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Fluid Mechanics
Kreith, F.; Berger, S.A.; et. al. “Fluid Mechanics”
Mechanical Engineering Handbook
Ed. Frank Kreith
Boca Raton: CRC Press LLC, 1999
c
1999byCRCPressLLC
Fluid Mechanics
*
Frank Kreith
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
University of Colorado
Stanley A. Berger
University of California, Berkeley
Stuart W. Churchill
University of Pennsylvania
J. Paul Tullis
Integral Relations for a Control Volume ¤ Reynolds Transport
Theorem ¤ Conservation of Mass ¤ Conservation of Momentum
¤ Conservation of Energy ¤ Differential Relations for Fluid
Motion ¤ Mass ConservationÏContinuity Equation ¤
Momentum Conservation ¤ Analysis of Rate of Deformation ¤
Relationship between Forces and Rate of Deformation ¤ The
NavierÏStokes Equations ¤ Energy Conservation Ð The
Mechanical and Thermal Energy Equations ¤ Boundary
Conditions ¤ Vorticity in Incompressible Flow ¤ Stream
Function ¤ Inviscid Irrotational Flow: Potential Flow
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
Dimensional Analysis ¤ Correlation of Experimental Data and
Theoretical Values
Thomas F. Irvine, Jr.
State University of New York, Stony Brook
Massimo Capobianchi
Basic Computations ¤ Pipe Design ¤ Valve Selection ¤ Pump
Selection ¤ Other Considerations
State University of New York, Stony Brook
Francis E. Kennedy
DeÝnition ¤ Uniform Flow ¤ Critical Flow ¤ Hydraulic Jump ¤
Weirs ¤ Gradually Varied Flow
Dartmouth College
E. Richard Booser
Consultant, Scotia, NY
Introduction and Scope ¤ Boundary Layers ¤ Drag ¤ Lift ¤
Boundary Layer Control ¤ Computation vs. Experiment
Donald F. Wilcock
Tribolock, Inc.
Robert F. Boehm
Introduction ¤ One-Dimensional Flow ¤ Normal Shock Wave
¤ One-Dimensional Flow with Heat Addition ¤ Quasi-One-
Dimensional Flow ¤ Two-Dimensional Supersonic Flow
University of Nevada-Las Vegas
Rolf D. Reitz
University of Wisconsin
Introduction ¤ Fundamentals ¤ GasÏLiquid Two-Phase Flow ¤
GasÏSolid, LiquidÏSolid Two-Phase Flows
Sherif A. Sherif
University of Florida
Bharat Bhushan
The Ohio State University
*
Nomenclature for Section 3 appears at end of chapter.
¨ 1999 by CRC Press LLC
3
-1
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3
-2
Section 3
Introduction ¤ ClassiÝcation 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
Introduction ¤ Sliding Friction and Its Consequences ¤
Lubricant Properties ¤ Fluid Film Bearings ¤ Dry and
Semilubricated Bearings ¤ Rolling Element Bearings ¤
Lubricant Supply Methods
Introduction ¤ Pumps ¤ Fans
Spray Characterization ¤ Atomizer Design Considerations ¤
Atomizer Types
Direct Methods ¤ Restriction Flow Meters for Flow in Ducts ¤
Linear Flow Meters ¤ Traversing Methods ¤ Viscosity
Measurements
Introduction ¤ Experimental Techniques ¤ Surface Roughness,
Adhesion, and Friction ¤ Scratching, Wear, and Indentation ¤
Boundary Lubrication
3.1 Fluid Statics
Stanley A. Berger
Equilibrium of a Fluid Element
If the sum of the external forces acting on a Þuid element is zero, the Þuid will be either at rest or
moving as a solid body Ð in either case, we say the Þuid element is in equilibrium. In this section we
consider Þuids in such an equilibrium state. For Þuids 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 deÝnition 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 Þuid 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 Ï
p
,
p
being the pressure.
Hydrostatic Pressure
If we carry out an equilibrium of forces on an elementary volume element
, the forces being
pressures acting on the faces of the element and gravity acting in the Ï
z
direction, we obtain
I
I
p
x
8 8
p
y
0, and
I
I
8Y 8Y
g
~
(3.1.1)
The Ýrst two of these imply that the pressure is the same in all directions at the same vertical height in
a gravitational Ýeld. The third, where
~
(
z
). For homogeneous Þuids, for which
=
constant, this last equation can be integrated immediately, yielding
¨ 1999 by CRC Press LLC
dxdydz
p
z
I
I
is the speciÝc weight, shows that the pressure increases with
depth in a gravitational Ýeld, the variation depending on
116951894.002.png
 
Fluid Mechanics
3
-3
pp gzz ghh
2
Y8Y Y
1
%
2
1
& 8Y
%
2
Y
1
&
(3.1.2)
or
pghpgh
2
'
2
8'
1
1
8
constant
(3.1.3)
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
h
p
atm
,
h
is usually measured from the ocean/atmosphere interface and
p
at any distance
below this interface differs from
p
atm
by an amount
pp gh
Y8
atm
(3.1.4)
Pressures may be given either as
absolute pressure,
pressure measured relative to absolute vacuum,
or
gauge pressure,
pressure measured relative to atmospheric pressure.
Manometry
The hydrostatic pressure variation may be employed to measure pressure differences in terms of heights
of liquid columns Ð 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
3.1.1 Ý lled with liquid of speciÝc 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 Ýgure, in the left leg
pghp
0
Y
2
8
atm
(3.1.5a)
and in the right leg
pghp
0
Y
1
8
(3.1.5b)
the difference being
pp
Y8Y
atm
ghh
%
1
Y
2
& 8Y 8Y
gd d
~
(3.1.6)
FIGURE 3.1.1
U-tube manometer.
¨ 1999 by CRC Press LLC
where
h
116951894.003.png
-4
Section 3
and determining
p
in terms of the height difference
d
=
h
1
Ï
h
2
between the levels of the Þuid in the
two legs of the manometer.
Hydrostatic Forces on Submerged Objects
We now consider the force acting on a submerged object due to the hydrostatic pressure. This is given by
F
8
MM MM
p
dA
8
p dA
n
8
MM
gh
dA
'
p
0
MM
dA
(3.1.7)
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
to a free surface shown i n Figure 3.1.2 . The force on one
side of the planar surface, from Equation (3.1.7), is
A
at an angle
Fn
8
ghdApA
A
MM
'
0
n
(3.1.8)
but
h
=
y
sin
, so
MM
hdA
8
sin
MM
ydAyA
8
c
sin
8
hA
c
(3.1.9)
A
A
where the subscript
c
indicates the distance measured to the centroid of the area
. Thus, the total force
(on one side) is
FAA
8
~
h
c
'
p
0
(3.1.10)
Thus, 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
A
in Equation (3.1.10) can be
dropped.
Since
p
is not evenly distributed over
A
, but varies with depth,
F
does not act through the centroid.
, can be determined by considering moments i n Figure
3.1.2 . The moment of the hydrostatic force acting on the elementary area
F
, called the
center of pressure
dA
about the axis perpendicular
to the page passing through the point 0 on the free surface is
ydFyy
8 %
~
sin
dA
& 8
~
y
2
sin
dA
(3.1.11)
so if
y
cp
denotes the distance to the center of pressure,
yF
8
~
sin
MM
ydA
2
8
~
sin
I x
(3.1.12)
cp
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
I
x
x
). Dividing by
F
=
~
h
A =
~ y c sin
A gives
¨ 1999 by CRC Press LLC
3
Consider the planar surface
A
0
The point action of
where
c

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