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Fluids: Eulerian View, Compressibility, Continuity, Stress Tensor, Slides of Classical Mechanics
An introduction to the study of fluids, focusing on the eulerian view, compressibility, fluid change, continuity equation, and stress. The eulerian view describes the points in space and their bulk properties, while compressibility measures the relationship between volume change and pressure. Fluid change is related to the divergence, and the continuity equation ensures mass conservation. Stress is a surface force per unit area, with normal and shear components, and the stress tensor represents the stress function as a symmetric tensor with six components.
Typology: Slides
2012/2013
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Fluids
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Eulerian
View
In
a
Lagrangian
view
each
body
is
described
at
each
point
in
space.
Difficult
for
a
fluid
with
many
particles.
In
an
Eulerian
view
the
points
in
space
are
described.
Bulk
properties
of
density
and
velocity
0
t
r
r
t r
t
r
v
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Fluid
Change
A
change
in
a
property
like
pressure
depends
on
the
view.
In
a
Lagrangian
view
the
total
time
derivative
depends
on
position
and
time.
An
Eulerian
view
is
just
the
partial
derivative
with
time.
Points
are
fixed
^
^
2
2
2
2
2
x
l
x l x l k F
p
v
p t
dp dt
dz^ dt
p z
dy dt
p y
dx dt
p x
p t
dp dt
^
v
t
d dt
p^ t
dp dt
const
r
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Volume
Change
Consider
a
fixed
amount
of
fluid
in
a
volume
V
Cubic,
Cartesian
geometry
Dimensions
x
,^
y
,
z
.
The
change
in
V
is
related
to
the
divergence.
Incompressible
fluids
must
have
no
velocity
divergence
z
v z
z
d dt
y
v y
y
d dt
x
v x
x
d dt
x y z
z
y
x
v z
v y
v x
V
d dt
z
y
x
V
v
V
d dt
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Stress
A
stress
measures
the
surface
force
per
unit
area.
A
normal
stress
acts
normal
to
a
surface.
A
shear
stress
acts
parallel
to
a
surface.
A
fluid
at
rest
cannot
support
a
shear
stress.
F
F
A
A
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Force
in
Fluids
Consider
a
small
prism
of
fluid
in
a
continuous
fluid.
Describe
the
stress
P
at
any
point.
Normal
area
vectors
S
form
a
triangle.
The
stress
function
is
linear.
1 S d
2 S d
2
1
S
d
S
d
2 S d
P
1 S d
P
2
1
S
d
S
d
P
S
d
P c
S
cd
P
S d P S d P
2
1
2
1
S d S d P S d P S d P
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Force
Density
The
force
on
a
closed
volume
can
be
found
through
Gauss’
law.
Use
outward
unit
vectors
A
force
density
due
to
stress
can
be
defined
from
the
tensor.
Due
to
differences
in
stress
as
a
function
of
position
next
S
dS n
F
P
V
dV
F
P
P
S f
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