1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

11.7 • Two-DI MENSI ONAL FLUID FLOW 4 7 3

by w = S ( z) is a one-to-one conformal mapping of the domain D consisting
of the z plane slit along the segment x = 0, -1 ::; y ::; 1 onto the domain G
consisting of thew plane slit along the segment - 1 ::; u::; 1, v = 0. The complex
potential for a uniform horizontal flow parallel to the slit in the w plane is given
by Fi (w) =Aw, where for convenience we choose A = 1 and where the slit lies

along the streamline \Ir (u, v) = c = 0. The composite function

l

F2 (z) = Fi (S(z)) = A (z^2 +1)^2

is the complex potential for a fluid flow in the domain D. The streamlines given
by ..p (x, y) = c for the flow in D are obtained by finding the preimage of the
streamline \Ir (u, v) = c in G given by the parametric equations

v=c and U= t, for - oo < t < oo.

The corresponding streamline in D is found by solving the equation

for x and y in terms oft. Squaring both sides of this equation yields

t^2 - c2 - 1 + i2ct = x^2 - y^2 + i2xy.

Equating the real and imaginary parts leads to the system of equations

x^2 - y^2 = t^2 - c^2 - 1 and xy = ct.

Eliminating the parameter t in the last two equations results in i? = ( x^2 + c2)

(y^2 - c^2 ), and we can solve for yin terms of x to obtain

y=c

1 + c^2 + x^2

c2 +x2

for streamlines in D. For large values of x, this streamline approaches the asymp-

tote y = c and approximates a horizontal flow, as shown in Figure 11.52.

Figure 11. 52 Flow a.round a segment.