1550251515-Classical_Complex_Analysis__Gonzalez_

512 Chapter^7

rotw = 'V X w = k(v., - uy)

where

T7 8. {), {)k
v=-1+-J+-
OX ay {)z
Hence the conditions div w = 0 and rot w = 0 are equivalent to the
Cauchy-Riemann equations u., = -vy, Uy = v., for iiJ = u - iv, and also to
the property of a fluid fl.ow to be both solenoidal and irrotational.
For a fluid motion to be possible it is necessary that the equation of
continui1~y, or conservation of mass, namely,

°;: + \7 • (pw) = 0

be satisfied. If p = constant, we must have

· '\7 • W = Ux +Vy = Q

Therefore, an incompressible flow is necessarily solenoidal.

(7.29-10)

In general, at a given instant a fluid element is subjected to a translation

as well as a rotation with angular velocity %( v., - Uy). If rot w = 0 in a

certain domain, then the fluid elements undergo translational motion (with

associated distortion) but no rotation. This explains the origin of the term

irrotational as applied to this type of motion.

The components U and V of the complex potential F are such that

U., = u = Vy and Uy = v = -Vx. As stated above, the function U is called

the real velocity potential since the velocity field can be derived from U.

In fact,

grad U = U.,i + Uyj = ui + vj = w (7.29-11)
Obviously, the vector grad U is orthogonal at each point to the equipotential
line through that point, and tangent to the corresponding streamline at the
same point, excepting stagnation points (Fig. 7.31).
From (7.29-10) and (7.29-11) it follows that

'\7^2 U = 'V • "VU = U.,., + Uyy = u., +Vy = 0

i.e., the real velocity potential satisfies the Laplace equation. This is clear

since U is the real part of the analytic function F. If the fl.ow is irrotational,

v., = Uy and we also have

'\7^2 V = v.,., + Vyy = -v., +Uy = 0

However, if the fl.ow is rotational, then '\7^2 V =f 0. In general, the value

w = v^2 v