Page 885 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 865

(1 −|t|)cos(πt/2) Im(T (z)T (it))
−1
7. u(x, y) = 2 dt
1 1 + sin (πt/2) 1 − 2Re(T (z)T (it)) +|T (z)| 2
Section 23.4 Models of Plane Fluid Flow
iθ
1. With a = Ke write
iθ
f (z) = az = Ke (x + iy) = K(x cos(θ) − y sin(θ)) + iK(x sin(θ) + y cos(θ)).
Equipotential curves: x cos(θ) − y sin(θ) = c.
Streamlines: x sin(θ) + y cos(θ) = k. There are no stagnation points, hence no sinks or sources.
3. f (z) = cos(x)cosh(y) − i sin(x)sinh(y). Equipotential curves are graphs of cos(x)cosh(y) = c, and streamlines are
graphs of sin(x)sinh(y) − k. Each point (nπ,0) with n any integer is a stagnation point.
5. f (z) = K log(z − z 0 ) = K ln|z − z 0 |+ i arg(z − z 0 ). Equipotential curves are graphs of K ln|z − z 0 |= c, concentric
circles about z 0 , streamlines are graphs of arg(z − z 0 ) = k, which are half-lines from z 0 . z 0 is a source if K > 0, a sink
if K < 0.
7. Write f (z) = K(x + iy + 1)/(x + iy) to obtain the equipotential curves as graphs of
2
2
Kx(x + y + 1)
= c
2
x + y 2
and streamlines as graphs of
2
2
ky(x + y − 1)
= k.
2
x + y 2
9. In polar coordinates, equipotential curves are graphs (in polar coordinates) of
b
2
K cos(θ)(r + 1) − rθ = c,r
2π
and streamlines are graphs of
b
2 2
K sin(θ)(r − 1) + r ln(r ) = c 2 r
4π
There are stagnation points where f (z) = 0, as

ib b 2
z =− ± 1 −
2
4Kπ 16π K 2
11. Compute
9aK 2
2
√ 2 √ 2
| f (z)| =
ia 3 ia 3
z − z +
2 2
Use the residue theorem to compute
2
4
1 18πa K ρ
2

A − Bi = iρ | f (z)| dz =− √ i,
2 γ 3 3a 3
yielding the vertical component B of the thrust.

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October 14, 2010 17:50 THM/NEIL Page-865 27410_25_Ans_p801-866