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# Chapter 2: Fluid Mechanics - Notes, Mechanical, Engineering, Semester Mechanical Engineering Notes | EduRev

## Mechanical Engineering : Chapter 2: Fluid Mechanics - Notes, Mechanical, Engineering, Semester Mechanical Engineering Notes | EduRev

``` Page 1

Kreith K., Berger S.A., Churchill S. W., Tullis J. P., White F. M., etal....“ Fluid Mechanics.”
The CRC Handbook of Thermal Engineering.
Ed. Frank Kreith
Boca Raton: CRC Press LLC, 2000
Page 2

Kreith K., Berger S.A., Churchill S. W., Tullis J. P., White F. M., etal....“ Fluid Mechanics.”
The CRC Handbook of Thermal Engineering.
Ed. Frank Kreith
Boca Raton: CRC Press LLC, 2000

2

-1

© 2000 by CRC Press LLC

2

Fluid Mechanics

2.1 Fluid Statics

Equilibrium of a Fluid Element • Hydrostatic Pressure •
Manometry • Hydrostatic Forces on Submerged Objects •
Hydrostatic Forces in Layered Fluids • Buoyancy • Stability
of Submerged and Floating Bodies • Pressure Variation in
Rigid-Body Motion of a Fluid

2.2 Equations of Motion and Potential Flow

Integral Relations for a Control Volume • Reynolds Transport
Theorem • Conservation of Mass • Conservation of Momentum •
Conservation of Energy • Differential Relations for Fluid
Motion • Mass Conservation–Continuity Equation •
Momentum Conservation • Analysis of Rate of Deformation •
Relationship between Forces and Rate of Deformation • The
Navier–Stokes Equations • Energy Conservation — The
Mechanical and Thermal Energy Equations • Boundary
Conditions • Vorticity in Incompressible Flow • Stream
Function • Inviscid Irrotational Flow: Potential Flow

2.3 Similitude: Dimensional Analysis and
Data Correlation

Dimensional Analysis • Correlation of Experimental Data and
Theoretical Values

2.4 Hydraulics of Pipe Systems

Basic Computations • Pipe Design • Valve Selection • Pump
Selection • Other Considerations

2.5 Open Channel Flow

De?nition • Uniform Flow • Critical Flow • Hydraulic Jump •

2.6 External Incompressible Flows

Introduction and Scope • Boundary Layers • Drag • Lift •
Boundary Layer Control • Computation vs. Experiment

2.7 Compressible Flow

Introduction • One-Dimensional Flow • Normal Shock Wave •
One-Dimensional Flow with Heat Addition • Quasi-One-
Dimensional Flow • Two-Dimensional Supersonic Flow

2.8 Multiphase Flow

Introduction • Fundamentals • Gas–Liquid Two-Phase Flow •
Gas–Solid, Liquid–Solid Two-Phase Flows

Frank Kreith, Editor

Engineering Consultant

Stanley A. Berger

University of California, Berkeley

Stuart W. Churchill

University of Pennsylvania

J. Paul Tullis

Utah State University

Frank M. White

University of Rhode Island

Alan T. McDonald

Purdue University

Ajay Kumar

NASA Langley Research Center

John C. Chen

Lehigh University

Thomas F. Irvine, Jr.

State University of New York,
Stony Brook

Massimo Capobianchi

Gonzaga University
Page 3

Kreith K., Berger S.A., Churchill S. W., Tullis J. P., White F. M., etal....“ Fluid Mechanics.”
The CRC Handbook of Thermal Engineering.
Ed. Frank Kreith
Boca Raton: CRC Press LLC, 2000

2

-1

© 2000 by CRC Press LLC

2

Fluid Mechanics

2.1 Fluid Statics

Equilibrium of a Fluid Element • Hydrostatic Pressure •
Manometry • Hydrostatic Forces on Submerged Objects •
Hydrostatic Forces in Layered Fluids • Buoyancy • Stability
of Submerged and Floating Bodies • Pressure Variation in
Rigid-Body Motion of a Fluid

2.2 Equations of Motion and Potential Flow

Integral Relations for a Control Volume • Reynolds Transport
Theorem • Conservation of Mass • Conservation of Momentum •
Conservation of Energy • Differential Relations for Fluid
Motion • Mass Conservation–Continuity Equation •
Momentum Conservation • Analysis of Rate of Deformation •
Relationship between Forces and Rate of Deformation • The
Navier–Stokes Equations • Energy Conservation — The
Mechanical and Thermal Energy Equations • Boundary
Conditions • Vorticity in Incompressible Flow • Stream
Function • Inviscid Irrotational Flow: Potential Flow

2.3 Similitude: Dimensional Analysis and
Data Correlation

Dimensional Analysis • Correlation of Experimental Data and
Theoretical Values

2.4 Hydraulics of Pipe Systems

Basic Computations • Pipe Design • Valve Selection • Pump
Selection • Other Considerations

2.5 Open Channel Flow

De?nition • Uniform Flow • Critical Flow • Hydraulic Jump •

2.6 External Incompressible Flows

Introduction and Scope • Boundary Layers • Drag • Lift •
Boundary Layer Control • Computation vs. Experiment

2.7 Compressible Flow

Introduction • One-Dimensional Flow • Normal Shock Wave •
One-Dimensional Flow with Heat Addition • Quasi-One-
Dimensional Flow • Two-Dimensional Supersonic Flow

2.8 Multiphase Flow

Introduction • Fundamentals • Gas–Liquid Two-Phase Flow •
Gas–Solid, Liquid–Solid Two-Phase Flows

Frank Kreith, Editor

Engineering Consultant

Stanley A. Berger

University of California, Berkeley

Stuart W. Churchill

University of Pennsylvania

J. Paul Tullis

Utah State University

Frank M. White

University of Rhode Island

Alan T. McDonald

Purdue University

Ajay Kumar

NASA Langley Research Center

John C. Chen

Lehigh University

Thomas F. Irvine, Jr.

State University of New York,
Stony Brook

Massimo Capobianchi

Gonzaga University

2

-2

© 2000 by CRC Press LLC

2.9 Non-Newtonian Flows

Introduction • Classi?cation of Non-Newtonian Fluids •
Apparent Viscosity • Constitutive Equations • Rheological
Property Measurements • Fully Developed Laminar Pressure
Drops for Time-Independent Non-Newtonian Fluids • Fully
Developed Turbulent Flow Pressure Drops • Viscoelastic Fluids

2.1 Fluid Statics

Stanley A. Berger

Equilibrium of a Fluid Element

If the sum of the external forces acting on a ?uid element is zero, the ?uid will be either at rest or
moving as a solid body — in either case, we say the ?uid element is in equilibrium. In this section we
consider ?uids in such an equilibrium state. For ?uids in equilibrium the only internal stresses acting
will be normal forces, since the shear stresses depend on velocity gradients, and all such gradients, by
the de?nition of equilibrium, are zero. If one then carries out a balance between the normal surface
stresses and the body forces, assumed proportional to volume or mass, such as gravity, acting on an
elementary prismatic ?uid volume, the resulting equilibrium equations, after shrinking the volume to
zero, show that the normal stresses at a point are the same in all directions, and since they are known
to be negative, this common value is denoted by –

p

,

p

being the pressure.

Hydrostatic Pressure

If we carry out an equilibrium of forces on an elementary volume element

dxdydz

, the forces being
pressures acting on the faces of the element and gravity acting in the –

z

direction, we obtain
(2.1.1)
The ?rst two of these imply that the pressure is the same in all directions at the same vertical height in
a gravitational ?eld. The third, where

?

is the speci?c weight, shows that the pressure increases with
depth in a gravitational ?eld, the variation depending on

?

(

z

). For homogeneous ?uids, for which

?

=
constant, this last equation can be integrated immediately, yielding
(2.1.2)
or
(2.1.3)
where

h

denotes the elevation. These are the equations for the hydrostatic pressure distribution.
When applied to problems where a liquid, such as the ocean, lies below the atmosphere, with a
constant pressure

p

atm

,

h

is usually measured (positive) downward from the ocean/atmosphere interface
and

p

at any distance

h

below this interface differs from

p

atm

by an amount
(2.1.4)
Pressures may be given either as

absolute pressure,

pressure measured relative to absolute vacuum,
or

gauge pressure,

pressure measured relative to atmospheric pressure.
?
?
?
?
?
?
??
p
x
p
y
p
z
g = = =- =- 0, and
pp gz z gh h
21 2 1 2 1
-=- -
()
=- -
()
??
pgh p gh
22 1 1
+=+ = ?? constant
pp gh -=
atm
?
Page 4

Kreith K., Berger S.A., Churchill S. W., Tullis J. P., White F. M., etal....“ Fluid Mechanics.”
The CRC Handbook of Thermal Engineering.
Ed. Frank Kreith
Boca Raton: CRC Press LLC, 2000

2

-1

© 2000 by CRC Press LLC

2

Fluid Mechanics

2.1 Fluid Statics

Equilibrium of a Fluid Element • Hydrostatic Pressure •
Manometry • Hydrostatic Forces on Submerged Objects •
Hydrostatic Forces in Layered Fluids • Buoyancy • Stability
of Submerged and Floating Bodies • Pressure Variation in
Rigid-Body Motion of a Fluid

2.2 Equations of Motion and Potential Flow

Integral Relations for a Control Volume • Reynolds Transport
Theorem • Conservation of Mass • Conservation of Momentum •
Conservation of Energy • Differential Relations for Fluid
Motion • Mass Conservation–Continuity Equation •
Momentum Conservation • Analysis of Rate of Deformation •
Relationship between Forces and Rate of Deformation • The
Navier–Stokes Equations • Energy Conservation — The
Mechanical and Thermal Energy Equations • Boundary
Conditions • Vorticity in Incompressible Flow • Stream
Function • Inviscid Irrotational Flow: Potential Flow

2.3 Similitude: Dimensional Analysis and
Data Correlation

Dimensional Analysis • Correlation of Experimental Data and
Theoretical Values

2.4 Hydraulics of Pipe Systems

Basic Computations • Pipe Design • Valve Selection • Pump
Selection • Other Considerations

2.5 Open Channel Flow

De?nition • Uniform Flow • Critical Flow • Hydraulic Jump •

2.6 External Incompressible Flows

Introduction and Scope • Boundary Layers • Drag • Lift •
Boundary Layer Control • Computation vs. Experiment

2.7 Compressible Flow

Introduction • One-Dimensional Flow • Normal Shock Wave •
One-Dimensional Flow with Heat Addition • Quasi-One-
Dimensional Flow • Two-Dimensional Supersonic Flow

2.8 Multiphase Flow

Introduction • Fundamentals • Gas–Liquid Two-Phase Flow •
Gas–Solid, Liquid–Solid Two-Phase Flows

Frank Kreith, Editor

Engineering Consultant

Stanley A. Berger

University of California, Berkeley

Stuart W. Churchill

University of Pennsylvania

J. Paul Tullis

Utah State University

Frank M. White

University of Rhode Island

Alan T. McDonald

Purdue University

Ajay Kumar

NASA Langley Research Center

John C. Chen

Lehigh University

Thomas F. Irvine, Jr.

State University of New York,
Stony Brook

Massimo Capobianchi

Gonzaga University

2

-2

© 2000 by CRC Press LLC

2.9 Non-Newtonian Flows

Introduction • Classi?cation of Non-Newtonian Fluids •
Apparent Viscosity • Constitutive Equations • Rheological
Property Measurements • Fully Developed Laminar Pressure
Drops for Time-Independent Non-Newtonian Fluids • Fully
Developed Turbulent Flow Pressure Drops • Viscoelastic Fluids

2.1 Fluid Statics

Stanley A. Berger

Equilibrium of a Fluid Element

If the sum of the external forces acting on a ?uid element is zero, the ?uid will be either at rest or
moving as a solid body — in either case, we say the ?uid element is in equilibrium. In this section we
consider ?uids in such an equilibrium state. For ?uids in equilibrium the only internal stresses acting
will be normal forces, since the shear stresses depend on velocity gradients, and all such gradients, by
the de?nition of equilibrium, are zero. If one then carries out a balance between the normal surface
stresses and the body forces, assumed proportional to volume or mass, such as gravity, acting on an
elementary prismatic ?uid volume, the resulting equilibrium equations, after shrinking the volume to
zero, show that the normal stresses at a point are the same in all directions, and since they are known
to be negative, this common value is denoted by –

p

,

p

being the pressure.

Hydrostatic Pressure

If we carry out an equilibrium of forces on an elementary volume element

dxdydz

, the forces being
pressures acting on the faces of the element and gravity acting in the –

z

direction, we obtain
(2.1.1)
The ?rst two of these imply that the pressure is the same in all directions at the same vertical height in
a gravitational ?eld. The third, where

?

is the speci?c weight, shows that the pressure increases with
depth in a gravitational ?eld, the variation depending on

?

(

z

). For homogeneous ?uids, for which

?

=
constant, this last equation can be integrated immediately, yielding
(2.1.2)
or
(2.1.3)
where

h

denotes the elevation. These are the equations for the hydrostatic pressure distribution.
When applied to problems where a liquid, such as the ocean, lies below the atmosphere, with a
constant pressure

p

atm

,

h

is usually measured (positive) downward from the ocean/atmosphere interface
and

p

at any distance

h

below this interface differs from

p

atm

by an amount
(2.1.4)
Pressures may be given either as

absolute pressure,

pressure measured relative to absolute vacuum,
or

gauge pressure,

pressure measured relative to atmospheric pressure.
?
?
?
?
?
?
??
p
x
p
y
p
z
g = = =- =- 0, and
pp gz z gh h
21 2 1 2 1
-=- -
()
=- -
()
??
pgh p gh
22 1 1
+=+ = ?? constant
pp gh -=
atm
?

2

-3

© 2000 by CRC Press LLC

Manometry

The hydrostatic pressure variation may be employed to measure pressure differences in terms of heights
of liquid columns — such devices are called manometers and are commonly used in wind tunnels and
a host of other applications and devices. Consider, for example the U-tube manometer shown in Figure
2.1.1 ?lled with liquid of speci?c weight

?

, the left leg open to the atmosphere and the right to the region
whose pressure

p

is to be determined. In terms of the quantities shown in the ?gure, in the left leg
(2.1.5a)
and in the right leg
(2.1.5b)
the difference being
(2.1.6)
which determines

p

in terms of the height difference

d

=

h

1

–

h

2

between the levels of the ?uid in the
two legs of the manometer.

Hydrostatic Forces on Submerged Objects

The force acting on a submerged object due to the hydrostatic pressure is given by
(2.1.7)
where

h

is the variable vertical depth of the element

dA

and

p

0

is the pressure at the surface. In turn we
consider plane and nonplanar surfaces.

Forces on Plane Surfaces

Consider the planar surface

A

at an angle

?

to a free surface shown in Figure 2.1.2. The force on one
side of the planar surface, from Equation (2.1.7), is
(2.1.8)

FIGURE 2.1.1

U-tube manometer.
pgh p
02
-= ?
atm
pgh p
01
-= ?
pp gh h gd d -=- -
()
=- =-
atm
???
12
F dA n dA dA == · = +
?? ?? ?? ??
p p dA gh p ?
0
Fn n =+
??
?ghdA pA
A
0
Page 5

Kreith K., Berger S.A., Churchill S. W., Tullis J. P., White F. M., etal....“ Fluid Mechanics.”
The CRC Handbook of Thermal Engineering.
Ed. Frank Kreith
Boca Raton: CRC Press LLC, 2000

2

-1

© 2000 by CRC Press LLC

2

Fluid Mechanics

2.1 Fluid Statics

Equilibrium of a Fluid Element • Hydrostatic Pressure •
Manometry • Hydrostatic Forces on Submerged Objects •
Hydrostatic Forces in Layered Fluids • Buoyancy • Stability
of Submerged and Floating Bodies • Pressure Variation in
Rigid-Body Motion of a Fluid

2.2 Equations of Motion and Potential Flow

Integral Relations for a Control Volume • Reynolds Transport
Theorem • Conservation of Mass • Conservation of Momentum •
Conservation of Energy • Differential Relations for Fluid
Motion • Mass Conservation–Continuity Equation •
Momentum Conservation • Analysis of Rate of Deformation •
Relationship between Forces and Rate of Deformation • The
Navier–Stokes Equations • Energy Conservation — The
Mechanical and Thermal Energy Equations • Boundary
Conditions • Vorticity in Incompressible Flow • Stream
Function • Inviscid Irrotational Flow: Potential Flow

2.3 Similitude: Dimensional Analysis and
Data Correlation

Dimensional Analysis • Correlation of Experimental Data and
Theoretical Values

2.4 Hydraulics of Pipe Systems

Basic Computations • Pipe Design • Valve Selection • Pump
Selection • Other Considerations

2.5 Open Channel Flow

De?nition • Uniform Flow • Critical Flow • Hydraulic Jump •

2.6 External Incompressible Flows

Introduction and Scope • Boundary Layers • Drag • Lift •
Boundary Layer Control • Computation vs. Experiment

2.7 Compressible Flow

Introduction • One-Dimensional Flow • Normal Shock Wave •
One-Dimensional Flow with Heat Addition • Quasi-One-
Dimensional Flow • Two-Dimensional Supersonic Flow

2.8 Multiphase Flow

Introduction • Fundamentals • Gas–Liquid Two-Phase Flow •
Gas–Solid, Liquid–Solid Two-Phase Flows

Frank Kreith, Editor

Engineering Consultant

Stanley A. Berger

University of California, Berkeley

Stuart W. Churchill

University of Pennsylvania

J. Paul Tullis

Utah State University

Frank M. White

University of Rhode Island

Alan T. McDonald

Purdue University

Ajay Kumar

NASA Langley Research Center

John C. Chen

Lehigh University

Thomas F. Irvine, Jr.

State University of New York,
Stony Brook

Massimo Capobianchi

Gonzaga University

2

-2

© 2000 by CRC Press LLC

2.9 Non-Newtonian Flows

Introduction • Classi?cation of Non-Newtonian Fluids •
Apparent Viscosity • Constitutive Equations • Rheological
Property Measurements • Fully Developed Laminar Pressure
Drops for Time-Independent Non-Newtonian Fluids • Fully
Developed Turbulent Flow Pressure Drops • Viscoelastic Fluids

2.1 Fluid Statics

Stanley A. Berger

Equilibrium of a Fluid Element

If the sum of the external forces acting on a ?uid element is zero, the ?uid will be either at rest or
moving as a solid body — in either case, we say the ?uid element is in equilibrium. In this section we
consider ?uids in such an equilibrium state. For ?uids in equilibrium the only internal stresses acting
will be normal forces, since the shear stresses depend on velocity gradients, and all such gradients, by
the de?nition of equilibrium, are zero. If one then carries out a balance between the normal surface
stresses and the body forces, assumed proportional to volume or mass, such as gravity, acting on an
elementary prismatic ?uid volume, the resulting equilibrium equations, after shrinking the volume to
zero, show that the normal stresses at a point are the same in all directions, and since they are known
to be negative, this common value is denoted by –

p

,

p

being the pressure.

Hydrostatic Pressure

If we carry out an equilibrium of forces on an elementary volume element

dxdydz

, the forces being
pressures acting on the faces of the element and gravity acting in the –

z

direction, we obtain
(2.1.1)
The ?rst two of these imply that the pressure is the same in all directions at the same vertical height in
a gravitational ?eld. The third, where

?

is the speci?c weight, shows that the pressure increases with
depth in a gravitational ?eld, the variation depending on

?

(

z

). For homogeneous ?uids, for which

?

=
constant, this last equation can be integrated immediately, yielding
(2.1.2)
or
(2.1.3)
where

h

denotes the elevation. These are the equations for the hydrostatic pressure distribution.
When applied to problems where a liquid, such as the ocean, lies below the atmosphere, with a
constant pressure

p

atm

,

h

is usually measured (positive) downward from the ocean/atmosphere interface
and

p

at any distance

h

below this interface differs from

p

atm

by an amount
(2.1.4)
Pressures may be given either as

absolute pressure,

pressure measured relative to absolute vacuum,
or

gauge pressure,

pressure measured relative to atmospheric pressure.
?
?
?
?
?
?
??
p
x
p
y
p
z
g = = =- =- 0, and
pp gz z gh h
21 2 1 2 1
-=- -
()
=- -
()
??
pgh p gh
22 1 1
+=+ = ?? constant
pp gh -=
atm
?

2

-3

© 2000 by CRC Press LLC

Manometry

The hydrostatic pressure variation may be employed to measure pressure differences in terms of heights
of liquid columns — such devices are called manometers and are commonly used in wind tunnels and
a host of other applications and devices. Consider, for example the U-tube manometer shown in Figure
2.1.1 ?lled with liquid of speci?c weight

?

, the left leg open to the atmosphere and the right to the region
whose pressure

p

is to be determined. In terms of the quantities shown in the ?gure, in the left leg
(2.1.5a)
and in the right leg
(2.1.5b)
the difference being
(2.1.6)
which determines

p

in terms of the height difference

d

=

h

1

–

h

2

between the levels of the ?uid in the
two legs of the manometer.

Hydrostatic Forces on Submerged Objects

The force acting on a submerged object due to the hydrostatic pressure is given by
(2.1.7)
where

h

is the variable vertical depth of the element

dA

and

p

0

is the pressure at the surface. In turn we
consider plane and nonplanar surfaces.

Forces on Plane Surfaces

Consider the planar surface

A

at an angle

?

to a free surface shown in Figure 2.1.2. The force on one
side of the planar surface, from Equation (2.1.7), is
(2.1.8)

FIGURE 2.1.1

U-tube manometer.
pgh p
02
-= ?
atm
pgh p
01
-= ?
pp gh h gd d -=- -
()
=- =-
atm
???
12
F dA n dA dA == · = +
?? ?? ?? ??
p p dA gh p ?
0
Fn n =+
??
?ghdA pA
A
0

2

-4

© 2000 by CRC Press LLC

but

h

=

y

sin

?

, so
(2.1.9)
where the subscript

c

indicates the distance measured to the centroid of the area

A

. The total force (on
one side) is then
(2.1.10)
Hence, the magnitude of the force is independent of the angle

?

, and is equal to the pressure at the
centroid,

?

h

c

+

p

0

, times the area. If we use gauge pressure, the term

p

0

A

in Equation (2.1.10) is dropped.
Since

p

is not evenly distributed over

A

, but varies with depth,

F

does not act through the centroid.
The point of action of

F

, called the

center of pressure

, can be determined by considering moments in
Figure 2.1.2. The moment of the hydrostatic force acting on the elementary area

dA

perpendicular to the page passing through the point 0 on the free surface is
(2.1.11)
so if

y

cp

denotes the distance to the center of pressure,
(2.1.12)
where

I

x

is the moment of inertia of the plane area with respect to the axis formed by the intersection
of the plane containing the planar surface and the free surface (say 0

x

). Dividing by

F

=

?

h

c

A

=

?

y

c

sin

?

A

gives
(2.1.13)

FIGURE 2.1.2

Hydrostatic force on a plane surface.
hdA y dA y A h A
AA
cc
?? ??
== = sin sin ??
FA A =+ ?hp
c 0
ydF y y dA y dA =()= ?? ? ? sin sin
2
yF y dA I
x cp
==
??
?? ?? sin sin
2
y
I
yA
x
c
cp
=
```
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