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Chapter 15
Fluid Mechanics
15.1 States of Matter

Solid
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Liquid

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Has a definite volume and shape
Has a definite volume but not a definite
shape
Gas – unconfined

Has neither a definite volume nor shape
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Fluids
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A fluid is a collection of molecules that
are randomly arranged and held
together by weak cohesive forces
between molecules and forces exerted
by the walls of a container.
Both liquids and gases are fluids
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Forces in Fluids
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A simplification model
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The fluids will be non viscous
The fluids do no sustain shearing forces
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The fluid cannot be modeled as a rigid object
The only type of force that can exist in a fluid
is the one perpendicular to a surface
The forces arise from the collisions of the fluid
molecules with the surface.
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Pressure

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The pressure, P, of the fluid at
the level to which the device has
been submerged is the ratio of
the force to the area
Pressure is a scalar and force is
a vector
The direction of the force
producing a pressure is
perpendicular to the area of
interest.
Units of pressure are Pascals
(Pa)
1 Pa  1
N
m2
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Atmospheric Pressure

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The atmosphere exerts a pressure on
the surface of the Earth and all objects
at the surface
Atmospheric pressure is generally taken
to be 1.00 atm = 1.013 x 105 Pa = Po
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15.2 Variation of Pressure
with Depth

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A fluid has pressure that varies with
depth
If a fluid is at rest in a container, all
portions of the fluid must be in static
equilibrium
All points at the same depth must be at
the same pressure

Otherwise, the fluid would not be in
equilibrium
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Pressure and Depth

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The darker region has a
cross-sectional area A and a
depth h.
Three external forces act on
the region
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Downward force on the top, PoA
Upward force on the bottom, PA
Gravity acting downward, mg


The mass can be found from
the density r of the fluid.
m = rV = rAh
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Pressure and Depth, 2

Since the fluid is in equilibrium,

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SFy = 0 gives PA – PoA – mg = 0
P = Po + rgh
The pressure P at a depth h below a
point in the liquid at which the pressure
is Po is greater by an amount rgh
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Pressure and Depth, final


If the liquid is open to the atmosphere,
and Po is the pressure at the surface of
the liquid, then Po is atmospheric
pressure
The pressure is the same at all points
having the same depth, independent of
the shape of the container
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Pascal’s Law

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Named for French scientist Blaise Pascal
The pressure in a fluid depends on depth and
on the value of Po
A change in the pressure applied to an
enclosed fluid is transmitted to every
point of the fluid and to the walls of the
container
This is the basis of Pascal’s Law
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Application of Pascal’s Law –
Hydraulic Press


A large output force
can be applied by
means of a small
input force
The volume of liquid
pushed down on the
left must equal the
volume pushed up on
the right
P1  P2
F1
F2

A1 A 2
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Pascal’s Law, Example cont.

Since the volumes are equal,
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A1 Dx1 = A2 Dx2
Combining the equations,
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F1 Dx1 = F2 Dx2 which means W1 = W2
This is a consequence of Conservation of
Energy
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15.3 Pressure Measurements:
Barometer

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Invented by Torricelli
A long closed tube is filled
with mercury and inverted in
a dish of mercury
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The closed end is nearly a
vacuum
Measures atmospheric
pressure as Po = rHggh
One 1 atm = 0.760 m (of Hg)
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Pressure Measurements:
Manometer

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A device for measuring the
pressure of a gas
contained in a vessel
One end of the U-shaped
tube is open to the
atmosphere
The other end is
connected to the pressure
to be measured
Pressure at B is Po+ rgh
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Absolute vs. Gauge Pressure

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P = Po + rgh
P is the absolute pressure
The gauge pressure is P – Po
 The gauge pressure is rgh
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This is what you measure in your tubes
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15.4 Buoyant Force
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The buoyant force is the upward force
exerted by a fluid on any immersed object
which is in equilibrium in the fluid.
The buoyant force is the resultant force due
to all forces applied by the fluid surrounding
the object.
The upward buoyant force must equal (in
magnitude) the downward gravitational force.
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Archimedes
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ca 289 – 212 BC
Greek mathematician,
physicist and engineer
Computed the ratio of a
circle’s circumference to
its diameter
Calculated the areas
and volumes of various
geometric shapes
Famous for buoyant
force studies
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Archimedes’ Principle

Any object completely or partially
submerged in a fluid experiences an
upward buoyant force whose magnitude
is equal to the weight of the fluid
displaced by the object.
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Archimedes’ Principle, cont

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The pressure at the top
of the cube causes a
downward force of PtopA
The pressure at the
bottom of the cube
causes an upward force
of Pbottom A
B = (Pbottom – Ptop) A =
Mg
M is the mass of the
fluid in the cube.
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Archimedes's Principle:
Totally Submerged Object
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An object is totally submerged in a fluid of
density rf
The upward buoyant force is B=rfgVf = rfgVo
Vf is the volume of the fluid displaced by the
object and Vo is the volume of the object.
The downward gravitational force of the
object is w =mg=rogVo
The net force is B-w=(rf-ro)gVo
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Archimedes’ Principle: Totally
Submerged Object, cont
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If the density of the object is less
than the density of the fluid, the
unsupported object accelerates
upward.
If the density of the object is greater
than the density of the fluid, the
unsupported object sinks.
The motion of an object in a fluid
is determined by the densities of
the fluid and the object.
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Archimedes’ Principle:
Floating Object
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The object is in static equilibrium
The upward buoyant force is balanced by the
downward force of gravity
Vf corresponds to the volume of the object beneath
the fluid level
The fraction of the volume of the object below the
fluid surface is equal to the ratio of the density of the
object to the fluid density.
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15.5 Fluid Dynamics –
Fluids in motion

Flow Characteristics: Laminar flow
 Steady flow
 Each particle of the fluid follows a smooth path so
that the paths of the different particles never cross
each other.
 The path taken by the particles is called a streamline.
The velocity of the fluid at any point remains
constant in time.
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Turbulent flow
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Above a certain critical speed, fluid flow
becomes turbulent.
An irregular flow characterized by small
whirlpool-like regions.
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Viscosity of a fluid
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Characterizing the degree of internal friction
in the fluid
This internal friction, viscous force, is
associated with the resistance that two
adjacent layers of fluid have to moving
relative to each other.
Since the viscous force is nonconservative,
part of the fluid’s kinetic energy is converted
to internal energy.
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Ideal Fluid – A simplified model of
fluids
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Four assumptions made to the complex
real fluids
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Nonviscous fluid– Internal friction is
neglected.
Incompressible fluid – The fluid density
remains constant.
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Ideal Fluid, cont
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Steady flow – The velocity of the fluid at
each point remains constant
Irrotational flow – The fluid has no
angular momentum about any point.
The first two assumptions are properties
of the ideal fluid and the last two are
descriptions of the way that the fluid
flows.
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15.6 Streamlines
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The path the particle
takes in steady flow
is a streamline
The velocity of the
particle is tangent to
a streamline
No two streamlines
can cross each
other.
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Equation of Continuity
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Consider a fluid moving through
a pipe of nonuniform size
(diameter)
The particles in the fluid move
along streamlines in steady flow
The volume of an incompressible
fluid is conserved
The mass that crosses A1 in
some time interval is the same
as the mass that crosses A2 in
that same time interval
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Equation of Continuity, cont
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A1v1 = A2v2
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This is called the volume flow rate, which is the
continuity equation for fluids
The product of the area and the fluid speed at all
points along a pipe is constant for an
incompressible fluid
The speed is high where the tube is
constricted (small A)
The speed is low where the tube is wide
(large A)
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15.7 Bernoulli’s Equation

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As a fluid moves through a region
where its speed and/or elevation above
the Earth’s surface changes, the
pressure in the fluid varies with these
changes
The relationship between fluid speed,
pressure and elevation was first derived
in 1738 by Daniel Bernoulli
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Daniel Bernoulli
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1700 – 1782
Swiss
mathematician and
physicist
Made important
discoveries involving
fluid dynamics
Also worked with
gases
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Bernoulli’s Equation
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Consider the two shaded
segments
The volumes of both
segments are equal
The net work done on the
segment is W=(P1 – P2) V
Part of the work changes
into the kinetic energy and
some changes into the
gravitational potential
energy
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Bernoulli’s Equation, 3
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The change in kinetic energy:
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DK = 1/2 m v22 - 1/2 m v12
There is no change in the kinetic energy of
the unshaded portion since we assume the
streamline flow
The masses of the two shaded segments
are the same since their volumes are the
same
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Bernoulli’s Equation, 3
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The change in gravitational potential energy:
 DU = mgy2 – mgy1
The work also equals the change in energy
Combining:
W = (P1 – P2)V=1/2 m v22 - 1/2 m v12 +
mgy2 – mgy1
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Bernoulli’s Equation, 4
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Rearranging and expressing in terms of density:
P1 + 1/2 r v12 + ρ g y1 = P2 + 1/2 r v22 + ρ g y2
This is Bernoulli’s Equation and is often
expressed as
P + 1/2 r v2 + ρ g y = constant
When the fluid is at rest, this becomes P1 – P2 =
rgh which is consistent with the pressure
variation with depth we found earlier
The general behavior of pressure with speed is
true even for gases

As the speed increases, the pressure decreases
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15.8 Applications of Fluid
Dynamics
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Streamline flow around
a moving airplane wing
Lift is the upward force
on the wing from the air
Drag is the resistance
The lift depends on the
speed of the airplane,
the area of the wing, its
curvature, the angle
between the wing and
the horizontal
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Lift – General
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In general, an object moving through a fluid
experiences lift as a result of any effect that
causes the fluid to change its direction as it
flows past the object
Some factors that influence lift are
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The shape of the object
Its orientation with respect to the fluid flow
Any spinning of the object
The texture of its surface
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Exercises
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13, 16, 22, 29, 35, 39, 51, 53, 71, 73
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