equilibrium of a body
balance
Sampson
hi
Rajeev
hello
Sampson
pls who can tell me more about Kirchoff's law?
muhammed
Hi all, love you all!!!
Giorgi Reply
Cool
Debabrata
How does resonance occur
Rahim Reply
plzz explanation of the equilibrium of a body
what is quantam
pamit Reply
quantum is a division of mechanics
Baje
what is friction
Muhammad Reply
a force act by surface between two bodies whose are always oppose the relative motion .....
Raghav
when two rough bodies are placed in contact and try to slip each other ... than a force act them and it's ippse the relative motion between them
Raghav
thats friction force and roughnes of both bodies is define friction of surface
Raghav
what is a progressive wave
sheriff-deen Reply
What is the wake for therapist
Ife Reply
can u like explain your question with clear detail
Chikamso
who would teach me vectors?
Tintin Reply
are you guys are physics student?
EL
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Rambo
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Rambo
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Katia
calculations get me confused sometimes
Beluchukwu
what's chemistry
Esther Reply
branch of science dt deals with the study of physical properties of matter and it's particulate nature
Josiah
Good
Daniel
actually
Nathz
Y acctually do u hav ur way of defining it? just bring ur iwn idear
Daniel
well, it deals with the weight of substances and reaction behind them as well as the behavior
Josiah
buh hope Esther, we've answered ur question
Josiah
what's ohms law
CHIJIOKE
ohms law states that, the current flowing through an electric circuit is directly proportional to the potential difference, provided temperature and pressure are kept constant
Josiah
what is sound
James
ohms law states that the resistance of a material is directly proportional to the potential difference between two points on that material, if temperature and other physical conditions become constant
Chikamso
How do I access the MCQ
Abraham Reply
As I think the best is, first select the easiest questions for you .and then you can answer the remaining questions.
lasitha
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Abraham
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Abraham
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lasitha
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Abraham
what is centripetal force
Don Reply
هي قوة ناتجة من الحركة الدائرية ويكون اتجاهها إلى المركز دائماً
meaning of vector quantity
Felix Reply
vector quantity is any quantity that has both magnitude in terms of number (units) and direction in terms of viewing the quantity from an origin using angles (degree) or (NEWS) method
LEWIS
vector quantity is physical quantity has magnitude and direction
vector is a quantity that is use in measuring size of physical properties and their direction
Bitrus
what difference and similarities between work,force,energy and power?
Anes Reply
I need the best answer
Anes
power
mehreen
power
saba
enery is the ability to do work. work is job done, force is a pull or push. power has to do with potential. they belong to different categories which include heat energy, electricity.
Andrew
force refers to a push or pull... energy refers to work done while power is work done per unit time
Shane
mathematically express angular velocity and angular acceleration
Mario Reply
it depends on the direction. an angular velocity will be linear and angular acceleration will be an angle of elevation.
Andrew
The sonic range finder discussed in the preceding question often needs to be calibrated. During the calibration, the software asks for the room temperature. Why do you suppose the room temperature is required?
Shaina Reply
Suppose a bat uses sound echoes to locate its insect prey, 3.00 m away. (See [link] .) (a) Calculate the echo times for temperatures of 5.00°C5.00°C and 35.0°C.35.0°C. (b) What percent uncertainty does this cause for the bat in locating the insect? (c) Discuss the significance of this uncertainty an
Shaina
give a reason why musicians commonly bring their wind instruments to room temperature before playing them.
Shaina
The ear canal resonates like a tube closed at one end. (See [link]Figure 17_03_HumEar[/link].) If ear canals range in length from 1.80 to 2.60 cm in an average population, what is the range of fundamental resonant frequencies? Take air temperature to be 37.0°C,37.0°C, which is the same as body tempe
Shaina
By what fraction will the frequencies produced by a wind instrument change when air temperature goes from 10.0°C10.0°C to 30.0°C30.0°C ? That is, find the ratio of the frequencies at those temperatures.
Shaina
Practice Key Terms 2

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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