Conformal Mapping and Flows
If $f$ is a sourcless and irrotational flow on a domain $D$ then there exists an analytic function $G$ such that
\[G'(z) = \overline{f(z)} \qquad z \in D\]$G$ is known as the complex potential of the flow. In this post, we will see that the level curves $\text{Im }G = c_0$ where $c_0$ is a constant are the paths followed by a particle in the flow. That is, these level curves are the streamlines of the flow.
Let $z_0$ be a point in $D$ such that $G(z_0) \ne 0$. Then $G’(z_0) = \overline{f(z_0)} \ne 0$ so $G$ is one-to-one in
\[\zeta_{\delta} = \left\{ z : |z-z_0| \lt \delta \right\}\]that is some disc centered at $z_0$ (see the discussion here). Since $G$ is one-to-one, it is invertible. Denote its inverse by $H$ so that $H(G(z)) = z$ at least for all $z$ in \(\zeta_{\delta}\). Then, $H(w)$ is an analytic function for \(w \in \Omega = \left\{ G(z) : \left\|z - z_0 \right\| \lt \delta\right\}\).
Now the level curve
\[\Gamma_0 = \left\{z: \text{Im }G(z) = \text{Im }G(z_0), |z-z_0| \lt \delta \right\}\]can be written as
\[\begin{align*} \Gamma_0 &= \left\{z: \text{Im }G(z) = \text{Im }G(z_0), |z-z_0| \lt \delta \right\}\\ &= \left\{H(w): \text{Im }w = \text{Im }G(z_0), w \in \Omega\right\}\\ &= \left\{H(\tau + ic_0): c_0 = \text{Im }G(z_0), \tau + ic_0 \in \Omega\right\} \end{align*}\]The tangent vector of $\Gamma_0$ is the derivative of $H(\tau + ic_0)$ with respect to $\tau$ which is also the derivative of $H$ with respect to $w$ because $H$ is analytic. Then, by the chain rule
\[1 = \frac{d}{dz}z = \frac{d}{dz}H(G(z)) = H'(G(z))G'(z) = H'(w)\overline{f(z)}\]So
\[\begin{equation} \frac{1}{H'(w)} = \overline{f(z)} \end{equation}\]and consequently,
\[\frac{H'(w)}{|H'(w)|} = \frac{|f(z)|}{f(z)} = \frac{\overline{f(z)}}{|f(z)|}\]So the unit tangent vector of $\Gamma_0$ is parallel to that of $f(z)$ for all $z$ in $\Gamma_0$.
Now if $u(t)$, $a \lt t \lt b$ is a continuously differentiable function with values on the line \(L = \left\{ \tau + ic_0: \tau + ic_0 \in \Omega, c_0 = \text{Im }G(z_0) \right\}\) such that
\[u'(t) = \left|H'(u(t))\right|^{-2}, \qquad a \lt t \lt b\]and define $\gamma (t) = H(u(t))$, $a \lt t \lt b$. Then the range of the curve $\gamma$ is
\[\begin{align*} \gamma(t) &= \left\{H(u(t)): \text{Im }G(z) = \text{Im }G(z_0), |z-z_0| \lt \delta \right\}\\ &= \left\{H(\tau + ic_0): c_0 = \text{Im }G(z_0), \tau + ic_0 \in \Omega\right\}\\ &= \left\{\tau + ic_0: \tau + ic_0 \in \Omega, c_0 = \text{Im }G(z_0)\right\}\\ &= \left\{z: c_0 = \text{Im }G(z_0)\right\} \end{align*}\]Which is precisely $\Gamma_0$. Then using (1) from above, the derivative of $\gamma$ is given by
\[\begin{align*} \gamma'(t) &= H'(u(t))u'(t)\\ &= \frac{H'(u(t))}{|H'(u(t))|^{2}}\\ &= \frac{1}{\overline{H'(u(t))}}\\ &= f(H(u(t))\\ &= f(\gamma(t)) \end{align*}\]for $a \lt t \lt b$. So the tangent vectors of $\gamma$ are precisely the paths that a particle in the flow will follow. That is, the level curves given by $\Gamma_0$ are exactly the streamlines of $f$.
This observation serves to motivate the Riemann Mapping theorem and Schwarz-Christoffel Transformations because it shows us why we might care to find specific conformal maps. Perhaps there will be a follow up post on these results. Working through this section and some of the computations for the examples, I realized I may need to have a look at The Geometry of Complex Numbers down the road.