April 10, 2006
1
Engineers 1
Engineers 1
Spring Semester 2006
Lecture 44
April 10, 2006
2
Review  Pascal
Review  Pascal
’s Principle
s Principle
!
Pascal’s Principle for incompressible
fluids
“When there is a change in pressure at any point in
a confined fluid, there is an equal change in
pressure at every point in the fluid”
!
Hydraulic advantage
F
out
=
F
in
A
out
A
in
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Ideal Fluid Motion
Ideal Fluid Motion
!
The motion of reallife fluids is complicated
!
Here we make some simplifying assumptions
!
Assume an “ideal fluid”, exhibiting
•
Laminar flow
• The velocity of the fluid at a given point in space does not change with time
•
Incompressible flow
• The density of the fluid does not change as the fluid flows
•
Nonviscous flow
• The fluid flows freely, no friction or losses
•
Irrotational flow
• No part of the fluid rotates about its center of mass; no turbulence
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Examples of Flow
Examples of Flow
Laminar Flow
Laminar to
Turbulent
Flow
Turbulent Flow
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5
Continuity Equation
Continuity Equation
!
Consider the motion of an ideal
fluid flowing with speed v in a
container with cross sectional
area A
!
The volume
!
V
that moves past a
point in a time
t
is given by
!
The volume per unit time is then
!
V
=
A
!
x
=
Av
!
t
!
V
!
t
=
Av
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Continuity
Continuity
Eqn
Eqn
(2)
(2)
!
Now consider a fluid flowing through a container
that changes cross sectional area
!
The fluid is initially flowing with speed
v
1
in a
container with area
A
and then enters a section of
the container where the fluid flows with speed
2
and cross sectional area
!
The volume of fluid entering this section of the
container must equal the volume of fluid leaving the
container
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Continuity
Continuity
Eqn
Eqn
(3)
(3)
!
The volume per unit time flowing in the first part of the
container is
!
The volume per unit time flowing in the second part of the
container is
!
Using the fact that the volume per unit time passing any
point must be the same we get
!
This equation is called the
continuity equation
!
V
!
t
=
A
1
v
1
!
V
!
t
=
A
2
v
2
A
1
v
1
=
A
2
v
2
April 10, 2006
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Bernoulli
Bernoulli
’s Equation
s Equation
!
We can express a constant volume flow rate
R
(=volume per time)
!
We can also express a constant mass flow rate
m
(=mass per time)
!
Now let’s consider the case where an incompressible fluid (constant
density
"
) is flowing at a steady rate through the container below
R
p
2
 pressure
v
2
 speed
y
2
 height
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 Spring '08
 Wolf
 Physics, Fluid Dynamics, Scientists&Engineers, Turbulent Flow, ideal fluid

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