problem 5.1

Complex velocity

Taking differential of a complex function F( z): d F = ? _x Fd x + ? _y Fd y, show the equalities of (5.55).

problem 5.2

Stagnation-point flow

Determine the stream-lines of two-dimensional velocity field given by v = ( u, v) = ( Ax, ? Ay) with A a real constant (Fig. 5.11). [The origin (0 , 0) is called the stagnation-point because v vanishes there.]

Figure 5.11: Stagnation-point flow ( ?: stream function, ?: velocity potential).

problem 5.3

Orthogonal net of a complex analytic function

The expressions (5.58) define two families of equi-potential lines and stream-lines. Show mutual orthogonal-intersection of the two families.

problem 5.4

Conformal property

Suppose that we are given two complex planes z = x + iy and Z = X + iY , and that there is a point z ₀ in the z plane and two points z ₁ and z ₂ infinitesimally close to z ₀. The two planes are related by a complex analytic function Z = F( z), and the three points z ₀ , z ₁ , z ₂ are mapped to Z ₀ , Z ₁ , Z ₂ in the Z plane by F( z), respectively. Show that the intersecting angle ?