≠
ω
constant
≠
s
0
≠
∇
s

Fluid Mechanics (Spring 2020) – Chapter 2 - U. Lei (
李雨
)
Equations in a non-inertial frame
So far we have derived our equations in an inertial frame.
In non-inertial frame, the continuity equation, the energy
equation, and the equations of state remain unchanged.
For the momentum equation, the acceleration term becomes:
dt
d
dt
d
dt
d
dt
d
c
I
u
r
Ω
r
Ω
Ω
u
Ω
u
u
+
×
+
×
×
+
×
+
=
)
(
2
↑
↑
Coriolis acceleration
centrifugal
acceleration
linear acceleration of the center
of the non-inertial coordinates
acceleration
observed in the
non-inertial frame
angular acceleration of the non-inertial frame

Vorticity and Circulation
(for understanding the movie “Vorticity” by Shapiro, from this to last page)
Vorticity,
, represents twice the spinning rate of an
infinitesimal fluid element.
Circulation,
Γ
, represent the strength of vorticity contained inside a
closed path
C
(
t
), defined as
= ∇ ×
u
ω
( )
C t
d
Γ =
⋅
u
r
( )
( )
( )
( )
.
( )
.
C t
A t
A t
A t
If
is regular inside C(t),
d
d
n
singula
dA
If
is
inside C
t ,
nd
r
A
Γ =
⋅
=
∇ ×
⋅
=
⋅
Γ ≠
⋅
u
u
r
u
A
u
ω
ω
Circulation
is helpful in understanding the force
on body, and in experimental quantification.

Example: Free vortex
(2D flow)
2
( )
0
( )
Velocity field in polar coordinates:
0,
.
(
= constant)
2
We have:
0,
,
and
0
because
is singular at
0.
r
C t
A t
B
u
u
B
r
d
u rd
B
ndA
u
r
θ
π
θ
θ
π
θ
=
=
∇ ×
=
Γ =
⋅
=
=
Γ ≠
⋅
=
=
u
u
r
ω =
ω
Note:
A fluid element may travel on a circular streamline while having zero vorticity.
Vorticity is proportional to the angular velocity of a fluid element about its principal
axes (
自轉
), not about an axis through some other reference point (
公轉
). Thus a fluid
particle which is traveling on a circular streamline will have zero vorticity,
provided that
it does not revolve about its center of gravity as it moves.

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- Spring '20
- Fluid Dynamics, Fluid Mechanics