Conformal Maps

Let $\zeta=\xi+{\rm i}\,\eta$ and $z=x +{\rm i}\,y$ , where $\xi$ , $\eta$ , $x$ , and $y$ are real. Suppose that $\zeta=f(z)$ , where $f$ is a well-behaved (i.e., single-valued, non-singular, and differentiable) function. We can think of $\zeta=f(z)$ as a map from the complex $z$ -plane to the complex $\zeta$ -plane. In other words, every point $(x$ , $y)$ in the complex $z$ -plane maps to a corresponding point $(\xi$ , $\eta)$ in the complex $\zeta$ -plane. Moreover, if $f(z)$ is indeed a well-behaved function then this mapping is unique, and also has a unique inverse. Suppose that the point $z=z_0$ in the $z$ -plane maps to the point $\zeta=\zeta_0$ in the $\zeta$ -plane. Let us investigate how neighboring points map. We have

$$\zeta_0+d\zeta' =f(z_0+dz'),$$ (6.53)

$$\zeta_0+d\zeta'' =f(z_0+dz'').$$ (6.54)

In other words, the points $z_0+dz'$ and $z_0+dz''$ in the complex $z$ -plane map to the points $\zeta_0+d\zeta'$ and $\zeta_0+d\zeta''$ in the complex $\zeta$ -plane, respectively. If $\vert dz'\vert$ , $\vert dz''\vert\ll 1$ then

$$d\zeta' = f'(z_0)\,dz',$$ (6.55)

$$d\zeta''&= f'(z_0)\,dz'',$$ (6.56)

where $f'(z)= df/dz$ . Hence,

$$\frac{d\zeta''}{d\zeta'} = \frac{dz''}{dz'}.$$ (6.57)

Thus, it follows that

$$\frac{\vert d\zeta''\vert}{\vert d\zeta'\vert} = \frac{\vert dz''\vert}{\vert dz'\vert},$$ (6.58)

and

$${\rm arg}(d\zeta'')-{\rm arg}(d\zeta') = {\rm arg}(dz'')-{\rm arg}(dz').$$ (6.59)

We can think of $dz'$ and $dz''$ as infinitesimal vectors connecting neighboring points in the complex $z$ -plane to the point $z=z_0$ . Likewise, $d\zeta'$ and $d\zeta''$ are infinitesimal vectors connecting the corresponding points in the complex $\zeta$ -plane. It is clear, from the previous two equations, that, in the vicinity of $z=z_0$ , the mapping from the complex $z$ -plane to the complex $\zeta$ -plane is such that the lengths of $dz'$ and $dz''$ expand or contract by the same factor, and the angle subtended between these two vectors remains the same. (See Figure 6.7.) This type of mapping is termed conformal.

Figure 6.7: A conformal map.

Suppose that $F(\zeta) = \phi(\xi,\eta)+{\rm i}\,\psi(\xi,\eta)$ is a well-behaved function of the complex variable $\zeta$ . It follows that $\nabla^{\,2}\phi=\nabla^{\,2}\psi=\nabla\phi\cdot\nabla\psi=0$ . Hence, the functions $\phi(\xi,\eta)$ and $\psi(\xi,\eta)$ can be interpreted as the velocity potential and stream function, respectively, of some two-dimensional, incompressible, irrotational flow pattern, where $\xi$ and $\eta$ are Cartesian coordinates. However, if $\zeta=f(z)$ , where $f(z)$ is well-behaved, then $F(\zeta)=F[f(z)] = G(z) = \widetilde{\phi}(x,y) + {\rm i}\,\widetilde{\psi}(x,y)$ , where $G(z)$ is also well-behaved. It follows that $\nabla^{\,2}\widetilde{\phi}=\nabla^{\,2}\widetilde{\psi}=\nabla\widetilde{\phi}\cdot\nabla\widetilde{\psi}=0$ . In other words, the functions $\widetilde{\phi}(x,y)$ and $\widetilde{\psi}(x,y)$ can be interpreted as the velocity potential and stream function, respectively, of some new, two-dimensional, incompressible, irrotational flow pattern, where $x$ and $y$ are Cartesian coordinates. In other words, we can use a conformal map to convert a given two-dimensional, incompressible, irrotational flow pattern into another, quite different, pattern. Incidentally, a conformal map converts a line source into a line source of the same strength, and a vortex filament into a vortex filament of the same intensity. (See Exercise 12.)

As an example, consider the conformal map

$$\zeta = {\rm i}\,{\rm e}^{\,\pi\,z/a}.$$ (6.60)

Writing $\zeta=r\,{\rm e}^{\,{\rm i}\,\theta}$ , it is easily demonstrated that $x=a\,\ln r/\pi$ and $y=a\,(\theta/\pi-1/2)$ . Hence, the positive $\xi$ -axis ($\theta=0$ ) maps to the line $y=-a/2$ , the negative $\xi$ -axis ( $\theta=\pi$ ) maps to the line $y=a/2$ , and the region $\eta>0$ ( $0\leq \theta\leq \pi$ ) maps to the region $-a/2<y<a/2$ . Moreover, the points $\zeta = (0$ , $\pm 1)$ map to the points $z=a\,(0$ , $-1/2\pm1/2)$ . (See Figure 6.8.) As we saw in Section 6.6, in the region $\eta>0$ , the velocity potential

$$F(\zeta) = {\rm i}\,\frac{\mit\Gamma}{2\pi}\,\ln\left(\frac{\zeta-{\rm i}}{\zeta+{\rm i}}\right)$$ (6.61)

corresponds to the flow pattern generated by a vortex filament of intensity ${\mit\Gamma}$ , located at the point $\zeta = (0$ , $1)$ , in the presence of a rigid plane at $\eta=0$ . Hence,

$$G(z)=F({\rm i}\,{\rm e}^{\,\pi\,z/a}) = {\rm i}\,\frac{\mit\Gamma}{2\pi} \ln\,\tanh\left(\frac{\pi\,z}{2\,a}\right),$$ (6.62)

corresponds to the flow pattern generated by a vortex filament of intensity ${\mit\Gamma}$ , located at the origin, in the presence of two rigid planes at $y=\pm a/2$ . This follows because the line $\eta=0$ is mapped to the lines $y=\pm a/2$ , and the point $\zeta = (0$ , $1)$ is mapped to the origin. Moreover, if the line $\eta=0$ is a streamline in the $\eta$ -plane then the lines $y=\pm a/2$ are also streamlines in the $z$ -plane. Thus, these lines could all correspond to rigid boundaries. The stream function associated with the previous complex velocity potential,

$$\psi(x,y)= \frac{{\mit\Gamma}}{\pi}\,\ln\left[\frac{\cosh(\pi\,x\,a)-\cos(\pi\,y/a) }{\cosh(\pi\,x/a)+\cos(\pi\,y/a)}\right],$$ (6.63)

is shown in Figure 6.9.

Figure: The conformal map $\zeta={\rm i}\,{\rm e}^{\,\pi\,z/a}$ .

Figure: Stream lines of the two-dimensional flow pattern due to a vortex filament at the origin in the presence of two rigid planes at $y=\pm a/2$ .

As a second example, consider the map

$$\zeta = z^{\,2}.$$ (6.64)

This maps the positive $\xi$ -axis to the positive $x$ -axis, the negative $\xi$ -axis to the positive $y$ -axis, the region $\eta>0$ to the region $x>0$ , $y>0$ , and the point $\zeta = (0$ , $2\,a^{\,2})$ to the point $z=(a$ , $a)$ . As we saw in Section 6.6, in the region $\eta>0$ , the velocity potential

$$F(\zeta) = \frac{Q}{2\pi}\,\ln(\zeta^{\,2}+4\,a^{\,4}),$$ (6.65)

corresponds to the flow pattern generated by a line source of strength $Q$ , located at the point $\zeta = (0$ , $2\,a^{\,2})$ , in the presence of a rigid plane at $\eta=0$ . Thus, the complex velocity potential

$$G(z)=F(z^{\,2}) = \frac{Q}{2\pi}\,\ln(z^{\,4}+4\,a^{\,4}),$$ (6.66)

corresponds to the flow pattern generated by a line source of strength $Q$ , located at the point $z=(a$ , $a)$ , in the presence of two orthogonal rigid planes at $y=0$ and $x=0$ . The stream function associated with the previous complex potential,

$$\psi(x,y) = -\frac{Q}{2\pi}\,\tan^{-1}\left[\frac{4\,x\,y\,(x^{\,2}-y^{\,2})}{x^{\,4}-6\,x^{\,2}\,y^{\,2}+y^{\,4}+4\,a^{\,4}}\right],$$ (6.67)

is shown in Figure 6.10.

Figure: Stream lines of the two-dimensional flow pattern due to a line source at $(a$ , $a)$ in the presence of two rigid planes at $x=0$ and $y=0$ .

As a final example, consider the map

$$z = \zeta+ \frac{l^{\,2}}{\zeta},$$ (6.68)

where $l$ is real and positive. Writing $\zeta=r\,{\rm e}^{\,{\rm i}\,\theta}$ , we find that

$$x = 2\,l\,\cosh[\ln(r/l)]\,\cos\theta = (r+l^{\,2}/r)\,\cos\theta,$$ (6.69)

$$y =2\,l\,\sinh[\ln(r/l)]\,\sin\theta = (r-l^{\,2}/r)\,\sin\theta.$$ (6.70)

Thus, the map converts the circle $\xi^{\,2}+\eta^{\,2}=c^{\,2}$ in the $\zeta$ -plane, where $c>l$ , into the ellipse

$$\left(\frac{x}{a}\right)^2+\left(\frac{y}{b}\right)^2 = 1$$ (6.71)

in the $z$ -plane, where

$$a = 2\,l\,\cosh[\ln(c/l)] = c + l^{\,2}/c,$$ (6.72)

$$b = 2\,l\,\sinh[\ln(c/l)] = c - l^{\,2}/c.$$ (6.73)

Note that the center of the ellipse lies at the origin, and its major and minor axes run parallel to the $x$ - and the $y$ -axes, respectively. As we saw in Section 6.4, in the $\zeta$ -plane, the complex velocity potential

$$F = - V_0\left(\zeta+\frac{c^{\,2}}{\zeta}\right),$$ (6.74)

represents uniform flow of unperturbed speed $V_0$ , running parallel to the $\xi$ -axis, around a circular cylinder of radius $c$ , centered on the origin. Thus, assuming that $c>l$ , in the $z$ -plane, the potential represents uniform flow of unperturbed speed $V_0$ , running parallel to the $x$ -axis [which follows because at large $\vert z\vert$ the map (6.68) reduces to $z=\zeta$ , and so the flow at large distances from the origin is the same in the complex $z$ - and $\zeta$ -planes], around an elliptical cylinder of major radius $a$ , aligned along the $x$ -axis, and minor radius $b$ , aligned along the $y$ -axis. Note that $l=(a^{\,2}-b^{\,2})^{1/2}/2$ and $c=(a+b)/2$ . The corresponding stream function in the $z$ -plane is

$$\psi(x,y) = - V_0\left(r -\frac{c^{\,2}}{r}\right)\sin\theta,$$ (6.75)

where

$$r = l\,\exp(\cosh^{-1}p),$$ (6.76)

$$\theta = \tan^{-1}\left(\frac{y}{x}\,\frac{p}{[p^{\,2}-1]^{1/2}}\right),$$ (6.77)

$$p =\left[\frac{x^{\,2}/l^{\,2}+y^{\,2}/l^{\,2}+4+\left([x^{\,2}/l^{\,2}+y^{\,2}/l^{\,2}+4]^2-16\,x^{\,2}/l^{\,2}\right)^{1/2}}{8}\right]^{1/2}.$$ (6.78)

Figure 6.11 shows the streamlines of the flow pattern calculated for $c=1.5\,l$ .

Figure: Stream lines of the two-dimensional flow pattern due to uniform flow parallel to the $x$ -axis around an elliptical cylinder.