Bernoulli’s theorem
i
Bernoulli’s theorem, in fluid dynamics, relation among the pressure, velocity, and elevation in a moving fluid (liquid or gas), the compressibility and viscosity (internal friction) of which are negligible and the flow of which is steady, or laminar. First derived (1738) by the Swiss mathematicianDaniel Bernoulli, the theorem states, in effect, that the total mechanical energy of the flowing fluid, comprising the energy associated with fluid pressure, the gravitational potential energy of elevation, and the kinetic energy of fluid motion, remains constant. Bernoulli’s theorem is the principle of energy conservation for ideal fluids in steady, or streamline, flow and is the basis for many engineering applications. Bernoulli’s theorem implies, therefore, that if the fluid flows horizontally so that no change in gravitational potential energy occurs, then a decrease in fluid pressure is associated with an increase in fluid velocity. If the fluid is flowing through a horizontal pipe of varying cross-sectional area, for example, the fluid speeds up in constricted areas so that the pressure the fluid exerts is least where the cross section is smallest. This phenomenon is sometimes called the Venturi effect, after the Italian scientist G.B. Venturi (1746–1822), who first noted the effects of constricted channels on fluid flow.

Application of Bernoulli's Theorem

When we blow air over a strip of paper as shown in the above figure, we find that the paper moves up. This is because, on blowing air, the velocity of air increases, creating low pressure above the paper and high pressure below the paper. This difference in pressure, lifts the paper

The working of spray-gun is based on Bernoulli's theorem

When the rubber bulb is squeezed, air is blown into the tube A, due to which, low pressure and high velocity is created. Since this pressure is less than the atmospheric pressure, the liquid is pushed up. This rising liquid is sprayed out of...

...bernoulli'stheorem
ABSTRACT / SUMMARY
The main purpose of this experiment is to investigate the validity of the Bernoulli equation when applied to the steady flow of water in a tape red duct and to measure the flow rate and both static and total pressure heads in a rigid convergent/divergent tube of known geometry for a range of steady flow rates. The apparatus used is Bernoulli’sTheorem Demonstration Apparatus, F1-15. In this experiment, the pressure difference taken is from h1- h5. The time to collect 3 L water in the tank was determined. Lastly the flow rate, velocity, dynamic head, and total head were calculated using the readings we got from the experiment and from the data given for both convergent and divergent flow. Based on the results taken, it has been analysed that the velocity of convergent flow is increasing, whereas the velocity of divergent flow is the opposite, whereby the velocity decreased, since the water flow from a narrow areato a wider area. Therefore, Bernoulli’s principle is valid for a steady flow in rigid convergent and divergent tube of known geometry for a range of steady flow rates, and the flow rates, static heads and total heads pressure are as well calculated. The experiment was completed and successfully conducted.
INTRODUCTION
In fluid dynamics, Bernoulli’s principle is best explained in the application that involves in...

...Experiment No. 1: Bernoulli’sTheorem
Object:
To verify Bernoulli'stheorem for a viscous and incompressible fluid.
Theory:
In our daily lives we consume a lot of fluid for various reasons. This fluid is
delivered through a network of pipes and fittings of different sizes from an overhead
tank. The estimation of losses in these networks can be done with the help of this
equation which is essentially principle of conservation of mechanical energy.
Formal Statement:
Bernoulli's Principle is essentially a work energy conservation principle which states that
for an ideal fluid or for situations where effects of viscosity are neglected, with no work
being performed on the fluid, total energy remains constant. Bernoulli's Principle is
named in honour of Daniel Bernoulli. 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.
Mathematical Description:
A+dA
ρ+dρ,V+dV
P+dP
τ =0
A
ds
dz
θ
ρ,V
CV
P
dW
Figure 1 Forces and Fluxes for Bernoulli’s Equation for frictionless flow along a streamline.
Applying the conservation of mass and momentum equation yields the following
equation
BournoulliEquation
1
∂V
dP
ds +
+ VdV + gdz = 0
(1.1)
∂t
ρ
Equation (1.1) is the...

...Report Bernoulli’sTheorem
Lubna Khan, BEng Architectural Engineering
Student ID No.: H00113999
Addressed to: Dr. Mehdi Nazirinia
Date: 22/12/2012
Lab Experiment held on: 28/11/2012
Table of Contents
Summary/Abstract Page 3
1.1. Introduction Page 4
1.2. Objective Page 5
2. Theory Page 5
2.1. Theoretical Background Page 5
2.1.1. Sample Calculations: Page 8
2.1.1.1. First experiment Page 8
3. Equipment Page 10
3.1. Preparing the Apparatus Page 10
3.2. Technical Data Page 11
4. Test Procedure Page 12
4.1. Bernoulli’sTheorem Test Page 12
5. Results and Discussions Page 12
5.1. Data Sheet Page 13
5.1.1. First Experiment Page 13
5.1.2. Second Experiment Page 13
5.1.3. Third Experiment Page 14
5.2.1. Velocity-Head Graph Page 14
5.2.2. Diff. between Mea. and The. Pressure-Head Graph Page 16
5.2.3. Total Head Loss Graph Page 17
5.2.4. Total Pressure Head Graph Page 18
Conclusion Page 19
References Page 21
Summary/Abstract:
The main objective of this experiment is to verify the Bernoulli Equation using a Venturi meter which is a converging-diverging tube of known geometry which is attached to a pipeline to help measure different quantities...

...HYDROLOGY & HYDRAULIC ENGINEERING I LABORATORY
REPORT 3
TITLE : BERNOULLI’STHEOREM APPARATUS
NAME :
ID. NO. :
SECTION : 02
EXPERIMENT DATE : 10th December 2009
SUBMISSION DATE : 17th December 2009
GROUP NO. : 2
GROUP MEMBERS :
LECTURER :
LAB INSTRUCTOR :
TABLE OF CONTENT
Content | Page |
Summary | 2 |
Objective | 2 |
Theory | 3 - 5 |
Equipment/ description of experimental apparatus | 6 |
Procedure | 6 |
Data and observation | 6a |
Analysis | 7,8 |
Discussion | 9 |
Conclusion | 10 |
References, Appendix | 10 |
Summary
The title of this experiment is the Bernoulli’stheorem apparatus. The objective of this experiment is to determine the relationship between pressure head, velocity head, and static head and compare it with Bernoulli’sTheorem. The total head of flowing liquid between two points remain s constant provided there is no loss due to friction no gain due to application of outside work between the two points. So basically the apparatus used for this experiment consist of a venture tube with different diameter at certain part, 8 tubes of water manometer, a rotameter, stainless steel tube and a stop watch to record the time of the water flow.
The procedures are simple, after connecting the hydraulics bench outlet to the test set, the pump is started. Then adjust the flow...

...Bernoulli's Principle states that for an ideal fluid (low speed air 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.
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. It is named after the Dutch/Swiss mathematician/scientist Daniel Bernoulli, though it was previously understood by Leonhard Euler and others. In fluid flow with no viscosity, and therefore, one in which a pressure difference is the only accelerating force, the principle is equivalent to Newton's laws of motion.
Incompressible flow
The original form, for incompressible flow in a uniform gravitational field, is:
[pic]
where:
v = fluid velocity along the streamline
g = acceleration due to gravity
h = height of the fluid
p = pressure along the streamline
ρ = density of the fluid
These assumptions must be met for the equation to apply:
• Inviscid flow − viscosity (internal friction) = 0
• Steady flow
• Incompressible flow − ρ = constant along a streamline. Density may vary from streamline to streamline, however.
• Generally, the equation applies along a streamline. For constant-density potential flow, it applies...

...CHAPTER 1
In modern world of technological advancement, there are a lot of applications that are used every day. For example, an airplane relies on Bernoulli’s Principle to generate lift on its wings. Rare cars employ the velocity and pressure dynamics specified by Bernoulli’s Principle to keep their rare wheels on the ground, even while zooming off at high speed. It is successfully employed in mechanism like the carburetor and the atomizer.
The study focuses on Bernoulli’sTheorem in Fluid Application. A fluid is any substance which when acted upon by a shear force, however small, cause a continuous or unlimited deformation, but at a rate proportional to the applied force. As a matter of fact, if a fluid is moving horizontally along a streamline, the increase in speed can be explained due the fluid that moves from a region of high pressure to a lower pressure region and so with the inverse condition with the decrease in speed.
Bernoulli’s Principle complies with the principle of conservation of energy. In a steady Flow, at all points of the streamline of a flowing fluid is the sum of all forms of mechanical energy along a streamline. It was first derived by the Swiss Mathematician Daniel Bernoulli; the theorem states that when a fluid flows from one place to another without friction, its total energy (kinetic+ potential+ pressure) remains constant.
Many...

...CHAPTER 1
INTRODUCTION:
Bernoulli's Principle is a physical phenomenon that was named after the Swiss scientist Daniel Bernoulli who lived during the eighteenth century. Bernoulli studied the relationship of the speed of a fluid and pressure.
The Swiss mathematician and physicist Daniel Bernoulli (1700-1782) discovered the principle that bears his name while conducting experiments concerning an even more fundamental concept: the conservation of energy. This is a law of physics that holds that a system isolated from all outside factors maintains the same total amount of energy, though energy transformations from one form to another take place.
The principle states that "the pressure of a fluid [liquid or gas] decreases as the speed of the fluid increases." Within the same fluid (air in the example of aircraft moving through air), high-speed flow is associated with low pressure, and low-speed flow is associated with high pressure.
OBJECTIVE:
*USE BERNOULLI’S PRINCIPLE TO EXPLAIN HOW THE ENERGY OF A FLUID AND ITS PRESSSURE ARE RELATED.
*EXPLAIN SOME SITUATIONS USING BERNOULLI’S PRINCIPLE.
SCOPE:
*the first fly of airplane
In 1899, after Wilbur Wright had written a letter of request to the Smithsonian Institution for information about flight experiments, the Wright Brothers designed their first aircraft: a small, biplane glider flown as a kite to test their solution for controlling the craft by wing warping. Wing warping...

...EXPERIMENT : - 2
EXPERIMENT Verification of Bernoulli’s Energy Equation
THEORY
For steady incompressible flow Bernoulli’s energy equation along a streamline is written as
[pic] constant
where
[pic] = pressure, [pic] = velocity and [pic] = height from datum
Purpose of this experiment is to verify this expression. In the special apparatus the pipe is tapered with the cross section decreasing in the direction of flow first and then increasing in the second half of its length.
Hydraulic Grade Line
The line which shows the sum of pressure head (p/() and the potential head z (i.e., p/(+z) is called the Hydraulic Grade Line.
Energy Grade Line
The line obtained by plotting the sum of pressure, elevation and velocity heads (i.e., p/(+z+V2/2g) along the pipe is called the Energy Grade Line.
[pic]
Figure BC4.1 Bernoulli’s Apparatus
OBJECTIVE
Verification of Bernoulli’s Equation.
APPARATUS
Bernoulli’s apparatus (refer to Figure BC4.1), stop watch, scale, measuring tank.
PROCEDURE
1. Measure cross section of pipe at inlet point, throat (mid section and outlet point). From this determine the cross section areas a to a at each piezometer tapping.
2. Start the water supply .The head under which the flow occurs can be regulated by a valve at B. Adjust the flow so that velocity is small....

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