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23.4 Models of Plane Fluid Flow 781

THEOREM 23.6

Let u and v be continuous with continuous first and second partial derivatives in a simply con-
nected domain D. Suppose ui + vj is irrotational (has curl O) and solenoidal. Then u and −v
satisfy the Cauchy-Riemann equations on D, and f (z) = u(x, y) − iv(x, y) is differentiable on
D. Conversely, if u and −v satisfy the Cauchy-Riemann equations on D, then ui + vj defines an
irrotational, solenoidal flow on D.

With zero curl, the fluid experiences no swirling, although there can be translations and
distortions in the motion. If the flow is solenoidal, then
∂u ∂v
div(ui + vj) = + = 0.
∂x ∂y
A further connection between flows and complex functions is provided by the following.

THEOREM 23.7

Let f be differentiable on a domain D. Then f (z) is an irrotational solenoidal flow on D. Con-

versely, if V = ui + vj is an irrotational, solenoidal vector field on a simply connected domain
D, then there is a differentiable complex function f defined on D such that f (z) = V (z).

Furthermore, if f (z) = ϕ(x, y) = iψ(x, y), then
∂ϕ ∂ψ ∂ϕ ∂ψ
= u = and = v =− .
∂x ∂y ∂y ∂x

In view of the fact that f (z) is the velocity of a flow, we call f a complex potential for the

flow. Theorem 23.7 implies that any differentiable function f (z) = ϕ(x, y) + iψ(x, y) defined
on a simply connected domain D determines an irrotational, solenoidal flow
∂ϕ ∂ψ

f (z) = + i = u(x, y) − iv(x, y) = u(x, y) + iv(x, y).
∂x ∂x
We call ϕ the velocity potential of the flow, and curves ϕ(x, y) = k are equipotential curves.The
function ψ is the stream function of the flow, and curves ψ(x, y) = c are the streamlines.

At any z at which f (z) = 0, f is a conformal mapping. A point at which f (z) = 0 is called
a stagnation point. Thinking of f as a mapping of the z-plane to the w-plane, we have
w = f (z) = ϕ(x, y) + iψ(x, y) = α + iβ.
Equipotential curves ϕ(x, y) = k map to vertical lines α = k, and streamlines ψ(x, y) = c map to
horizontal lines β = c. Because these vertical and horizontal lines are orthogonal in the w-plane
and f is conformal, the streamlines and equipotential curves in the z-plane also form orthogonal
families. Every streamline is orthogonal to each equipotential curve at any point of intersection.
This condition fails at a stagnation point, where the mapping may not be conformal.
Along an equipotential curve ϕ(x, y) = k,
∂ϕ ∂ϕ
dϕ = dx + dy = udx + v dy = 0.
∂x ∂y
Now ui + vj is the velocity of the flow at (x, y), and x (s)i + y (s)j is a unit tangent to the

equipotential curve through (x, y). Since the dot product of these two vectors is zero (from the
fact that dϕ = 0 along the equipotential curve), we conclude that the velocity is orthogonal to the
equipotential curve through (x, y)—provided that (x, y) is not a stagnation point.

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