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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

a particle projected from origin moving on x-y plane passes through P & Q having consituents (9,7) , (18,4) respectively.find eq. of trajectry.
rahul Reply
definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
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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