Inviscid Incompressible Flows
3.5 The Method of Conformal Mapping
Separating the real and imaginary parts in these equations, show that the
‘mass centre’ of the pair, whose coordinates are defined as xc= Γ1x1+ Γ2x2
Γ1+ Γ2
, yc= Γ1y1+ Γ2y2
Γ1+ Γ1
,
remains at the same position at all times. Show further that the distance between the vortices remains constant. Hence, conclude that the vortices travel around one other along concentric circular trajectories.
12. What is the trajectory and speed of a single vortex placed at height h above a flat surface as shown in Figure 3.40.
x y
Γ ih
z
Fig. 3.40: Vortex above a flat surface.
3.5 The Method of Conformal Mapping
One of the most powerful tools in the theory of two-dimensional inviscid flows is the method of conformal mapping. Before turning to fluid-dynamic applications of the method, we shall discuss basic mathematical aspects of the theory of conformal mapping.
Any complex function ζ = f (z) serves the purpose of defining the value of ζ = ξ+iη for a given value of the argument z = x + iy. It may therefore be thought of as a mapping of points in the z-plane into the corresponding points in the ζ-plane.
3.5.1 Mapping with a linear function
To start with, we shall consider the mapping with a linear function
ζ = az + b, (3.5.1)
where a and b are complex numbers. If, in particular, a = 1, then, denoting the real and imaginary parts of b by brand bi, respectively, we have
ξ = x + br, η = y + bi.
This shows that mapping with the function ζ = z + b represents parallel translation of all the points in the complex plane.
Let us now consider the case where b in (3.5.1) is zero and a is an arbitrary complex number different from zero and infinity. It may then be represented in the exponential form
a = κeiδ, (3.5.2)
where κ is the modulus of a and δ its argument. With a given by (3.5.2) and b = 0, the mapping (3.5.1) takes the form
ζ = κeiδz.
Since z may also be written in the exponential form (see Figure 3.41a) z = reiϑ,
we have
ζ = κeiδreiϑ = (κr)ei(δ+ϑ).
Thus, the mapping ζ = az leads to magnification of|z| by a factor κ = |a| and rotation through an angle δ = arg a (see Figure 3.41b).
Let us now return to the general linear function (3.5.1) and consider two points z′ and z′′ in the z-plane. Their images in the ζ-plane are
ζ′= az′+ b, ζ′′= az′′+ b, and we see that
ζ′′− ζ′= a(z′′− z′). (3.5.3) If we again use the exponential form (3.5.2) for a and represent z′′− z′ as
z′′− z′=|z′′− z′| eiϑ, then (3.5.3) becomes
ζ′′− ζ′= κeiδ|z′′− z′|eiϑ= κ|z′′− z′| ei(δ+ϑ).
This shows that mapping with the linear function (3.5.1) rotates a segment of a straight line through an angle δ and stretches it κ times (compresses, if κ < 1).
x
y z
z ϑ r O
(a) z-plane.
ξ η
δ + ϑ ζ ζ
z κr
O
(b) ζ-plane.
Fig. 3.41: Mapping with the function ζ = az.
3.5. The Method of Conformal Mapping 183
x
y z
θ
O
L1 L2
(a) z-plane.
ξ η
θ ζ
O
L1
L2
(b) ζ-plane.
Fig. 3.42: Preservation of the angle θ between intersecting lines L1 and L2 when mapping with the linear function (3.5.1).
Since the angle of rotation does not depend on the initial orientation of the segment in the z-plane or on its length, the following two statements are valid: (i) a straight line in the z-plane is mapped by the linear function (3.5.1) onto a straight line in the ζ-plane; (ii) if two lines in the z-plane make an angle θ at the point of their intersection, then this angle is preserved in the course of the mapping with a linear function; see Figure 3.42.
We can further prove the following theorem.
Theorem 3.5 The linear function
ζ = az + b,
where a6= 0, transforms any circle on the z-plane into a circle on the ζ-plane.
Proof Let us consider a circle Cz of radius r centred at point z0 in the z-plane (see Figure 3.43). We denote its image in the ζ-plane by Cζ. For any point z lying on Cz, the following equation is valid:
|z − z0| = r.
Using z0 instead of z′ and z instead of z′′ in (3.5.3), we have
ζ− ζ0= a(z− z0). (3.5.4)
Here ζ0 is the image of the centre z0 and ζ lies on the image Cζ of the circle Cz. It follows from (3.5.4) that
|ζ − ζ0| = |a| |z − z0| = kr,
which proves that Cζ is indeed a circle. ✷
x y
z z
O
r z0
Cz
Fig. 3.43: Mapping of a circle.
3.5.2 Conformal mapping
We shall use the following definition of conformal mapping.
Definition 3.2 The mapping ζ = f (z) is said to be conformal at a point z0 if the function f (z) is analytic at z0 and f′(z0)6= 0.
Remember that a function f (z) is called analytic at a point z0 if it is differentiable at this point, i.e. there exists the limit
f′(z0) = lim
∆z→0
f (z0+ ∆z)− f(z0)
∆z ,
which is independent of the orientation of ∆z in the complex plane. This means that in a small neighbourhood of z0,
f (z0+ ∆z)− f(z0) = f′(z0) ∆z + α(z0, ∆z) ∆z, where the function α(z0, ∆z) is such that
∆z→0lim α(z0, ∆z) = 0.
Therefore, if we restrict our attention to a small neighbourhood of z0, then we can write
f (z0+ ∆z)− f(z0) = f′(z0) ∆z.
Finally, denoting z = z0+ ∆z and taking into account that ζ = f (z) and ζ0 = f (z), we have
ζ− ζ0= f′(z0) (z− z0), or, equivalently,
ζ = az + b, where
a = f′(z0), b = ζ0− f′(z0) z0. (3.5.5) We can conclude that any conformal mapping behaves locally as a linear mapping.
In particular, it maps small circles onto small circles and preserves the angles between
3.5. The Method of Conformal Mapping 185 intersecting lines. It further follows from (3.5.2) and the first of equations (3.5.5) that
|f′(z)| is the magnification factor κ of the mapping ζ = f(z), while the angle of rotation δ is given by
δ = arg
df dz
. (3.5.6)
3.5.3 Mapping with the power function The power function is given by
ζ = zα, (3.5.7)
with α being a real constant. The function (3.5.7) is analytic in the whole complex plane except at z = 0 and z =∞. Therefore if one wants to deal with a single-valued analytic branch of the power function, a branch cut should be made in the z-plane connecting points z = 0 and z =∞.
The derivative of (3.5.7)
dζ
dz = αzα−1
remains finite at all finite z. As z→ 0, it tends to zero for all α > 1. If, on the other hand, α < 1, then dζ/dz becomes infinite at z = 0. This suggests that the mapping performed by (3.5.7) preserves angles at all points of the complex plane, except z = 0.
As an example let us consider the corner made of two rays OA and OB emerging from point O at an angle π− θ to one another; see Figure 3.44(a). For our purposes, it is convenient to place the coordinate origin at the point O and draw the real axis along one of the rays, say, OA.
Representing z in the exponential form
z = reiϑ (3.5.8)
and substituting (3.5.8) into (3.5.7) yields ζ = rαeiαϑ.
This shows that the mapping with the power function (3.5.7) increases all the angles by a factor α. It obviously leaves the first ray OA at the original place. The second ray OB is rotated around point O, changing its angle from π− θ to α(π − θ). We see
x y
z z
r
θ ϑ
O A
B
(a) z-plane.
ξ η
ζ ζ
rα αϑ O
B A
(b) ζ-plane.
Fig. 3.44: Mapping with the power function (3.5.7).
that if, for example, we need to map the region above the corner in the z-plane onto the upper half of the ζ-plane, then we have to set
α(π− θ) = π.
We can conclude that the power function (3.5.7) performs the desired mapping, pro-vided that
α = π π− θ. 3.5.4 Linear fractional transformation The mapping
ζ = az + b
cz + d, (3.5.9)
where a, b, c, and d are complex constants such that ad− bc 6= 0, is called a linear fractional transformation.
If c = 0, then (3.5.9) reduces to the linear transformation (3.5.1). If, on the other hand, c6= 0, then (3.5.9) may be written in the form
ζ = a
c +bc− ad c
1
cz + d, (3.5.10)
which shows that the condition ad− bc 6= 0 is necessary to ensure that the linear fractional transformation (3.5.9) is not a constant function mapping all the points in the z-plane into just one point in the ζ-plane.
When cleared of fractions, equation (3.5.9) takes the form
c ζz + dζ− az − b = 0, (3.5.11)
which is linear in z and linear in ζ; i.e. it is bilinear in z and ζ. Hence, another name for the linear fractional transformation (3.5.9) is a bilinear transformation.
Solving equation (3.5.11) for z, we find z = −dζ + b
cζ− a . (3.5.12)
It follows from (3.5.9) and (3.5.12) that each point in the z-plane (except possibly z =−d/c) has one and only one image point in the ζ-plane. Conversely, each point in the ζ-plane (except possibly ζ = a/c) has one and only one image point in the z-plane.
In order to include the points z =−d/c and ζ = a/c in our considerations, we adopt the following conventions for the complex number ∞:
1. If a is a finite number, then
a
∞ = 0.
2. If a6= 0, then
a 0 =∞.
3.5. The Method of Conformal Mapping 187 It is easily seen that for large but finite z, equation (3.5.9) may be written as
ζ = a + b/z c + d/z. For this reason, we will say that
ζ = a
c at z =∞. (3.5.13)
If z approaches−d/c, then (3.5.9) gives
ζ = b− ad/c
0 ,
and we will say that
ζ =∞ at z = −d
c. (3.5.14)
With the extensions (3.5.13) and (3.5.14) the linear fractional function (3.5.9) performs a one-to-one mapping of the extended z-plane onto the extended ζ-plane.
Definition 3.3 The complex z-plane with included infinite number z = ∞ is called the extended z-plane.
Let us now return to formula (3.5.10). It shows that the linear fractional mapping (3.5.9) can be obtained by the superposition of the following three mappings:
z1= cz + d, (3.5.15a)
ζ1= 1 z1
, (3.5.15b)
ζ = a
c +bc− ad
c ζ1. (3.5.15c)
The first and third are linear mappings, the properties of which we already know.
Hence, we only need to clarify the properties of the second mapping, (3.5.15b). Chang-ing notation slightly, we write
ζ = 1
z. (3.5.16)
If we use the exponential form for z, namely z =|z|eiϑ, then (3.5.16) gives ζ = 1
|z|e−iϑ.
We see that the transformation (3.5.16) consists of (i) reflection of point z in the circle of unit radius (in this reflection, the image of z stays on the same radius but its modulus changes to 1/|z|) and (ii) reflection in the real axis; see Figure 3.45.
An important property of the linear fractional transformation is the circle property, which is expressed by the following theorem.
x y
z z
ζ (i)
ϑ (ii)
−ϑ r=
1
Fig. 3.45: Two steps of the transformation (3.5.16).
Theorem 3.6 The linear fractional transformation ζ = az + b
cz + d
maps any circle on the extended z-plane into a circle on the extended ζ-plane.
Proof Consider, first, the mapping with the function (3.5.16). The inverse to (3.5.16) is written as
z = 1 ζ.
Expressing z and ζ via their real and imaginary parts z = x + iy, ζ = ξ + iη, we have
x + iy = 1
ξ + iη = ξ− iη ξ2+ η2. Hence,
x = ξ
ξ2+ η2, y =− η
ξ2+ η2. (3.5.17)
Any circle in the z-plane may be written as
A(x2+ y2) + Bx + Cy + D = 0, (3.5.18) where A, B, C, and D are real numbers. Substitution of (3.5.17) into (3.5.18) leads to D(ξ2+ η2) + Bξ− Cη + A = 0, (3.5.19) which represents a circle in the ζ-plane.