1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 509

Let r = (x1,x 2 ,x3) and v = (v 1 ,v 2 ,v 3 ). The fluid motion is said to be

two-dimensional or plane-parallel if a Cartesian coordinate system exists

with respect to which v is independent of x 3 and v 3 = 0. In this case it

is customary to write

Xl = X, Xz = y, V1 = u, Vz = V

Hence a stationary plane-parallel fl.ow of an ideal fluid is characterized by

a vector function of the variables x and y, or, alternatively, by a complex

function

w = f(z) = u(x, y) + iv(x, y) (7.29-1)

continuous except possibly at isolated points. The function u(x, y) gives

at each point the u-component of the velocity of the fluid, and v( x, y)

gives its v-component. Points at which w = 0 (the zeros off) are called

stagnation points.

In what follows we refer only to stationary, plane-parallel flows of an

ideal fluid. Suppose that the function f in (7.29-1) is defined in some

region D, and let

C: z=z(s)=x(s)+iy(s),

be a smooth simple arc with graph contained in D, the parameters being
the arc length measured from the initial point z 0 = z(O). Then

r( s) = z' ( s) = x' ( s) + iy' ( s) = cos e + i sine

represents the unit tangent vector to C at z( s) [in magnitude and orien-

tation, but not in position; r( s) will be in position if it is laid off from

z(s)]. Also,

v( s) = -iz' ( s) = y' ( s) - ix' ( s) = sine - i cos e

represents the unit normal vector to C at z( s ).
The component of the fl.ow w = u + iv tangential to C at z is given
by the scalar product

Wr = UX^1 ( S) + vy^1 ( S) = U COS 0 + V sin 0 (7.29-2)

and the component of the fl.ow normal to C at z( s) is given by

Wv = uy^1 (s)-vx'(s) = usinO-vcosO

as shown in Fig. 7.30.
Next let C be a simple closed piecewise smooth contour in D. Then Wr
and Wv may not be defined at a finite number of points of [O, L]. However,