3Dynamics
IB Fluid Dynamics
3.6 Linear flows
Suppose we have a favorite point
x
0
. Near the point
x
0
, it turns out we can break
up the flow into three parts — uniform flow, pure strain, and pure rotation.
To do this, we take the Taylor expansion of u about x
0
:
u(x) = u(x
0
) + (x − x
0
) · ∇u(x
0
) + ···
= u
0
+ r · ∇u
0
,
with
r
=
x −x
0
and
u
0
=
u
(
x
0
). This is a linear approximation to the flow field.
We can do something more about the
∇u
term. This is a rank-2 tensor, i.e.
a matrix, and we can split it into its symmetric and antisymmetric parts:
∇u =
∂u
i
∂x
j
= E
ij
+ Ω
ij
= E + Ω,
where
E
ij
=
1
2
∂u
i
∂x
j
+
∂u
j
∂x
i
,
Ω
ij
=
1
2
∂u
i
∂x
j
−
∂u
j
∂x
i
,
We can write the second part in terms of the vorticity. Recall we have defined
the vorticity as
ω = ∇ × u.
Then we have
ω × r = (∇ × u) × r = r
j
∂u
i
∂x
j
−
∂u
j
∂x
i
= 2Ω
ij
r
j
.
So we can write
u = u
0
+ Er +
1
2
ω × r.
The first component is uniform flow; the second is the strain field; and the last
is the rotation component.
uniform flow
pure strain pure rotation
Since we have an incompressible fluid, we have
∇ · u
= 0. So
E
has zero trace,
i.e. if
E =
E
1
0 0
0 E
2
0
0 0 E
3
,
then E
1
+ E
2
+ E
3
= 0. This justifies the picture above.