FLUID_DYNAMICS

0

FLUID DYNAMICS

AND

IT’S APPLICATION –

BERNOULLI’S THEOREM

Submitted to:

Amity School of Engineering and Technology

Under the guidance of: Submitted by:

Mr. Rajan Raman Vishwambaram Siddharth Sharma 8001

Hoorain Manhas 8020

Kanchan Tanya 8022

AMITY UNIVERSITY,

UTTAR PRADESH

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Acknowledgement

We feel immense pleasure in submitting this Project on Fluid Dynamics and its

Application- Bernoulli’s Theorem. The valuable guidance of the teaching staff made this

study possible. They have been a constant source of encouragement throughout the

completion of this report.

We would sincerely like to thank Mr. Rajan Raman Vishwambaram for their help and

support during the making of this report. This report would not have been successful without

their guidance and the valuable time that they had spent with us during the report

development stages. We would like to express our immense gratitude to our parents, who

have always lent us a helping hand during tough times and stood by us throughout this project

and this life.

Siddharth Sharma

Hoorain Manhas

Kanchan Tanya

Certificate

2

This is to certify that Mr. Siddharth Sharma, Ms.Hoorain Manhas, and Ms. Kanchan Tanya, students

of B.Tech(Civil Engineering) have carried out the work presented in the project entitled“Fluid

Dynamics and its Application- Bernoulli’s Theorem” as a part of Second Year programme of

Bachelor of Technology in Civil from Amity School of Engineering and Technology, Amity

University, Noida, Uttar Pradesh under my supervision.

Mr. Rajan Raman Vishwambaram

Department of Mechanical Engineering

ASET, Noida

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INDEX

Page No.

1. Introduction 42. Fluid dynamics 53. Equations of Fluid dynamics 64. Terminology in Fluid dynamics 105. Bernoulli’s Theorem- An Important Application Of Fluid dynamics 136. Incompressible Flow Equation 147. Compressible Flow Equation 188. Derivation of Bernoulli’s Theorem 209. Real World Application 2510. Misunderstandings in Generation of Lift 2611. Conclusion 2712. Bibliography 28

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INTRODUCTION

A fluid is a substance that continually deforms (flows) under an applied shear stress, no matter how

small. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some

extent, plastic solids.

In common usage, "fluid" is often used as a synonym for "liquid", with no implication that gas could

also be present. For example, "brake fluid" is hydraulic oil and will not perform its required function

if there is gas in it. This colloquial usage of the term is also common in medicine and in nutrition

("take plenty of fluids").

Liquids form a free surface (that is, a surface not created by the container) while gases do not. The

distinction between solids and fluid is not entirely obvious. The distinction is made by evaluating

the viscosity of the substance. Silly Putty can be considered to behave like a solid or a fluid,

depending on the time period over which it is observed. It is best described as a viscoelastic fluid.

There are many examples of substances proving difficult to classify. A particularly interesting one is

pitch, as demonstrated in the pitch drop experiment

Fluids display such properties as:

not resisting deformation, or resisting it only lightly (viscosity), and

the ability to flow (also described as the ability to take on the shape of the container).This also

means that all fluids have the property of fluidity.

These properties are typically a function of their inability to support a shear stress in

static equilibrium.

Solids can be subjected to shear stresses, and to normal stresses—both compressive and tensile. In

contrast, ideal fluids can only be subjected to normal, compressive stress which is called pressure.

Real fluids display viscosity and so are capable of being subjected to low levels of shear stress.

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FLUID DYNAMICS

Fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural

science of fluids (liquids and gases) in motion. It has several subdisciplines itself,

including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study

of liquids in motion). Fluid dynamics has a wide range of applications, including

calculating forces and moments on aircraft, determining the mass flow rate of petroleum through

pipelines, predicting weather patterns, understanding nebulae in interstellar space and reportedly

modeling fission weapon detonation. Some of its principles are even used in traffic engineering,

where traffic is treated as a continuous fluid.

Fluid dynamics offers a systematic structure that underlies these practical disciplines, that embraces

empirical and semi-empirical laws derived from flow measurement and used to solve practical

problems. The solution to a fluid dynamics problem typically involves calculating various properties

of the fluid, such as velocity, pressure, density, and temperature, as functions of space and time.

Historically, hydrodynamics meant something different than it does today. Before the twentieth

century, hydrodynamics was synonymous with fluid dynamics. This is still reflected in names of some

fluid dynamics topics, like magnetohydrodynamics and hydrodynamic stability—both also applicable

in, as well as being applied to, gases.

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EQUATIONS OF FLUID DYNAMICS

The foundational axioms of fluid dynamics are the conservation laws, specifically, conservation of

mass, conservation of linear momentum (also known as Newton's Second Law of Motion),

and conservation of energy (also known as First Law of Thermodynamics). These are based

onclassical mechanics and are modified in quantum mechanics and general relativity. They are

expressed using the Reynolds Transport Theorem.

In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are

composed of molecules that collide with one another and solid objects. However, the continuum

assumption considers fluids to be continuous, rather than discrete. Consequently, properties such as

density, pressure, temperature, and velocity are taken to be well-defined at infinitesimally small

points, and are assumed to vary continuously from one point to another. The fact that the fluid is

made up of discrete molecules is ignored.

For fluids which are sufficiently dense to be a continuum, do not contain ionized species, and have

velocities small in relation to the speed of light, the momentum equations for Newtonian fluids are

theNavier-Stokes equations, which is a non-linear set of differential equations that describes the

flow of a fluid whose stress depends linearly on velocity gradients and pressure. The unsimplified

equations do not have a general closed-form solution, so they are primarily of use in Computational

Fluid Dynamics. The equations can be simplified in a number of ways, all of which make them easier

to solve. Some of them allow appropriate fluid dynamics problems to be solved in closed form.

In addition to the mass, momentum, and energy conservation equations,

a thermodynamical equation of state giving the pressure as a function of other thermodynamic

variables for the fluid is required to completely specify the problem. An example of this would be

the perfect gas equation of state:

where p is pressure, ρ is density, Ru is the gas constant, M is the molar mass and T is temperature.

Compressible vs incompressible flow

Viscous vs inviscid flow

Steady vs unsteady flow

Laminar vs turbulent flow

Newtonian vs non-Newtonian fluids

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Compressible vs incompressible flowAll fluids are compressible to some extent, that is changes in pressure or temperature will result in

changes in density. However, in many situations the changes in pressure and temperature are

sufficiently small that the changes in density are negligible. In this case the flow can be modeled as

an incompressible flow. Otherwise the more general compressible flow equations must be used.

Mathematically, incompressibility is expressed by saying that the density ρ of a fluid parcel does not

change as it moves in the flow field, i.e.,

where D / Dt is the substantial derivative, which is the sum of local and convective derivatives.

This additional constraint simplifies the governing equations, especially in the case when the

fluid has a uniform density.

For flow of gases, to determine whether to use compressible or incompressible fluid dynamics,

the Mach number of the flow is to be evaluated. As a rough guide, compressible effects can be

ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible

assumption is valid depends on the fluid properties (specifically the critical pressure and

temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow

pressure becomes). Acoustic problems always require allowing compressibility, since sound

waves are compression waves involving changes in pressure and density of the medium through

which they propagate.

Viscous vs inviscid flowViscous problems are those in which fluid friction has significant effects on the fluid motion.

The Reynolds number, which is a ratio between inertial and viscous forces, can be used to evaluate

whether viscous or inviscid equations are appropriate to the problem.

Stokes flow is flow at very low Reynolds numbers, Re<<1, such that inertial forces can be neglected

compared to viscous forces.

On the contrary, high Reynolds numbers indicate that the inertial forces are more significant than

the viscous (friction) forces. Therefore, we may assume the flow to be an inviscid flow, an

approximation in which we neglect viscosity completely, compared to inertial terms.

This idea can work fairly well when the Reynolds number is high. However, certain problems such as

those involving solid boundaries, may require that the viscosity be included. Viscosity often cannot

be neglected near solid boundaries because the no-slip condition can generate a thin region of large

strain rate (known as Boundary layer) which enhances the effect of even a small amount of viscosity,

and thus generating vorticity. Therefore, to calculate net forces on bodies (such as wings) we should

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use viscous flow equations. As illustrated by d'Alembert's paradox, a body in an inviscid fluid will

experience no drag force. The standard equations of inviscid flow are the Euler equations. Another

often used model, especially in computational fluid dynamics, is to use the Euler equations away

from the body and the boundary layer equations, which incorporates viscosity, in a region close to

the body.

The Euler equations can be integrated along a streamline to get Bernoulli's equation. When the flow

is everywhere irrotational and inviscid, Bernoulli's equation can be used throughout the flow field.

Such flows are called potential flows.

Steady vs unsteady flowWhen all the time derivatives of a flow field vanish, the flow is considered to be a steady flow.

Steady-state flow refers to the condition where the fluid properties at a point in the system do not

change over time. Otherwise, flow is called unsteady. Whether a particular flow is steady or

unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a sphere is

steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference

that is stationary with respect to a background flow, the flow is unsteady.

Turbulent flows are unsteady by definition. A turbulent flow can, however, be statistically stationary.

According to Pope:[3]

The random field U(x,t) is statistically stationary if all statistics are invariant under a shift in time.

This roughly means that all statistical properties are constant in time. Often, the mean field is the

object of interest, and this is constant too in a statistically stationary flow.

Steady flows are often more tractable than otherwise similar unsteady flows. The governing

equations of a steady problem have one dimension less (time) than the governing equations of the

same problem without taking advantage of the steadiness of the flow field.

Laminar vs turbulent flowTurbulence is flow characterized by recirculation, eddies, and apparent randomness. Flow in which

turbulence is not exhibited is calledlaminar. It should be noted, however, that the presence of eddies

or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be

present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds

decomposition, in which the flow is broken down into the sum of an average component and a

perturbation component.

It is believed that turbulent flows can be described well through the use of the Navier–Stokes

equations. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it

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possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the

power of the computer used and the efficiency of the solution algorithm. The results of DNS agree

with the experimental data.

Most flows of interest have Reynolds numbers much too high for DNS to be a viable option[4], given

the state of computational power for the next few decades. Any flight vehicle large enough to carry

a human (L > 3 m), moving faster than 72 km/h (20 m/s) is well beyond the limit of DNS simulation

(Re = 4 million). Transport aircraft wings (such as on an Airbus A300 or Boeing 747) have Reynolds

numbers of 40 million (based on the wing chord). In order to solve these real-life flow problems,

turbulence models will be a necessity for the foreseeable future. Reynolds-averaged Navier–Stokes

equations (RANS) combined with turbulence modeling provides a model of the effects of the

turbulent flow. Such a modeling mainly provides the additional momentum transfer by the Reynolds

stresses, although the turbulence also enhances the heat and mass transfer. Another promising

methodology is large eddy simulation (LES), especially in the guise of detached eddy simulation (DES)

—which is a combination of RANS turbulence modeling and large eddy simulation.

Newtonian vs non-Newtonian fluidsSir Isaac Newton showed how stress and the rate of strain are very close to linearly related for many

familiar fluids, such as water and air. These Newtonian fluids are modeled by a coefficient

calledviscosity, which depends on the specific fluid.

However, some of the other materials, such as emulsions and slurries and some visco-elastic

materials (e.g. blood, some polymers), have more complicated non-Newtonian stress-strain

behaviours. These materials include sticky liquids such as latex, honey, and lubricants which are

studied in the sub-discipline of rheology.

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TERMINOLOGY IN FLUID DYNAMICS

The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure

can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or

not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other

methods.

Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar

areas of study. In particular, some of the terminology used in fluid dynamics is not used in fluid

statics.

Terminology in incompressible fluid dynamicsThe concepts of total pressure and dynamic pressure arise from Bernoulli's equation and are

significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense—

they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential

ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure to

distinguish it from total pressure and dynamic pressure. Static pressure is identical to pressure and

can be identified for every point in a fluid flow field.

In Aerodynamics, L.J. Clancy writes[5]: 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.

A point in a fluid flow where the flow has come to rest (i.e. speed is equal to zero adjacent to some

solid body immersed in the fluid flow) is of special significance. It is of such importance that it is

given a special name—a stagnation point. The static pressure at the stagnation point is of special

significance and is given its own name—stagnation pressure. In incompressible flows, the stagnation

pressure at a stagnation point is equal to the total pressure throughout the flow field.

Terminology in compressible fluid dynamicsIn a compressible fluid, such as air, the temperature and density are essential when determining the

state of the fluid. In addition to the concept of total pressure (also known as stagnation pressure),

the concepts of total (or stagnation) temperature and total (or stagnation) density are also essential

in any study of compressible fluid flows. To avoid potential ambiguity when referring to temperature

and density, many authors use the terms static temperature and static density. Static temperature is

identical to temperature; and static density is identical to density; and both can be identified for

every point in a fluid flow field.

The temperature and density at a stagnation point are called stagnation temperature and stagnation

density.

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A similar approach is also taken with the thermodynamic properties of compressible fluids. Many

authors use the terms total (or stagnation) enthalpy and total (or stagnation) entropy. The terms

static enthalpy and static entropy appear to be less common, but where they are used they mean

nothing more than enthalpy and entropy respectively, and the prefix "static" is being used to avoid

ambiguity with their 'total' or 'stagnation' counterparts. Because the 'total' flow conditions are

defined by isentropically bringing the fluid to rest, the total (or stagnation) entropy is by definition

always equal to the "static" entropy.

Lagrangian and Eulerian Specifications

There are two ways of describing a fluid motion. If we follow a particle through a flow, we can describe flow variables by F(xo, t), xo being the particles position at time to and t being how much time has elapsed. For example, the position of a particle is expressed as x(xo, t). This is called a Lagrangian description. We can deduce characteristics of the flow field by putting several Lagrangian tracer particles in a flow, and watch their positions, velocities, and accelerations over time.

If instead we focus on spatial points in the flow, instead of particles, we have a Eulerian description. Variables in this description are given by F(x,t), x being the spatial position, and t being time. We can measure the velocity and acceleration of particles going past a specific spatial point, for example.

The Material Derivative

When using Lagrangian coordinates, it is easy to find the velocity and acceration of the particle; they are simply the partial time derivatives:

and

In Eulerian coordinates, this is a little more complicated. The partial time derivatives only give us the rate of change of the variable at that specific spatial point. To help describe this, let’s take the example of an airplane taking off from a runway early in the morning. As the sun comes out, and begins to heat the airport, there is a localchange in temperature at the location of the airport. As the plane takes off and begins to accelerate into the sky, it is traveling through a temperature gradient, so the plane will experience a change in temperature due to its change in altitude. This is the advective part of the temperature change. The change in temperature with respect to time will depend on the local and advective terms. The equation for the Material Derivative, which takes both of these into account, is shown below.

As you can see, the material derivative, denoted by the capital D, is made up of the local derivative plus the advective term.

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Streamline

At an instant in time, there is a velocity vector at every point in a flow. A curve that is everywhere tangent to the direction of the velocity vectors is called a streamline. Below is a photograph of streamlines for laminar flow around an object.

A path line is the trajectory of a fluid particle over time.

A streak line is physical line of particles that have passed through some position in the flow field. Think of injecting a continuous stream of dye at one point in the flow. This line can theoretically never be crossed, as it is a physical, continuous line of particles. In a steady flow, streamlines, streaklines, and path lines coincide. Below is a diagram illustrating particle trajectories, stream lines, and streak lines in a time dependent fluid flow.

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BERNOULLI’S THEOREM- AN IMPORTANT APPLICATION OF FLUID DYNAMICS

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 inpressure 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 forcompressible 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.

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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. A common form of Bernoulli's equation, valid at

any arbitrary point along a streamline where gravity is constant, is:

Where:

is the fluid flow speed at a point on a streamline,

is the acceleration due to gravity,

is the elevation of the point above a reference plane, with the positive z-direction pointing upward

— so in the direction opposite to the gravitational acceleration,

is the pressure at the point, and

is the density of the fluid at all points in the fluid.

For conservative force fields, Bernoulli's equation can be generalized as:

where Ψ is the force potential at the point considered on the streamline. E.g. for the Earth's gravity Ψ = gz.

The following two assumptions must be met for this Bernoulli equation to apply

the fluid must be incompressible — even though pressure varies, the density must remain

constant along a streamline;

friction by viscous forces has to be negligible.

By multiplying with the fluid density ρ, equation (A) can be rewritten as:

or:

where:

is dynamic pressure,

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is the piezometric head or hydraulic head (the sum of the elevation z and

the pressure head and

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:

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:

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.

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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.[11] 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.

If the fluid flow at some point along a stream line is brought to rest, this point is called a stagnation

point, and at this point the total pressure is equal to the stagnation pressure.

Applicability of incompressible flow equation to flow of gasesBernoulli'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 is entropic (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 flowThe 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:[12]

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.

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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.[12]

Further f(t) can be made equal to zero by incorporating it into the velocity potential using the

transformation

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).

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

forces,

(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

(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:

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where

p0 is the total pressure

ρ0 is the total density

Compressible flow in thermodynamicsAnother useful form of the equation, suitable for use in thermodynamics, is:

[15]

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 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:

where w0 is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly

proportional to the temperature, and this leads to the concept of the total (or stagnation)

temperature.

When shock waves are present, in a reference frame in which the shock is stationary and the flow is

steady, 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.

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Derivation Of Bernoulli’s Theorem

(i) For incompressible fluids

The Bernoulli equation for incompressible fluids can be derived by integrating the Euler equations, or applying the law ofconservation 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

In steady flow, v = v(x) so

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

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or

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.[16] In the form of the work-energy theorem, stating that[17]

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

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 s1and 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:

The work done by the forces consists of two parts:

The work done by the pressure acting on the area's A1 and A2

The work done by gravity: the gravitational potential energy in the volume A1 s1 is lost, and

at the outflow in the volumeA2 s2 is gained. So, the change in gravitational potential energy

ΔEpot,gravity in the time interval Δt is

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

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force times change in elevation—will be negative for a positive elevation change Δz = z2 − z1, while

the corresponding potential energy change is positive.[18] So:

And the total work done in this time interval Δt is

The increase in kinetic energy is

Putting these together, the work-kinetic energy theorem W = ΔEkin gives:[16]

or

After dividing by the mass Δm = ρ A1 v1 Δt = ρ A2 v2 Δt the result is:[16]

or, as stated in the first paragraph:

(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:

(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

when arriving at elevation z = 0. Or when we rearrange it as a head:

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

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, with p0 some reference pressure, or when we rearrange it as

a head:

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.

(Eqn. 2b)

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

pressure terms:

(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.

(ii) For Compressible FluidsThe 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:

.

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 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,

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

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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 mechanicalp dV work:

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 ΔE2 may easily be constructed. So now setting 0 = ΔE1 − ΔE2:

which can be rewritten as:

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

obtain

which is the Bernoulli equation for compressible flow.

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REAL WORLD APPLICATIONS

In modern everyday life there are many observations that can be successfully explained by

application of Bernoulli's principle, even though no real fluid is entirely inviscid and a small viscosity

often has a large effect on the flow.

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[21]—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 underside. 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 Pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft.

These two devices are connected to the airspeed indicator which determines the dynamic

pressure of the airflow past the aircraft. Dynamic pressure is the difference between stagnation

pressure and static pressure. Bernoulli's principle is used to calibrate the airspeed indicator so

that it displays theindicated airspeed appropriate to the dynamic pressure.[22]

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

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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.

In open-channel hydraulics, a detailed analysis of the Bernoulli theorem and its extension were

recently developed. It was proved that the depth-averaged specific energy reaches a minimum

in converging accelerating free-surface flow over weirs and flumes. Further, in general, a channel

control with minimum specific energy in curvilinear flow is not isolated from water waves, as

customary state in open-channel hydraulics.

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, propeller blades, 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 the Bernoulli principle to connect the flow kinematics to the flow-

induced pressures. In cases of incorrect (or partially correct) explanations of lift, also relying at some

stage on the Bernoulli principle, the errors generally occur in the assumptions on the flow

kinematics, and how these are produced. It is not the Bernoulli principle itself that is questioned

because this principle is well established.

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CONCLUSION

This research on fluid dynamic and its application provides us concept about motion of fluid and its application. The following thesis gives us idea of use of motion of fluid in different field of engineering and day to day life.

This project emphasis and briefly discusses about the main application of fluid dynamics i.e. Bernoulli’s equation and its contribution in various field of engineering.

This thesis explains all Bernoulli’s theorem with its equations, assumptions, description of all terms related equation, formula, with examples and provides clear understanding of the topic with proper explains.

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BIBLIOGRAPHY

www.wikipedia.org

www.google.co.in

faculty.trinityvalleyschool.org/.../Lesson%2061-Derivation%20of%20Bernoullis%20Equation.pdf

www.4physics.com/phy_demo/bernoulli-effect-equation.html