2
1
1
2
2
1
r
u
r
r
u
r
r
D
Ω
r
r
D
Ω
r
r
D
r

Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
Physical meaning of the deformation tensor

shear
(2)
Then
2
sin
cos
1
cos
1
d
d
d
ds
d
ds
d
1
2
r
D
r
=
+
θ
θ
θ
θ
1
2
2
2
1
1
ds
ds
dt
dt
ds
dt
ds
⋅
⋅
−
/
d
ds
=
r
e
/
e
r
=
ds
d
e
e
⊥
Let
with
instantaneously
1
1
,
1
2
2
2
2
1
90 ,
o
θ
=
cos
θ
= 0,
and sin
θ
= 1.
Let
with
instantaneously,
Then
21
1
2
2
2
D
e
e
dt
d
−
=
⋅
⋅
−
=
D
θ
or
dt
d
D
D
θ
2
1
21
12
−
=
=
(represents a pure shear motion)
Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
Example 1
Given velocity field :
kx
u
=
0
v
=
0
=
w
We have :
11
1
,
2
u
u
u
D
k
x
x
x
∂
∂
∂
=
+
=
=
∂
∂
∂
0
31
23
12
33
22
=
=
=
=
=
D
D
D
D
D
A pure straining (if
k
> 0) motion in the xdirection:
Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
Example 2
Given velocity field :
,
u
ky
=
0,
v
=
0.
w
=
We have :
12
21
1
1
1
,
2
2
2
d
v
u
D
D
k
dt
x
y
θ
∂
∂
=
= −
=
+
=
∂
∂
0
31
23
33
22
11
=
=
=
=
=
D
D
D
D
D
(a pure shearing motion in the
xy
plane,
with
k
the shear rate)
Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
Rigid or deformable motion
If
D
= 0, the motion is called rigid, i.e., the distance between any
two fluid particles remains unchanged during the fluid motion.
(Example :
the steady state fluid motion within a cylindrical tank,
which is rotating about its principal axis at a constant rate.)
If
D
≠
0, we have stretching and shearing motion. However, it is
l
ibl
t
fi d th
di
l
di
ti
( t
i t)
always possible to find three perpendicular directions (at a point)
along which there is straining and such that the angles between
them are instantaneously rigid the axes along these directions are
them are instantaneously rigid, the axes along these directions are
called the principal axes
. We have
1
2
3
',
',
')
(e
e
e
'
'
'
'
'
'
3
3
3
2
2
2
1
1
1
e
e
e
e
e
e
D
λ
λ
λ
+
+
=
Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
To find the principal values and directions
Refer to the eigenvalue problem for a square matrix
e
e
D
λ
=
⋅
0
=
⋅
−
e
I)
(D
λ
or
0
13
12
11
−
λ
λ
D
D
D
D
D
D
or
33
23
13
23
22
12
=
−
−
λ
D
D
D
eigenvalues :
λ
1
2
3
(
,
,
)
λ
λ
λ
Principal strain rate
eigenvectors (efunctions) :
e
,
1
2
3
(
',
',
')
e
e
e
Principal axes
Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
Vorticity tensor and vorticity vector (1)
Vorticity tensor :
∂
∂
Ω
i
j
u
u
1
Vorticity tensor :
∂
−
∂
=
j
i
ij
x
x
2
1
u
u
∂
∂
Vorticity vector :
(
)
,
=
∇ ×
ω
u
i
2
j
k
ijk
j
k
e
x
x
ω
=
−
∂
∂
Relationship :
(
)
(
)
(
)
i
1
1
1
2
2
2
j
k
ijk
j
k
ijk
jk
u
u
x
x
e
e
ω
∂
∂
=
−
=
∇ ×
⋅
+
∇ ×
⋅
=
∇ ×
⋅
∂
∂
=
Ω
u
i
u
i
u
i
i
ω
ilm
jk
ijk
ilm
e
e
e
=
Ω
Fluid Mechanics (Spring 2019) – Chapter 2  U. Lei (
李雨
)
Vorticity tensor and vorticity vector (2)
,
ilm
ijk
lj
mk
lk
mj
e
e
δ δ
δ δ
=
−
With
We have
i
2
(
)
lj
mk
lk
mj
jk
ilm
l
l
m
m
m
l
e
δ δ
δ δ
ω
=
−
Ω
= Ω
=
Ω
− Ω
or
1
1
0
−
−
=
Ω
Ω
−
Ω
You've reached the end of your free preview.
Want to read all 127 pages?