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EGM 6812 Fluid Mechanics 1  Fall 2004
3
Kinematics of Local Fluid Motion
Kinematics is a branch of mechanics that treats motion without reference to forces causing that
motion.
3.1 Lagrangian and Eulerian Methods of Description
3.1.1 Lagrangian Viewpoint
General form of conservation law (system, “convected,” material region)
Attention is focused on material particles
as they move through the flow.
It is natural extension
of particle mechanics.
Each particle in the flow is formally labeled and identified by
•
Original position
0
i
x
•
Time
t
$
There are 2 independent variables in the Lagrangian method:
0
ˆ
,
i
x t
Define:
i
r
as the position of a material point as it travels through space
•
(
29
0
,
i
i
i
r
r x t
∴ =
$
•
Velocity:
i
r
t
$
•
Acceleration:
2
2
i
r
t
$
2
r
1
r
0
i
x
@
i
r
t
$
3.1.2 Eulerian Viewpoint
Control region or field
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Focuses on the properties of the flow at a given point in the flow (move convenient for fluid
flow)
All flow properties such as
,
,etc.
i
i
i
r v a
are functions of the following independent variables in the
Eulerian method of description
•
Space:
i
x
•
Time:
t
3.2 Streamlines, Streaklines, Pathlines, and Timelines
Line Patterns to Visualize Flow (*all equal for steady flow except timeline*)
•
Pathlines
consider one particle (Lagranian) and the path that it traces out in time.
To find the pathline:
(
29
,
dx
V x t
dt
=
r
r
r
, integrate w.r.t. time, then determine constants by
specifying a location at a given time, i.e., particle passes through
0
x
r
at
0
t
.
•
Streaklines
It is a locus of particles that have passed through a given point
To find the streakline:
(
29
,
dx
V x t
dt
=
r
r
r
, integrate w.r.t. time, then determine constants by
specifying an initial condition.
•
Streamlines
a line that is everywhere tangent to the velocity vector at any instant in time
d x
r
V
r
0
example:
1
dx
dy
dz
V
d x
u
v
w
x
V
i
j
t
=
=
=
=
+
+
r
r
r
To find the equation for the streamline:
(
29
(
29
0
0
0
1
0
1
1
ln
or
y y
t
dy
v
t
dx
u
x
y
y
t
x x
x
x e

+
+
=
=

=
+
=
•
Timelines
a set of particles (Lagranian) that form a line at a given instant in time.
EGM 6812 Fluid Mechanics 1  Fall 2004
3.3 Substantial Derivative
(“material derivative”)
Express the time rate of change of a particle property or material property in Eulerian variable.
Let
F
be property of the flow under consideration
•
“Lagrangian”
(
29
0
ˆ
,
L
i
F
F
x t
=
•
“Eulerian”
(
29
,
E
i
F
F
x t
=
where
(
29
0
ˆ
,
i
i
i
x
r x t
=
Transformation “Law” between Lagrangian, Eulerian methods of descriptions
(
29
(
29
(
29
(
29
(
29
0
0
0
ˆ
ˆ
,
ˆ
ˆ
ˆ
,
,
,
ˆ
ˆ
ˆ
i
i
i
L
i
E
i
i
i
L
E
i
E
E
E
i
i
i
E
E
t
t
r x t
x
F
F
x t
F
x
r x t
t
t
F
F dx
F dt
F
F
v
t
x dt
t dt
t
x
F
V
F
t
=
=
=
=
=
=
=
+
=
+
=
+
r
Stokes gave this special notation
(
29
(
29
(
29
(
29
(
29
(
29
(
29
Index:
Gibbs:
i
i
D
u
Dt
t
x
D
V
Dt
t
=
+
=
+
For acceleration
F
V
=
(
29
Index:
Gibbs:
j
j
j
i
i
Du
u
u
u
Dt
t
x
DV
V
V
V
Dt
t
=
+
=
+
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This note was uploaded on 01/17/2012 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.
 Fall '09
 RENWEIMEI
 Fluid Mechanics

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