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23.4 Models of Plane Fluid Flow 779

C is the boundary of the right half-plane, which is the imaginary axis. Parametrize C as ξ =
(0,t) = it with t varying from ∞ to −∞ for positive orientation (if we walk down this axis, the
right half-plane is over our left shoulder). Now
1 −∞ itz − 1 −1
f (z) = u(0,t) idt
πi ∞ it − z 1 + t 2
1 ∞ itz − 1 1
= u(0,t) dt.
π it − z 1 + t 2
−∞
2
The solution is the real part of this integral. Now t, u(0,t) and 1/(1 + t ) are real, so we must
pull out the real part of the terms containing i and z = x + iy in the integral. Write
itz − 1 itx − ty − 1 itx − ty − 1 −it − x + iy
= =
it − z it − x − iy it − x − iy −it − x + iy
2
tx(t − y) − itx + ity(t − y) + txy + i(t − y) + x
= .
x + (t − y) 2
2
The real part of this expression is
2
x(1 + t )
.
2
x + (t − y) 2
Therefore,
1 ∞ x(1 + t ) 1
2
u(x, y) = Re( f (z)) = u(0,t) dt
2
2
π −∞ x + (t − y) 1 + t 2
1 ∞ x
= g(t) dt.
2
π x + (t − y) 2
−∞
SECTION 23.3 PROBLEMS

In each of Problems 1 through 6, use equation (23.4) to 5. The unit disk if u(x, y) = x − y for (x, y) on the
solve the Dirichlet problem for the given domain with the boundary circle.
given boundary data.
6. The unit disk with
1. Upper half-plane: u(x,0) = f (x).
1for 0 ≤ θ ≤ π/4
iθ
u(e ) =
2. Right quarter plane: Re(z)> 0, u(x,0) = f (x),and 0for π/4 <θ < 2π.
u(0, y) = 0.
7. Solve the Dirichlet problem for the strip −1 < Im(z)<
3. The disk |z − z 0 | < R if u(x, y) = xy for (x, y) on the
1, Re(z)> 0 with the boundary conditions
boundary.
u(x,1) = u(x,−1) = 0for x > 0
4. Right half-plane with boundary condition:
and

1 for −1 ≤ x ≤ 1
u(0, y) = u(0, y) = 1 −|y| for − 1 ≤ y ≤ 1.
0 for |y| > 1.

23.4 Models of Plane Fluid Flow

This section is an introduction to complex function models of fluid flow. Suppose an incompress-
ible fluid moves with velocity field V(x, y). By assuming that the velocity depends only on two
variables, we are taking the flow to be the same in all planes parallel to the complex plane. Such

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October 14, 2010 15:39 THM/NEIL Page-779 27410_23_ch23_p751-788