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Stream lines, quasilines and holomorphic motions
 Gaven J. Martin^{1}Email author
 Received: 18 February 2015
 Accepted: 19 February 2015
 Published: 15 September 2015
Abstract
We give a new application of the theory of holomorphic motions to study the distortion of level lines of harmonic functions and stream lines of ideal planar fluid flow. In various settings, we show they are in fact quasilines—the quasiconformal images of the real line. These methods also provide quite explicit global estimates on the geometry of these curves.
Mathematics Subject Classification
 30C62
 37F30
 30C75
1 Background
The theory of holomorphic motions, introduced by ManéSadSullivan [1] and advanced by Slodkowski [2], has had a significant impact on the theory of quasiconformal mappings. A reasonably thorough account of this is given in our book [3]. In [4, 5] we established some classical distortion theorems for quasiconformal mappings and used the theory to develop connections between Schottky’s theorem and Teichmüller’s theorem. We also gave sharp estimates on the distortion of quasicircles which in turn gave estimates for the distortion of extensions of analytic germs as studied in [6]. Here we consider the geometry of stream lines for ideal fluid flow in a domain and establish bounds on their distortion in terms of a reference line. These bounds come from an analysis of the geometry of the level lines of the hyperbolic metric and seem to be of independent interest. When the reference line is known to be a quasiline—the image of \({\mathbb R}\) under a quasiconformal map of \({\mathbb C}\)—which occurs for instance when there is some symmetry about, it follows that all level lines are quasilines and it is possible to give explicit distortion estimates which contains global geometric information—such as bounded turning—for the curve, see for instance (8) below. As such these estimates will have implications for parabolic linearisations.
We first recall the two basic notions we need here.
1.1 Quasiconformal mappings
1.2 Holomorphic motions
The theorem quoted below, known as the extended \(\lambda\)lemma and first proved by Slodkowski [2], is key in what follows. The distortion estimate of K in terms of the hyperbolic metric was observed by Bers & Royden earlier and it is from this that we will be able to make our explicit distortion estimates below. See [7] for a discussion. A complete and accessible proof can be found in [3, Chapter 12]. First, the definition of a holomorphic motion.

For any fixed \(a\in X\), the map \(\lambda \mapsto \Phi (\lambda ,a)\) is holomorphic.

For any fixed \(\lambda \in {\mathbb D}\), the map \(a\mapsto \Phi (\lambda ,a)\) is an injection.

\(\Phi (0,a)=a\) for all \(a\in X\).
Theorem 1
Remark
We will use this formulation when \(\Omega\) is a strip.
2 Geometry of hyperbolic level lines
Theorem 2

\(f(\gamma )=\alpha\),

\(f\partial \Omega = {{identity}}\),

\(K \le e^{c}\).
Proof
Theorem 3
Remark
Proof
Remark
We now have the next two corollaries.
Corollary 1
Let \(\Omega\) and h be as in Theorem 3. If for some value of \(a\in (0,1)\) the level line \(\{h=a\}\) is a quasiarc, then all level lines are quasiarcs.
Corollary 2
Proof
The hypotheses imply that \(\{h=\frac{1}{2}\}\) is a segment of the real axis and thus a quasiarc with distortion \(K=1\). The results follow once we extend the mapping f given by Theorem 3 by the identity outside \(\Omega\) and note the resulting map is quasiconformal. \(\square\)
3 Ideal fluid flow
Knowing geometric information about the level lines of harmonic functions has many applications. Here we give a couple which are simple and direct and concern ideal fluid flow in a channel and in particular two examples where computational results are known [10, 11]. In some ways the regularity results we derive (showing level lines are quasi lines) justify the computational results.
A channel is the conformal image of the strip \(\mathcal{S}\), \(\varphi :{\mathcal{S}}\rightarrow {\mathbb C}\), with the property that \(\varphi (x+iy)\rightarrow \pm \infty\) as \(x\rightarrow \pm \infty\). A channel is not a Jordan domain, but the reader can easily see that the above results apply without modification to this situation.
Conformal invariance shows us that the stream lines of the fluid flow starting with a source at \(\infty\) and flowing to a sink at \(+\infty\) are the level curves of a realvalued nonconstant harmonic function h which is constant on the boundary. We can normalise so that these numbers are \(h=+1\) on \(\varphi (x+i\pi /2)\) and \(h=1\) on \(\varphi (xi\pi /2)\). With this normalisation there is a central stream line \(\alpha _0=\{h=0\}\). We have the following two corollaries.
Corollary 3
Of course \(\alpha _0\) need not be a quasiline itself—the channel could be a regular neighbourhood of a smoothly embedded real line which is not quasiconformally equivalent to \({\mathbb R}\) by a mapping of \({\mathbb C}\). The results say that every stream line is the bounded geometric image of the central line. Next, if \(\mathcal{C}\) is symmetric about the real line then \(\alpha _0={\mathbb R}\) and we have the following:
Corollary 4
Theorem 4
Proof
The modulus of the ring \({\mathcal{S}}{\setminus} [r,r]\) tends to infinity as \(r\searrow 0\) and 0 for \(r=\infty .\) This modulus is continuous and so the intermediate value theorem gives us an r so that \(\mathrm{Mod}({\mathcal{S}}{\setminus} [r,r])=\mathrm{Mod}({\mathcal{S}}{\setminus} \Omega )\). For this r there is a conformal mapping \(\varphi : {\mathcal{S}}{\setminus} [r,r] \rightarrow {\mathcal{S}}{\setminus} \Omega\). The stream lines for ideal fluid flow in \({\mathcal{S}}{\setminus} [r,r]\) are simply the lines \(\{{\mathbb R}+iy: 0< y \le \pi /2 \}\). The images of these lines under \(\varphi\) are the stream lines for flow in \({\mathcal{S}}{\setminus} \Omega\). We can also use the Carathéodory/Schwarz extension/reflection principle to extend \(\varphi\) to a map \(\tilde{\varphi } : \{z:0< y < \pi \}\rightarrow {\mathbb C}\), \(\tilde{\varphi }({\mathbb R}+i\pi /2)={\mathbb R}+i\pi /2\). It follows from Corollary 4 that for \(0<y<\pi /2\), the stream line \(\tilde{\varphi }({\mathbb R}+iy)\) is a quasiline with distortion \(e^{d_{hyp}(\tilde{\varphi }({\mathbb R}+iy), {\mathbb R}+i\pi /2)}\)—with the metric here being that of \(\tilde{\varphi }(\{0<y<\pi \}).\) \(\square\)
It is not difficult to further refine this estimate upon consideration of the harmonic function involved following the arguments given for Corollary 2. What is remarkable here is that the global geometric estimates one achieves on the stream lines do not depend on the complexity of the object. Indeed the fact that the hyperbolic metric increases under inclusion implies that we can use a slightly larger smooth approximation to the object to get the estimates we require. So the boundary being highly irregular (say Hausdorff dimension \(>1\)) does not matter for estimating the distortion unless the stream line comes very close to the boundary. Further, the same argument (with the same estimates) works for flow around multiple objects, though there are issues when there is more than one obstacle as further conformal invariants may appear. In the simplest case, the bounds on the distortion of the stream lines apply for flow around \(\varphi ({\mathcal{S}}{\setminus} \cup I_i)\subset {\mathcal{S}}\), where \(\{I_i\}\) is any disjoint collection of closed intervals of \({\mathbb R}\), and \(\varphi\) is conformal with \(\varphi (\{\mathfrak {I}m(z)=\pm \frac{\pi }{2}\}) = \{\mathfrak {I}m(z)=\pm \frac{\pi }{2}\}\)
Declarations
Acknowledgements
GJM research supported by the Marsden Fund.
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Authors’ Affiliations
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