if and only if it has vanishing divergence:
∇ ·
v
=
∂u
∂x
+
∂v
∂y
= 0
.
(7
.
36)
Incompressibility means that the fluid volume does not change as it flows. Most liquids,
including water, are, for all practical purposes, incompressible.
On the other hand, the
flow is
irrotational
if and only if it has vanishing curl:
∇ ×
v
=
∂v
∂x
−
∂u
∂y
= 0
.
(7
.
37)
Irrotational flows have no vorticity, and hence no circulation. A flow that is both incom
pressible and irrotational is known as an
ideal fluid flow
. In many physical regimes, liquids
(and, although less often, gases) behave as
ideal fluids
.
Observe that the two constraints (7.36–37) are almost identical to the Cauchy–Riemann
equations (7.18); the only difference is the change in sign in front of the derivatives of
v
.
But this can be easily remedied by replacing
v
by its negative
−
v
. As a result, we establish
a profound connection between ideal planar fluid flows and complex functions.
Theorem 7.14.
The velocity vector field
v
= (
u
(
x,y
)
,v
(
x,y
))
induces an ideal fluid
flow if and only if
f
(
z
) =
u
(
x,y
)
−
i
v
(
x,y
)
(7
.
38)
is a complex analytic function of
z
=
x
+ i
y
.
Thus, the components
u
(
x,y
) and
−
v
(
x,y
) of the velocity vector field for an ideal
fluid flow are necessarily harmonic conjugates. The corresponding complex function (7.38)
is, not surprisingly, known as the
complex velocity
of the fluid flow. When using this result,
do not forget
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 Olver
 Fluid Dynamics, Differential Equations, Equations, Partial Differential Equations, Ideal fluid flow, Peter J. Olver, velocity vector field

Click to edit the document details