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2025Equivalent local force conditions minimizing the Frank free energy for topological defect equilibria in nematic liquid crystalsProc. R. Soc. A.48120240569http://doi.org/10.1098/rspa.2024.0569

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Equivalent local force conditions minimizing the Frank free energy for topological defect equilibria in nematic liquid crystals

Hiroyuki Miyoshi

https://orcid.org/0000-0002-3678-1641

Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, 7-3-1, Bunkyo-Ku, Tokyo 113-8656, Japan

[email protected]

Contribution: Conceptualization, Formal analysis, Funding acquisition, Methodology, Writing – original draft

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Darren G. Crowdy

https://orcid.org/0000-0002-7162-0181

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK

Contribution: Conceptualization, Formal analysis, Project administration, Supervision, Writing – review and editing

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Hiroki Miyazako

https://orcid.org/0000-0002-2795-271X

Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, 7-3-1, Bunkyo-Ku, Tokyo 113-8656, Japan

Contribution: Conceptualization, Funding acquisition, Project administration, Resources, Writing – review and editing

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Takaaki Nara

Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, 7-3-1, Bunkyo-Ku, Tokyo 113-8656, Japan

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Hiroyuki Miyoshi

https://orcid.org/0000-0002-3678-1641

Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, 7-3-1, Bunkyo-Ku, Tokyo 113-8656, Japan

[email protected]

Contribution: Conceptualization, Formal analysis, Funding acquisition, Methodology, Writing – original draft

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Darren G. Crowdy

https://orcid.org/0000-0002-7162-0181

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK

Contribution: Conceptualization, Formal analysis, Project administration, Supervision, Writing – review and editing

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Hiroki Miyazako

https://orcid.org/0000-0002-2795-271X

Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, 7-3-1, Bunkyo-Ku, Tokyo 113-8656, Japan

Contribution: Conceptualization, Funding acquisition, Project administration, Resources, Writing – review and editing

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Takaaki Nara

Department of Information Physics and Computing, Graduate School of Information Science and Technology, The University of Tokyo, Hongo, 7-3-1, Bunkyo-Ku, Tokyo 113-8656, Japan

Contribution: Conceptualization, Funding acquisition, Methodology, Project administration, Writing – review and editing

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Published:19 February 2025https://doi.org/10.1098/rspa.2024.0569

Abstract

Recently, Miyoshi et al. (Miyoshi et al. 2024 Proc. R. Soc. A 480, 20240405 (doi:10.1098/rspa.2024.0405)) derived analytical formulas for the Frank free energy associated with multiple topological defects in nematic liquid crystals confined to an arbitrary simply connected domain. The energy formulas, derived using an analogy with the so-called Kirchhoff–Routh path function in point vortex dynamics, required the evaluation of contour integrals involving analytical formulas associated with the crystal alignment field. Equilibria for topological defects were then obtained by finding local extrema of this Frank free energy. By contrast, in vortex dynamics, it is rare to find point vortex equilibria by minimizing the Kirchhoff–Routh path function that is a regularized kinetic energy; more commonly, a local ‘non-self-induction rule’ is used that is equivalent to each point vortex being free of net force. It is shown here that an analogous equivalent set of local force conditions can also be used to find equilibria for topological defects in nematic liquid crystals in confined domains, and without any need to evaluate the Frank free energy. This observation is significant theoretically and also because, at a technical level, the new local force conditions are easier to formulate and solve.

1. Introduction

There has been much recent interest in finding the equilibrium configurations of defects in liquid crystals. Identifying and controlling equilibria is important in bioengineering because topological defects regulate biological processes [1–3]. Kawaguchi et al. [3] reported that topological defects drive the aggregation of cultured murine neural progenitor cells. They found that the cells concentrate around defects with a charge of $+ 1 / 2$ but escape from defects with a charge of $- 1 / 2$ . Ienaga et al. [4] studied the existence and transition of topological defect formulations in so-called doublet or triplet regions to control the positions of defects. They showed that the topological defect configuration has significant dependence on the boundary shape (e.g. convexity, connectedness). It is therefore important to find effective mathematical techniques for the investigation of such defect equilibria.

Based on liquid crystal theory and the one-constant approximation [5], the theory of two-dimensional harmonic functions has been exploited for localizing and finding the equilibria of topological defects in confined geometries [6–9]. Duclos et al. [6] investigated the positions of topological defects in a circular domain of radius $R$ and confirmed both experimentally and mathematically that two defects with a charge of $+ 1 / 2$ are located at a distance of $5^{- 1 / 4} R$ from the centre of the disc. Since the alignment angles are harmonic functions in two dimensions, it is also natural to use a complex analysis formulation. Chandler & Spagnolie [10] used Schwarz–Christoffel mappings to find the optimal positions of topological defects on the boundary of polygonal-shaped immersed bodies with finite anchoring boundary conditions. They later extended their theory to study two-body interactions including consideration of body forces and torques [9]. Tang et al. [8] showed that the Frank free energy can be classified into torque-free energy and orientation energy and found that torque determines the dynamics of topological defects. Miyazako & Nara [7] formulated the notion of a complex function for the cell alignment angles based on complex analysis to represent the orientational order of cells. Using conformal mappings, they derived an analytical formula that describes the orientation of cells with topological defects at arbitrary positions strictly inside arbitrary closed domains (i.e. not on the boundaries). In addition, by utilizing the complex form of Green’s theorem, they evaluated the energy associated with the crystals in the domain as line integrals and avoided the divergence of the energy at the singular points associated with the defects. Miyazako et al. also estimated equilibrium distributions of the defects using a Markov chain Monte Carlo method and compared mousemyoblast (C2C12) cells experimental data [11]. Recently, this approach using line integrals has been extended to doubly connected domains [12] using the prime function machinery developed by Crowdy [13] relevant to solving problems involving multiply connected domains.

Recently, Miyoshi et al. [14] have derived explicit formulas for the Frank free energy associated with nematic liquid crystals with multiple topological defects confined in simply connected two-dimensional domains. This result can be seen as an extension of Čopar & Kos [15], who derived explicit expressions for the Frank free energy of defects in an infinite domain without any objects. Miyoshi et al. evaluated the line integrals in the free energy, up to $O (ϵ \log ϵ)$ , where $ϵ$ is a cut-off parameter that approximates the size of a defect core [14], by pointing out and exploiting a mathematical analogy with the Hamiltonians describing the dynamics of point vortex singularities; these Hamiltonians are regularized kinetic energy functionals that have become known in the field of vortex dynamics as Kirchhoff–Routh path functions [16–19]. Miyoshi et al. [14] derived formulas for the Frank free energy making use of conformal mappings from canonical pre-image domains to the physical domain of interest and involving the evaluation of line integrals around circles that enclose the defects and the boundary of the crystal domain. Equilibrium configurations for topological defects were then obtained by finding the extrema of this free energy by setting to zero derivatives of the free energy with respect to the pre-images of the topological defects in the conformal pre-image plane. In the literature on nematic liquid crystals to date, this direct minimization of the Frank free energy has been the tool used to obtain the positions of the equilibria of topological defects [7–10,20].

The aim of the present paper is to offer a new perspective. As will be shown here, the aforementioned analogy with point vortex dynamics and the Kirchhoff–Routh path function turns out to have even more ramifications for the theory of two-dimensional nematic liquid crystals. In particular, in the field of vortex dynamics, it is rare to calculate a point vortex equilibrium by calculating and then minimizing the Kirchhoff–Routh path function; in other words, it is rare in vortex dynamics to do what is always done for liquid crystals. Rather, the more usual approach in point vortex dynamics is to write down a complex potential for the fluid flow associated with a given point vortex distribution and, in terms of it, isolate a set of local conditions at each point vortex that are equivalent to that vortex being free of net force: it is therefore natural to refer to these as local force conditions. Saffman [16] gives a careful derivation of this local force condition at each vortex. Some of the vortex dynamics literature refers to this condition as the ‘non-self-induction rule’ since it involves subtracting off a singular term in the complex velocity (the very term causing the singular flow at the point vortex itself) and allowing that point vortex to move with the local velocity induced by anything surrounding it (e.g. other point vortices, or sources of flow not associated with the point vortex itself). Routh [21] is generally agreed to be the first to carefully establish this rule although it is implicit in many other works. Llewellyn Smith [22] has given a careful account of the long history of this local force condition in the field of vortex dynamics. The key contribution of the present paper is to demonstrate that there is an analogous set of equivalent local force conditions that can be used to find the equilibria of topological defects in nematic liquid crystals without having to compute the Frank free energy at all. This important observation is significant both theoretically and practically because, at a technical level, it is easier to write down and solve these equivalent local force conditions than to calculate and minimize the Frank free energy.

This paper is organized as follows. Section 2 introduces the mathematical formulation of the alignment angle field with topological defects and the Frank free energy. Section 3 reviews the recent work of Miyoshi et al. [14] who derived explicit formulas for the Frank free energy in terms of the defect locations. Sections 4 and 5 contain the main new results of this paper and derive the new local force conditions, in simply and doubly connected regions respectively, that are equivalent to finding extrema of the Frank free energy. These sections also include calculations confirming that previous results obtained using Frank energy minimization are retrieved by the new local force conditions. The paper closes with some general remarks in §6.

Figure 1 summarizes the contents of this paper, with the new contributions highlighted in blue for clarity. Using the analytical formulas for the energy, the equivalent local force conditions (4.4), (4.6) for simply connected domains and (5.17) and (5.18) for doubly connected domains that minimize the Frank free energy are derived. In §6, the local force on a defect is calculated using the residue of the squares of the derivatives of the complex functions associated with alignment angles.

Figure 1. Summary of the paper, with the new contributions highlighted in blue. Using the analytical formulas for the energy, we derive the equivalent local force conditions (4.4), (4.6) for simply connected domains and (5.17) and (5.18) for doubly connected domains that minimize the Frank free energy. In §6, the local force on a defect is calculated by evaluating the residue of the squares of the derivatives of the complex functions associated with alignment angles.

2. Topological defects and the Frank free energy

Nematic liquid crystals are assumed to be spindle-shaped and infinitely small relative to a characteristic length scale associated with the containing domain. They can thus be treated as a continuum of rod-like particles with that located at $(x, y)$ assumed to be aligned in the direction of unit vector $n_{d}$ as follows:

\begin{array}{ll} n_{d} = (\cos ϕ (x, y), \sin ϕ (x, y))^{⊤}, \end{array}

(2.1)

where $ϕ (x, y)$ is the alignment angle field for $(x, y)$ . Since $n_{d}$ and $- n_{d}$ are indistinguishable, the arbitrary angle $ϕ$ is equivalent to $ϕ \in [- π / 2, π / 2)$ [7]; that is, the angles $ϕ_{1}$ and $ϕ_{2}$ are equivalent if there exists an integer $n \in ℤ$ such that $ϕ_{1} = ϕ_{2} + n π$ .

Let $Ω$ now denote the two-dimensional liquid crystal domain viewed as a subdomain of a complex $z = x + i y$ -plane and let its boundary be denoted by $\partial Ω$ . There is no need for $Ω$ to be simply connected, indeed it will be allowed to have $M$ internal boundaries viewed as the boundaries of $M$ ‘holes’ in the liquid crystal domain. The alignment angle $ϕ$ satisfies the two-dimensional Laplace’s equation except at the defects based on the one-constant approximation of the Frank energy [6], and has a tangential anchoring boundary condition. Assume also that there exist $N$ topological defects at $z_{k} = x_{k} + i y_{k}$ , $k = 1, \dots, N$ , strictly inside $Ω$ , whose topological charges are $q_{k}$ . The charges are half-integer or full integer except $0$ ; that is,

\begin{array}{ll} q_{k} \in {n / 2 | n \in Z, n \neq 0} . \end{array}

(2.2)

The set of defect positions is denoted by $Z \equiv {z_{k} | k = 1,2, \dots, N}$ . In a domain $Ω$ , for any subdomain $D_{k} \subset Ω$ , $k = 1, \dots, N$ containing only the $k$ -th defect, the following holds:

\begin{array}{lrr} \oint_{\partial D_{k}} \frac{\partial ϕ}{\partial s} d s = 2 π q_{k} . \end{array}

(2.3)

Thus, on encircling the defect $z = z_{k}$ once, the alignment angle changes by $2 π q_{k}$ . The quantity (2.3) can be seen as the rotation number of a topological defect located at $z = z_{k}$ .

Let $ψ$ define the harmonic conjugate of $ϕ$ . Consider the following surface integral in a domain $Ω$ except subdomain $\cup_{k = 1}^{N} D_{k}$ :

\begin{array}{lrlr} 0 & = \int_{Ω \ (\cup_{k = 1}^{N} D_{k})} \nabla^{2} ψ (x, y) d x d y = \oint_{\partial Ω} \frac{\partial ψ}{\partial n} d s - \sum_{k = 1}^{N} \oint_{\partial D_{k}} \frac{\partial ψ}{\partial n} d s \\ = - \oint_{\partial Ω} \frac{\partial ϕ}{\partial s} d s + \sum_{k = 1}^{N} \oint_{\partial D_{k}} \frac{\partial ϕ}{\partial s} d s = 2 π (- 1 + M + \sum_{k = 1}^{N} q_{k}), \end{array}

(2.4)

where we used (2.3) and a tangential boundary condition on $ϕ$ on $\partial Ω$ . This result (2.4) suggests that the sum of charges is equal to $1 - M$ , i.e.

\begin{array}{lrr} \sum_{k = 1}^{N} q_{k} = 1 - M . \end{array}

(2.5)

A multiply connected generalization of Riemann’s mapping theorem due to Koebe (see [13]) guarantees the existence of a conformal map $z = g (ζ)$ from a pre-image domain $D_{ζ}$ in a complex $ζ = ξ + i η$ -plane to the multiply connected region $Ω$ in the $z$ -plane. A canonical choice for $D_{ζ}$ is a multiply connected circular domain comprising the unit $ζ$ -disc with $M$ smaller circular discs excised [13]. Let the conformal map $z = g (ζ)$ transplant $D_{ζ}$ to $Ω$ . The pre-images of defect positions $z_{k}$ under $g (ζ)$ are denoted as ${ζ_{k} | k = 1, \dots, N}$ so that

\begin{array}{lcr} z_{k} = g (ζ_{k}), k = 1, \dots, N . \end{array}

(2.6)

Using the above formulation for a simply connected $Ω$ for which $D_{ζ}$ is just the unit disc (no holes), Miyazako & Nara [7] derived an explicit formula for the alignment angle field $ϕ (x, y)$ . When $Ω = D_{ζ}$ , so that the conformal map is just the identity map, the alignment angle field is given by

\begin{array}{lrr} ϕ (ξ, η) = Re [f_{D_{ζ}} (ζ)], f_{D_{ζ}} (ζ) = \sum_{k = 1}^{N} q_{k} G (ζ, ζ_{k}) - \frac{π}{2}, \end{array}

(2.7)

where

\begin{array}{lrr} G (ζ, ζ_{k}) \equiv - i (\log (ζ - ζ_{k}) + \log (1 - ζ_{k} ζ)), \end{array}

(2.8)

where $ζ_{k}$ is the complex conjugate of the complex number $ζ_{k}$ . For an arbitrarily shaped domain $Ω$ , when $z = g (ζ)$ is no longer trivial, the complex function $f_{Ω} (z)$ in $Ω$ must be modified slightly to

f_{Ω} (z) = f_{D_{ζ}} (ζ) - i l o g g^{'} (ζ) = f_{D_{ζ}} (g^{- 1} (z)) - i l o g g^{'} (g^{- 1} (z)),' \equiv \frac{d}{d ζ} .

(2.9)

The sum of topological charges $q_{k}$ should satisfy condition (2.5) with $M = 0$ .

3. Frank free energy formula in simply connected domains

The positions of static topological defects $z_{k}$ are determined where the energy is a local minimum [6,7]. Miyazako & Nara [7] used the complex form of Green’s theorem to represent the energy as a line integral as follows:

\begin{aligned} (2) & F = \frac{K}{2} \int_{Ω (ϵ)} | \nabla ϕ (x, y) |^{2} d x d y = \frac{K}{4 i} \oint_{\partial Ω (ϵ)} \frac{\partial f_{Ω}}{\partial z} \bar{f_{Ω}} d z, \frac{\partial}{\partial z} \equiv \frac{1}{2} (\frac{\partial}{\partial x} - i \frac{\partial}{\partial y}), \end{aligned}

(3.1)

where $Ω (ϵ) \equiv Ω \ D (ϵ)$ , $D (ϵ) \equiv \cup_{k = 1}^{N} D_{k} (ϵ)$ and $D_{k} (ϵ)$ is a small disc around the $k$ -th defect with radius $ϵ$ , defined by

\begin{array}{ll} D_{k} (ϵ) \equiv {z_{k} + r e^{i θ} | 0 \leq r \leq ϵ, 0 \leq θ < 2 π}, k = 1, \dots, N . \end{array}

(3.2)

The regions $D_{k} (ϵ)$ are excluded when calculating the energy (3.1) in order that the energy is not unbounded. The integration path in a simply connected domain is shown on the left side of figure 2. Since $F$ is conformally invariant, the energy $F$ can be evaluated using the line integral along the unit circle with the boundaries of the excluded zones $\tilde{D} (ϵ)$ in the unit disc. Using the line integral formula (3.1), Miyazako et al. [7] numerically confirmed that in the case of the unit disc, the energy is minimized when defects exist at positions $z = \pm 5^{- 1 / 4}$ . This result was initially obtained by Duclos et al. [6].

In figure 2, $l_{k}^{+}$ , $l_{k}^{-}$ denote the paths of the line integral from $\partial D_{ζ}$ to $\partial {\tilde{D}}_{k} (ϵ)$ used to avoid the jumps of the branch cuts of $\log (ζ - ζ_{k})$ . This is crucial as the function $Re [f_{Ω}]$ differs by $2 π q_{k}$ across the branch cut and thus the energy remains undefined unless we select a path that does not intersect the branch cut. The explanation for the integral path for doubly connected domain is given in appendix A and for simply connected domain in [14].

Miyoshi et al. [14] later derived an explicit formula for the Frank free energy for the unit disc and arbitrarily shaped domains up to $O (ϵ \log ϵ)$ . By evaluating the contour integral as shown on the left side of figure 2, they showed that the energy for the unit disc that excludes small regions $D (ϵ) \equiv \cup_{k = 1}^{N} D_{k} (ϵ)$ is given by

\begin{array}{ll} F = \frac{K}{2} (- 2 π \sum_{k = 1}^{N} q_{k}^{2} \log ϵ + 2 π F_{d}) + O (ϵ \log ϵ), \end{array}

(3.3)

where the first term in the brackets depends only on $q_{k}$ and $ϵ$ (and not the defect locations) while the second term is given by

\begin{array}{ll} F_{d} = \sum_{k = 1}^{N} q_{k}^{2} I m [R (ζ_{k}, ζ_{k})] + \sum_{k = 1}^{N} \sum_{l \neq k}^{N} q_{k} q_{l} I m [G (ζ_{k}, ζ_{l})], \end{array}

(3.4)

where $G (., .)$ is given in equation (2.8) and

\begin{array}{lrr} R (ζ, ζ_{k}) \equiv G (ζ, ζ_{k}) + i \log (ζ - ζ_{k}) . \end{array}

(3.5)

The function $R (ζ, ζ_{k})$ is analytic at $ζ = ζ_{k}$ and is single-valued. It is obtained by removing the contribution of the logarithmic singularity at $ζ = ζ_{k}$ .

Miyoshi et al. [14] also calculated the energy for an arbitrarily shaped simply connected domain $Ω$ mapped from the unit disc $D_{ζ}$ by a conformal mapping $z = g (ζ)$ . They assumed that the boundary of the domain is sufficiently smooth and $g^{'} (ζ) \neq 0$ for $ζ \in \partial D_{ζ}$ . The total energy $F$ for the region $Ω \ D (ϵ)$ is written as follows [14]:

\begin{array}{ll} F = \frac{K}{2} (- 2 π \sum_{k = 1}^{N} q_{k}^{2} \log ϵ + 2 π F_{d} + F_{g} + O (ϵ \log ϵ)), \end{array}

(3.6)

where

\begin{array}{ll} F_{d} \equiv \sum_{k = 1}^{N} (q_{k}^{2} - 2 q_{k}) \log | g^{'} (ζ_{k}) | + \sum_{k = 1}^{N} q_{k}^{2} I m [R (ζ_{k}, ζ_{k})] + \sum_{k = 1}^{N} \sum_{l \neq k}^{N} q_{k} q_{l} I m [G (ζ_{k}, ζ_{l})], \end{array}

(3.7)

\begin{array}{lrlr} F_{g} \equiv \int_{D_{ζ}} {| \frac{g^{''} (ζ)}{g^{'} (ζ)} |}^{2} d x d y + 4 π \log | g^{'} (0) | . \end{array}

(3.8)

Now the energy $F_{d}$ depends both on $ζ_{k}$ and on the conformal map $g (ζ)$ , whereas $F_{g}$ is independent of $ζ_{k}$ . To obtain the extrema of $F$ , only the derivatives of $F_{d}$ with respect to $ζ_{k}$ need to be considered; that is

\begin{array}{lrr} \frac{\partial F_{d}}{\partial ζ_{k}} = 0, k = 1, \dots, N . \end{array}

(3.9)

The equilibria for topological defects are obtained by finding ${ζ_{k} | k = 1, . . ., N}$ satisfying (3.9). A similar minimization technique was applied to find the locations of topological defects by Chandler & Spagnolie [9] and Duclos et al. [6].

An important observation is that $Im [G (ζ_{k}, ζ_{l})]$ and $Im [R (ζ_{k}, ζ_{l})]$ satisfy reciprocity conditions; that is

\begin{array}{lrlr} Im [G (ζ_{k}, ζ_{l})] = - (\log | ζ_{k} - ζ_{l} | + \log | 1 - ζ_{l} ζ_{k} |) = Im [G (ζ_{l}, ζ_{k})], \end{array}

(3.10)

\begin{array}{lrlr} Im [R (ζ_{k}, ζ_{l})] = - \log | 1 - ζ_{l} ζ_{k} | = Im [R (ζ_{l}, ζ_{k})] . \end{array}

(3.11)

Using (3.10) and (3.11), the following relations can be established:

\begin{array}{ll} {\frac{\partial R}{\partial ζ} (ζ, ζ_{k}) |}_{ζ = ζ_{k}} = i \frac{\partial}{\partial ζ_{k}} I m [R (ζ_{k}, ζ_{k})], {\frac{\partial G}{\partial ζ} (ζ, ζ_{l}) |}_{ζ = ζ_{k}} = 2 i \frac{\partial}{\partial ζ_{k}} I m [G (ζ_{k}, ζ_{l})] . \end{array}

(3.12)

These properties will be important in the next section.

4. Local force conditions for nematic liquid crystals

This section contains the main new results of this paper and builds further on connections to point vortex dynamics and the Kirchhoff–Routh path function pointed out by Miyoshi et al. [14]. To give context to the following results it is worth surveying some results from the theory of point vortex motion.

The reciprocity relations (3.10) and (3.11) written down at the end of the previous section are reminiscent of similar reciprocity relations satisfied by the so-called hydrodynamic Green’s function [13,17–19,23] relevant when studying the dynamics of point vortices in two-dimensional fluid domains. For a simply connected domain, the hydrodynamic Green’s function coincides with the standard first-type Green’s function in potential theory: the function harmonic everywhere in the domain except for a single logarithmic singularity and vanishing everywhere on the domain boundary. In a multiply connected domain, the hydrodynamic Green’s function also has a single logarithmic singularity, vanishes on the outer boundary and has constant non-zero values on the internal boundaries of the holes; these values are chosen in order to ensure zero circulation conditions around the holes. The problem of point vortex dynamics in free space (no boundaries) was shown to be Hamiltonian by Kirchhoff [16] with the relevant Hamiltonian given by a regularized kinetic energy of the fluid. This regularized kinetic energy is akin to the regularized Frank free energy discussed in the previous section where small cut-off regions around the defects are incorporated in order to render the free energy finite. Lin [17,18] later showed that the motion of a finite set of point vortices in an arbitrary multiply connected domain is also Hamiltonian; he proved this using the existence of the hydrodynamic Green’s function in a multiply connected domain and wrote down the Hamiltonian in terms of this Green’s function and its finite part (or Robin function), as well as the conformal mapping from a canonical region. Lin’s work was of an abstract nature and features no example calculations of point vortex trajectories. Crowdy & Marshall [19] later made Lin’s abstract theory constructive by finding explicit expressions for the hydrodynamic Green’s function in the class of multiply connected circular domains $D_{ζ}$ mentioned earlier. The main tool in the theory of Crowdy & Marshall [19] is the prime function associated with $D_{ζ}$ ; this is a very general mathematical object that can be associated with any multiply connected domain and has a wide range of applications across the physical sciences [13]. Using this mathematical framework it is easy to exhibit a wide range of point vortex trajectories [19,24,25].

It is important to point out that while Kirchhoff–Routh theory can be used to write down the Hamiltonians (or Kirchhoff–Routh path functions) associated with point vortex motion in bounded multiply connected domains, it is not necessary to use it and, as mentioned earlier, there are local force conditions at each point vortex that will produce the same dynamics. Restricting interest now to equilibria, which means in the point vortex problem that the arrangement of point vortices sits in steady equilibrium without moving, one can choose either to minimize the Kirchhoff–Routh path function (i.e. find where its partial derivatives with respect to all the point vortex locations vanishes) to find the equilibrium configuration or one can choose to write down and solve the local force conditions. Both will lead to the same point vortex equilibria but are mathematically very different processes.

The key point is that there is an analogy between point vortex dynamics and the theory of topological defects in nematic liquid crystals and, so far, all studies to find defect equilibria rely on minimizing the Frank free energy (corresponding to solving point vortex problems using Kirchhoff–Routh path functions). But there are also analogous local force conditions in the liquid crystal problem that can be used instead, and these appear to have gone unnoticed until now.

To make all this more precise, and to actually construct the relevant local force conditions in the nematic liquid crystal problem, it is convenient to define a function ${\hat{f}}_{k} (ζ)$ that is a complex function $f_{D_{ζ}} (ζ)$ except for a logarithmic singularity at $ζ = ζ_{k}$ ; that is

\begin{array}{ll} {\hat{f}}_{k} (ζ) \equiv f_{D_{ζ}} (ζ) + i q_{k} \log (ζ - ζ_{k}) . \end{array}

(4.1)

In the presence of multiple vortices, the equilibria of point vortices can be obtained under the condition that the vortex speed at the point vortex is equal to zero. For general geometries, it is necessary to consider an additional term that comes from conformal mappings. The additional term under conformal mappings is obtained from a local analysis around $ζ = ζ_{k}$ . We prove that the same technique can be used for finding the energy extrema in the presence of topological defects in the unit disc and arbitrarily shaped domains.

(a) Local force conditions in the unit disc

In this case, the complex function and the energy are given by (2.7) and (3.3), respectively. Taking the derivative of $F$ in (3.3) with respect to $ζ_{k}$ gives

\begin{array}{ll} \frac{\partial F}{\partial ζ_{k}} = \frac{K}{2} \cdot 2 π q_{k} (q_{k} \frac{\partial}{\partial ζ_{k}} I m [R (ζ_{k}, ζ_{k})] + 2 \sum_{l \neq k}^{N} q_{l} \frac{\partial}{\partial ζ_{k}} I m [G (ζ_{k}, ζ_{l})]), \end{array}

(4.2)

where the reciprocity condition $Im [G (ζ_{k}, ζ_{l})] = Im [G (ζ_{l}, ζ_{k})]$ is used. The derivative of (4.1) with respect to $ζ$ evaluated at $ζ = ζ_{k}$ gives

\begin{array}{ll} {\frac{\partial {\hat{f}}_{k}}{\partial ζ} |}_{ζ = ζ_{k}} & = q_{k} {\frac{\partial R}{\partial ζ} (ζ, ζ_{k}) |}_{ζ = ζ_{k}} + \sum_{l \neq k}^{N} q_{l} {\frac{\partial G}{\partial ζ} (ζ, ζ_{l}) |}_{ζ = ζ_{k}} \\ = i (q_{k} \frac{\partial}{\partial ζ_{k}} I m [R (ζ_{k}, ζ_{k})] + 2 \sum_{l \neq k}^{N} q_{l} \frac{\partial}{\partial ζ_{k}} I m [G (ζ_{k}, ζ_{l})]), \end{array}

(4.3)

where the reciprocity conditions (3.10) and (3.11) are used. From (4.3) and (4.2),

\begin{array}{ll} \frac{\partial F}{\partial ζ_{k}} = - i π q_{k} K {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}} . \end{array}

(4.4)

This relation is important: it means that the extrema of the free energy, i.e. solving for $\partial F / \partial ζ_{k} = 0$ , can be obtained by finding zeros of the derivatives of the set of regularized complex functions ${\hat{f}}_{k} (ζ)$ at each of the defect locations. This is entirely analogous to the situation in the field of vortex dynamics where this equivalence is better known.

(b) Local force conditions for arbitrary simply connected domains

Using a conformal map $z = g (ζ)$ from $ζ$ to $z$ , the complex function for the alignment angle is given by (2.9). The derivative of the energy formula (3.6) with respect to $ζ_{k}$ gives

\begin{array}{ll} \frac{\partial F}{\partial ζ_{k}} & = \frac{K}{2} \cdot 2 π (\frac{1}{2} (q_{k}^{2} - 2 q_{k}) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + q_{k}^{2} \frac{\partial}{\partial ζ_{k}} I m [R (ζ_{k}, ζ_{k})] + 2 q_{k} \sum_{l \neq k}^{N} q_{l} \frac{\partial}{\partial ζ_{k}} I m [G (ζ_{k}, ζ_{l})]) \\ = π q_{k} K \cdot ((\frac{q_{k}}{2} - 1) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + q_{k} \frac{\partial}{\partial ζ_{k}} I m [R (ζ_{k}, ζ_{k})] + 2 \sum_{l \neq k}^{N} q_{l} \frac{\partial}{\partial ζ_{k}} I m [G (ζ_{k}, ζ_{l})]) . \end{array}

(4.5)

Through direct calculation, it can be demonstrated that

\begin{array}{ll} \frac{\partial F}{\partial z_{k}} = - i π q_{k} K {\frac{\partial {\hat{f}}_{Ω, k}}{\partial z} (z) |}_{z = z_{k}} = - \frac{i π q_{k} K}{g^{'} (ζ_{k})} (i (\frac{q_{k}}{2} - 1) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}}), \end{array}

(4.6)

where ${\hat{f}}_{k} (ζ)$ is defined in (4.1) and

\begin{array}{lrr} {\hat{f}}_{Ω, k} (z) \equiv f_{Ω} (z) + i q_{k} \log (z - z_{k}) . \end{array}

(4.7)

The derivation of (4.6) is explained in appendix A.

Therefore, to find the equilibria of topological defects, it is sufficient to find the set of ${ζ_{k}}$ that satisfies the set of local force conditions

\begin{array}{ll} {\hat{V}}_{k} (ζ_{k}) = \frac{1}{g^{'} (ζ_{k})} (i (\frac{q_{k}}{2} - 1) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + {\frac{\partial {\hat{f}}_{k}}{\partial ζ} |}_{ζ = ζ_{k}}) = 0, k = 1, \dots, N . \end{array}

(4.8)

It is not necessary to deal with the Frank free energy directly. We use this fact to find the energy extrema of some examples in the next subsections and to corroborate the results by cross-checking with previous work that has used the Frank free energy.

(c) Example 1: equilibria in a unit disc

First let $Ω$ be the unit disc in the complex $ζ$ -plane. Under the assumption that the configuration of static topological defects is symmetrical, there exist two topological defects with a charge of $+ 1 / 2$ located at $ζ_{1} = + r$ and $ζ_{2} = - r$ , $0 < r < 1$ , $r \in ℝ$ , respectively. The result (4.3) implies that a local condition for equilibrium is

\begin{array}{ll} {\frac{\partial {\hat{f}}_{1}}{\partial ζ} |}_{ζ = ζ_{1}} = i [\frac{r}{2 (1 - r^{2})} - \frac{1}{4 r} - \frac{r}{2 (1 + r^{2})}] = i \frac{5 r^{4} - 1}{4 (1 - r^{4})} = 0, \end{array}

(4.9)

with a similar condition holding, by symmetry, at $ζ_{2}$ . This clearly gives $r = 5^{- 1 / 4}$ , a result verified experimentally by Duclos et al. [6]. It is important to emphasize that this result has been retrieved here without any consideration of the Frank free energy.

(d) Example 2: equilibria in a quadrature domain

A doublet or triplet [4,14] is the name given to a domain made up of multiple intersecting discs. In Chapter 11 of [13] Crowdy shows how a class of domains known as quadrature domains can be viewed geometrically as collections of circular discs that have merged so their contact points have been smoothed out into thin necks. A doublet domain $Ω$ can therefore be approximated by the following conformal map from a unit $ζ$ disc to a simply connected quadrature domain:

\begin{array}{lrr} g (ζ) = \frac{R ζ}{ζ^{2} - a^{2}}, a > 1, g^{'} (ζ) = \frac{R}{ζ^{2} - a^{2}} - \frac{2 R ζ^{2}}{(ζ^{2} - a^{2})^{2}} = - \frac{R (ζ^{2} + a^{2})}{(ζ^{2} - a^{2})^{2}}, \end{array}

(4.10)

where

\begin{array}{lrr} R = \frac{a^{4} - 1}{1 - a + a^{2}} \end{array}

(4.11)

is chosen so that $Ω$ is approximately a domain made up of two merged discs with radius $1$ . Due to the symmetry of $Ω$ , assume that there exist topological defects with a charge of $+ 1 / 2$ located at $ζ_{1} = r$ and $ζ_{2} = - r$ , $0 < r < 1$ in the $ζ$ -plane. From (4.6) it follows that the equilibrium condition at $ζ = ζ_{1}$ is

\begin{array}{lrlr} i (\frac{q_{1}}{2} - 1) \frac{g^{''} (ζ_{1})}{g^{'} (ζ_{1})} + {\frac{\partial {\hat{f}}_{1}}{\partial ζ} |}_{ζ = ζ_{1}} & = - \frac{i}{2} [- \frac{ζ_{1}}{1 - ζ_{1} ζ_{1}} + \frac{1}{ζ_{1} - ζ_{2}} - \frac{ζ_{2}}{1 - ζ_{2} ζ_{1}}] - \frac{3 i}{4} \frac{g^{''} (ζ_{1})}{g^{'} (ζ_{1})} \\ = - \frac{i}{2} \cdot \frac{1 - 5 r^{4}}{2 r (1 - r^{4})} - \frac{3 i}{4} \frac{g^{''} (r)}{g^{'} (r)} = 0, \end{array}

(4.12)

with a similar condition derived at $ζ_{2}$ . This gives the following condition to determine $r$ :

\begin{aligned} \frac{1 - 5 r^{4}}{2 r (1 - r^{4})} - \frac{3 r (r^{2} + 3 a^{2})}{r^{4} - a^{4}} = 0 \end{aligned},

(4.13)

which coincides with a polynomial formula obtained by considering the derivative of the free energy as considered recently in [14].

5. Local force conditions for annular regions

The local force conditions are independent of domain connectivity so it is possible to extend the approach to domains with higher connectivity. To demonstrate this requires showing that imaginary parts of the functions $G (., .)$ and $R (., .)$ satisfy the same reciprocity conditions as for the simply connected case; see (5.7) and (5.8) to follow.

Following Miyazako & Sakajo [7], the following analytical formula for the complex function associated with the alignment angle field with topological defects $N$ located at $ζ = ζ_{k}$ , $k = 1, \dots, N$ is given, as a function of a variable $ζ$ in a concentric annular region $D_{ζ} = {ζ | ρ < | ζ | < 1}$ , by

\begin{array}{lrr} f_{D_{ζ}} (ζ) = \sum_{k = 1}^{N} q_{k} G (ζ, ζ_{k}) + (- i + \frac{Q_{n}}{\log ρ}) \log ζ - \frac{π}{2}, \end{array}

(5.1)

where

\begin{array}{lrr} G (ζ, ζ_{k}) \equiv - i (\log P (ζ / ζ_{k}, ρ) + \log P (ζ ζ_{k}, ρ) + \log (- ζ_{k})), \end{array}

(5.2)

where $P (., .)$ is closely related to the prime function for $D_{ζ}$ [13]. One explicit infinite product representation of it (there are others—see [13]) is

\begin{array}{lrr} P (ζ, ρ) = (1 - ζ) \hat{P} (ζ, ρ), \hat{P} (ζ, ρ) \equiv \prod_{n = 1}^{\infty} (1 - ρ^{2 n} ζ) (1 - ρ^{2 n} ζ^{- 1}) . \end{array}

(5.3)

Due to the existence of the inner region, the sum of the topological charges $q_{k}$ should satisfy

\begin{array}{lrr} \sum_{k = 1}^{N} q_{k} = 0 . \end{array}

(5.4)

This is (2.5) with $M = 1$ . The value $Q_{n}$ corresponds to the circulation (torque) around the inner boundary, which is calculated using an arbitrary integer $n$ and the arguments of $ζ_{k}$ as follows:

\begin{array}{ll} Q_{n} = n π - \sum_{k = 1}^{N} q_{k} I m [\log ζ_{k}] . \end{array}

(5.5)

It is easy to prove that the function (5.1) satisfies the tangential boundary condition on both inner and outer boundaries.

Note that the real part of $- i \log ζ$ in (5.1) has a branch cut between the inner and outer boundaries. However, both the director field and alignment angle in annular regions is continuous as the angles $ϕ_{1}$ and $ϕ_{2}$ are equivalent if there exists an integer $m \in ℤ$ such that $ϕ_{1} = ϕ_{2} + m π$ , as mentioned in §2.

For simplicity, define a function $R (ζ, ζ_{k})$ that is analytic at $ζ = ζ_{k}$ as follows:

\begin{array}{lrr} R (ζ, ζ_{k}) \equiv G (ζ, ζ_{k}) + i \log (ζ - ζ_{k}) = - i (\log \hat{P} (ζ / ζ_{k}, ρ) + \log P (ζ ζ_{k}, ρ)) . \end{array}

(5.6)

Similar to the case of a simply connected domain, the imaginary parts of the functions $G (., .)$ and $R (., .)$ satisfy the following reciprocity relations:

\begin{array}{lrlr} Im [G (ζ_{l}, ζ_{k})] & = - (\log | P (ζ_{l} / ζ_{k}, ρ) | + \log | P (ζ_{l} ζ_{k}, ρ) | + \log | ζ_{k} |) \\ = - (\log | ζ_{l} / ζ_{k} \cdot P (ζ_{k} / ζ_{l}, ρ) | + \log | P (ζ_{k} ζ_{l}, ρ) | + \log | ζ_{k} |) = Im [G (ζ_{k}, ζ_{l})], \end{array}

(5.7)

\begin{array}{lrlr} Im [R (ζ_{l}, ζ_{k})] & = - (\log | \hat{P} (ζ_{l} / ζ_{k}, ρ) | + \log | P (ζ_{l} ζ_{k}, ρ) |) = Im [R (ζ_{k}, ζ_{l})], \end{array}

(5.8)

where the following property of the $P (., .)$ functions [13] are used:

\begin{array}{ll} P (ζ^{- 1}, ρ) = - ζ^{- 1} P (ζ, ρ), \hat{P} (ζ^{- 1}, ρ) = \hat{P} (ζ, ρ) . \end{array}

(5.9)

Due to the reciprocity conditions (5.7) and (5.8), it can be seen that

\begin{array}{lrr} {\frac{\partial R}{\partial ζ} (ζ, ζ_{k}) |}_{ζ = ζ_{k}} = i \frac{\partial}{\partial ζ_{k}} Im [R (ζ_{k}, ζ_{k})], {\frac{\partial G}{\partial ζ} (ζ, ζ_{l}) |}_{ζ = ζ_{k}} = 2 i \frac{\partial}{\partial ζ_{k}} Im [G (ζ_{k}, ζ_{l})] . \end{array}

(5.10)

The Frank free energy in annular regions is calculated in a similar manner to the simply connected case except on a branch cut between the inner and outer boundaries. The path of the contour integral is shown on the right in figure 2. The branch cut between $C_{0}$ and $C_{1}$ must be selected such that it does not intersect the cuts between $\partial {\tilde{D}}_{k}$ and $C_{0}$ . The integral path is also chosen not to intersect the branch cut, as mentioned in appendix A. Using the same method as that used for simply connected domains in [14] and considering the effect on the branch cut, the Frank free energy for $D_{ζ}$ excluding small discs $D_{k} (ϵ)$ with radius $ϵ$ around $ζ = ζ_{k}$ is given by

\begin{array}{ll} F = \frac{K}{2} (- 2 π \sum_{k = 1}^{N} q_{k}^{2} \log ϵ + 2 π F_{d} + O (ϵ \log ϵ)), \end{array}

(5.11)

where

\begin{array}{ll} F_{d} & = - (1 + \frac{Q_{n}^{2}}{(\log ρ)^{2}}) \log ρ - 2 \sum_{k = 1}^{N} q_{k} R e [\log ζ_{k}] \\ + \sum_{k = 1}^{N} q_{k}^{2} I m [R (ζ_{k}, ζ_{k})] + \sum_{k = 1}^{N} \sum_{l \neq k}^{N} q_{k} q_{l} I m [G (ζ_{k}, ζ_{l})] . \end{array}

(5.12)

We also consider an arbitrarily shaped doubly connected domain $Ω$ in the $z$ -plane mapped from the annular region $D_{ζ}$ by conformal mapping $z = g (ζ)$ , where the boundary of the domain is sufficiently smooth. Using the same technique as that used for simply connected domains (2.9), the complex function for the doubly connected domains $f_{Ω} (z)$ is given by

\begin{array}{lrr} f_{Ω} (z) = f_{D_{ζ}} (ζ) - i \log g^{'} (ζ) = f_{D_{ζ}} (g^{- 1} (z)) - i \log g^{'} (g^{- 1} (z)) . \end{array}

(5.13)

The total energy $F$ of the region $Ω \ D (ϵ)$ , $D (ϵ) \equiv \cup_{k = 1}^{N} D_{k} (ϵ)$ , can be written as follows:

\begin{array}{ll} F = \frac{K}{2} (- 2 π \sum_{k = 1}^{N} q_{k}^{2} \log ϵ + 2 π F_{d} + F_{g} + O (ϵ \log ϵ)), \end{array}

(5.14)

where

\begin{array}{ll} F_{d} & \equiv - (1 + \frac{Q_{n}^{2}}{(\log ρ)^{2}}) \log ρ - 2 \sum_{k = 1}^{N} q_{k} R e [\log ζ_{k}] \\ + \sum_{k = 1}^{N} q_{k}^{2} I m [R (ζ_{k}, ζ_{k})] + \sum_{k = 1}^{N} \sum_{l \neq k}^{N} q_{k} q_{l} I m [G (ζ_{k}, ζ_{l})] + \sum_{k = 1}^{N} (q_{k}^{2} - 2 q_{k}) \log | g^{'} (ζ_{k}) |, \end{array}

(5.15)

\begin{array}{ll} F_{g} & \equiv \int_{D_{ζ}} {| \frac{g^{″} (ζ)}{g^{'} (ζ)} |}^{2} d S . \end{array}

(5.16)

The detailed derivations of (5.12) and (5.14) are given in appendix B.

Using the expression (5.12), it can be shown that for a concentric annulus,

\begin{array}{ll} \frac{\partial F}{\partial ζ_{k}} = - i π q_{k} K {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}}, \end{array}

(5.17)

where the regularized function ${\hat{f}}_{k} (ζ) \equiv f_{D_{ζ}} (ζ) + i q_{k} \log (ζ - ζ_{k})$ with $f_{D_{ζ}} (ζ)$ defined in (5.1). For arbitrarily shaped domains, by direct calculations of the derivative of ${\hat{f}}_{Ω, k} (z) \equiv f_{Ω} (z) + i q_{k} \log (z - z_{k})$ and of the energy (5.14), it can be shown that

\begin{array}{ll} \frac{\partial F}{\partial z_{k}} = - i π q_{k} K {\frac{\partial {\hat{f}}_{Ω, k}}{\partial z} (z) |}_{z = z_{k}} . \end{array}

(5.18)

The derivations are explained in appendices C and D. The equations (5.17) and (5.18) are the equivalent local force conditions that minimize the energy for doubly connected domains. Some examples are given in the next subsection.

(a) Example 3: equilibria of pairs of topological defects in annular regions

An important check is to verify that the local force conditions in a concentric annulus just derived retrieve the same equilibria as computed recently by Miyazako & Sakajo [12] by minimizing the Frank free energy. The first case is where the $- 1 / 2$ defect is located at $ζ = r_{1} \in ℝ$ and the $+ 1 / 2$ defect is located at $ζ = r_{2} \in ℝ$ , where $ρ < r_{1} < r_{2} < 1$ in concentric annular regions with inner radius $ρ$ [12]. It is convenient to define the derivative of the logarithm of $P (., .)$ with respect to $ζ$ :

\begin{array}{ll} K (ζ, ρ) \equiv ζ \frac{\partial}{\partial ζ} \log P (ζ, ρ) = - \frac{ζ}{1 - ζ} + \sum_{n = 1}^{\infty} [- \frac{ρ^{2 n} ζ}{1 - ρ^{2 n} ζ} + \frac{ρ^{2 n} ζ^{- 1}}{1 - ρ^{2 n} ζ^{- 1}}], \end{array}

(5.19)

\begin{array}{ll} \hat{K} (ζ, ρ) \equiv ζ \frac{\partial}{\partial ζ} \log \hat{P} (ζ, ρ) = \sum_{n = 1}^{\infty} [- \frac{ρ^{2 n} ζ}{1 - ρ^{2 n} ζ} + \frac{ρ^{2 n} ζ^{- 1}}{1 - ρ^{2 n} ζ^{- 1}}] . \end{array}

(5.20)

To find the equilibria of these defects, the locations $ζ = r_{1}$ and $ζ = r_{2}$ are found based on the local force conditions, that is,

\begin{aligned} \frac{\partial {\hat{f}}_{1}}{\partial ζ} (r_{1}) = i (\frac{K (r_{1}^{2}, ρ)}{2 r_{1}} - \frac{1}{2 r_{1}} [K (r_{1} / r_{2}, ρ) + K (r_{1} r_{2}, ρ)] - \frac{1}{r_{1}}) = 0, \end{aligned}

(5.21)

\begin{aligned} \frac{\partial {\hat{f}}_{2}}{\partial ζ} (r_{2}) = i (- \frac{K (r_{2}^{2}, ρ)}{2 r_{2}} + \frac{1}{2 r_{2}} [K (r_{2} / r_{1}, ρ) + K (r_{2} r_{1}, ρ)] - \frac{1}{r_{2}}) = 0, \end{aligned}

(5.22)

where the fact that $\hat{K} (1, ρ) = 0$ has been used. The locations $r_{1}$ and $r_{2}$ can be found using a standard nonlinear solver.

Figure 3 shows the equilibria of topological defects in annular regions with $ρ = 0.1$ , $ρ = 0.15$ and the colour maps of alignment angles $Re [f_{D_{ζ}} (ζ)]$ . The locations of the topological defects were calculated from the conditions (5.21) and (5.22) for $ρ = 0.1$ and $ρ = 0.15$ . Figure 4 shows the values of $r_{1}$ and $r_{2}$ . It is numerically confirmed that when $ρ$ is greater than approximately $0.1530$ , there is no equilibrium for defect pairs at locations $r_{1}$ and $r_{2}$ . This transition was calculated by Miyazako & Sakajo [12]. The numerical results obtained using the local force conditions for annular regions are consistent with their results (when a parameter $n$ in their notation is chosen to be zero).

annular region with pair of topological defects. — Figure 3. Colour maps of $Re [f_{D_{ζ}} (ζ)]$ in an annular region with a pair of topological defects. A defect with a charge of $- 1 / 2$ is located at $ζ = r_{1}$ and a defect with a charge of $+ 1 / 2$ is located at $ζ = r_{2}$ , where $ρ < r_{1} < r_{2} < 1$ . The locations of topological defects can be using the conditions (5.21) and (5.22). When $ρ = 0.1$ and $ρ = 0.15$ , the two defects are in equilibrium as the conditions are satisfied at these locations. When $ρ ≳ 0.1530$ , our numerical calculation shows that no equilibria can be found.

Figure 4. The values of $r_{1}$ and $r_{2}$ that satisfy local force conditions in an annular region with a pair of topological defects. The values $r_{1}$ and $r_{2}$ are found from conditions (5.21) and (5.22). It is found that no equilibrium can be obtained for a pair of defects located at $ζ = r_{1}$ and $ζ = r_{2}$ when $ρ ≳ 0.1530$ . The results obtained using the local force conditions are the same as those reported by Miyazako & Sakajo [12].

It is important to check that when $ρ \to 0$ and $r_{1} \to 0$ , the location of the other defect $r_{2}$ is determined by the equilibrium of only one defect with a charge of $+ 1 / 2$ in the presence of a defect with a charge of $+ 1 / 2$ fixed at $r_{1} = 0$ . The local force condition at $ζ = r_{2}$ is

\begin{aligned} {\frac{\partial {\hat{f}}_{2}}{\partial ζ} |}_{ζ = r_{2}} = i (\frac{r_{2}}{2 (1 - r_{2}^{2})} - \frac{1}{2 r_{2}}) = i (\frac{2 r_{2}^{2} - 1}{2 r_{2} (1 - r_{2}^{2})}) = 0, \end{aligned}

(5.23)

which gives $r_{2} = 1 / \sqrt{2}$ . This is consistent with the numerical results shown in figure 4 and the value predicted by Miyazako & Sakajo [12].

The second case is where two $- 1 / 2$ defects are located at $ζ = r_{1}, - r_{1}$ , respectively, and two $+ 1 / 2$ defects are located at $ζ = r_{2}, - r_{2}$ , respectively, where $ρ < r_{1} < r_{2} < 1$ . The third case is where three $- 1 / 2$ defects are located at $ζ = r_{1}, r_{1} ω, r_{1} ω^{2}$ , respectively, and three $+ 1 / 2$ defects are located at $ζ = r_{2}, r_{2} ω, r_{2} ω^{2}$ , respectively, where $ρ < r_{1} < r_{2} < 1$ and $ω \equiv \exp (2 π i / 3)$ . Both cases were calculated by Miyazako & Sakajo using the contour integral for the free energy [12]. To find the equilibria of these defects, we consider the equivalent local force conditions. Figure 5 shows the results for two pairs of topological defects. When $ρ = 0.1$ and $ρ = 0.14$ , four defects are in equilibrium, as equilibrium conditions are satisfied at these locations. However, when $ρ ≳ 0.144$ , no solutions for $r_{1}$ and $r_{2}$ can be found so no equilibrium exists in this case. This result is numerically verified as shown in the colour map on the right side of figure 5. We have also confirmed that it is impossible to obtain any formations of equilibria for the three pairs of $+ 1 / 2$ and $- 1 / 2$ topological defects with threefold symmetry as shown in figure 6.

annular region with two pairs. — Figure 5. Colour maps of $Re [f_{D_{ζ}} (ζ)]$ in an annular region with two pairs of $+ 1 / 2$ and $- 1 / 2$ topological defects. Two defects with a charge of $- 1 / 2$ are located at $ζ = r_{1}, - r_{1}$ , and two defects with a charge of $+ 1 / 2$ are located at $ζ = r_{2}, - r_{2}$ , where $ρ < r_{1} < r_{2} < 1$ . The optimal values of $r_{1}$ and $r_{2}$ can be found by the local force conditions. When $ρ = 0.1$ and $ρ = 0.14$ , the two defects are in equilibrium as equilibrium conditions are satisfied at these locations.

annular region with inner radius. — Figure 6. Colour map of $Re [f_{D_{ζ}} (ζ)]$ in an annular region with inner radius $ρ$ in the presence of three pairs of topological defects. Accounting for the symmetry, three defects with a charge of $- 1 / 2$ are located at $ζ = r_{1}, r_{1} ω, r_{1} ω^{2}$ , and three defects with a charge of $+ 1 / 2$ are located at $ζ = r_{2}, r_{2} ω, r_{2} ω^{2}$ , where $ρ < r_{1} < r_{2} < 1$ and $ω \equiv \exp (2 π i / 3)$ . This case is shown not to admit an equilibrium for any $ρ$ .

6. Final remarks

The analytical formulas derived in this paper give local conditions for the equilibria of topological defects that, we have shown, are equivalent to minimizing the Frank free energy. Throughout the paper these have been referred to as local force conditions and it is important to justify use of this terminology mainly because they have been derived here purely mathematically without reference to any physical interpretation.

One reason for associating the local conditions with forces is due to the analogy with point vortex dynamics where this local condition at each point vortex is indeed known to be equivalent to that point vortex having no net force on it; see the arguments given in Saffman [16] based on Newton’s second law of mechanics.

However, it is also possible to use the principle of virtual work to argue that the analogous local conditions at defects in nematic liquid crystals are also associated with local forces. We assume that the energy increase $δ F$ caused by small displacements of $z_{k}$ is given by

\begin{array}{ll} δ F = \frac{\partial F}{\partial z_{k}} d z_{k} + \frac{\partial F}{\partial \bar{z_{k}}} d \bar{z_{k}} . \end{array}

(6.1)

By the principle of virtual work,

\begin{array}{ll} δ F = - (f_{x} d x_{k} + f_{y} d y_{k}) = - (f_{x} - i f_{y}) \frac{d z_{k}}{2} - (f_{x} + i f_{y}) \frac{d \bar{z_{k}}}{2}, \end{array}

(6.2)

where $d z_{k} = d x_{k} + i d y_{k}$ and $(f_{x}, f_{y})$ is the force on the defect located at $z_{k}$ . By equating (6.1) and (6.2), the force on the defect located at $z = z_{k}$ can be associated with the local derivative of the complex function as follows:

\begin{array}{ll} f_{x} - i f_{y} = - 2 \frac{\partial F}{\partial z_{k}} = 2 i π q_{k} K {\frac{\partial {\hat{f}}_{Ω, k}}{\partial z} |}_{z = z_{k}} = \frac{2 i π q_{k} K}{g^{'} (ζ_{k})} (i (\frac{q_{k}}{2} - 1) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}}), \end{array}

(6.3)

where, importantly, we used (4.6) for the second equality. Because this is the force on a topological defect in a confined geometry it is natural to refer to our equivalent set of local conditions for minimizing the Frank free energy as local force conditions.

Chandler & Spagnolie [10] calculated the forces on immersed bodies with tangential or weak anchoring boundary conditions also using the principle of virtual work. Defect dynamics governed by the field have also been studied [26]. The results were compared with those obtained using Toner-Tu models based on statistical mechanics. In view of this, it is worth pointing out another observation: the same local force conditions derived here can be derived in a quite different way by modelling the defect core with radius $ϵ$ as an immersed impenetrable body. Chandler & Spagnolie [9] derived a formula for the forces on an immersed body, a formula akin to the Blasius force formula in fluid mechanics (those authors also discuss several other analogies with fluid dynamics, but they do not touch on the analogy with point vortex dynamics explored in the present paper or in [14]). Following their theory, the force on a defect with defect core modelled as a small immersed circular body $D_{k} (ϵ)$ of radius $ϵ ≪ 1$ is calculated as

\begin{array}{ll} f_{x} - i f_{y} & = \frac{i K}{2} \oint_{\partial D_{k} (ϵ)} {(\frac{\partial f_{Ω}}{\partial z})}^{2} d z \\ = \frac{i K}{2} \oint_{\partial D_{k} (ϵ)} {[- \frac{i q_{k}}{z - z_{k}} + {\frac{\partial {\hat{f}}_{Ω, k}}{\partial z} (z) |}_{z = z_{k}} + O (z - z_{k})]}^{2} d z \\ = i K \cdot (2 π q_{k}) \cdot {\frac{\partial {\hat{f}}_{Ω, k}}{\partial z} (z) |}_{z = z_{k}}, \end{array}

(6.4)

where the residue theorem has been used in the final step. This expression is the same as (6.3).

It should be noted that another derivation for (6.3) was given by Denniston [27]. There the alignment angle field $ϕ$ is decomposed into $ϕ_{q}$ and $ϕ_{ex}$ , where $ϕ_{ex}$ denotes the external field of the function harmonic at the defect core and $ϕ_{q}$ satisfies the following boundary condition on the boundary of the defect core with a small radius:

\begin{array}{lrr} ϕ_{q} \to q_{k} θ + const \end{array}

(6.5)

as $z \to z_{k} + ϵ e^{i θ}$ . The main result of Denniston [27] is the force given as the charge multiplied by the gradient of $ϕ_{ex}$ evaluated at $z = z_{k}$ times a rotational matrix. The final formula (6.3) can be seen as a complex form of the result and a simple extension of it to multiply connected domains with tangential anchoring boundary conditions. The equation for forces on topological defects (6.3) has also been derived by Giomi et al. [28]. They formulated the force on a defect influenced by the disclination of the other defects in the presence of an external fluid. Nevertheless, the formula derived here is new in the sense that we show the force on the defect is also expressed by the rotational number $2 π q_{k}$ multiplied by the derivative of the complex function based on the analogy with point vortex dynamics that has given rise to the new results.

While the focus here has been on equilibria, the dynamics of defects are expressed in terms of the gradient of the Frank free energy for topological charges and torques for defects. Tang et al. considered both the charge and torque for defects and calculated the free energy-driven dynamics [8]. This approach was extended to calculate the dynamics of defects located above a wall and under some active stress [20]. The orientation of defects has also been considered [29] to capture the force field of two interacting defects. Čoper & Kos also derived a system of equations for torques associated with many defects and calculated the forces in unbounded domains [15]. Future research will focus on using the torque to describe the complex behaviour of defects in confined geometries.

Finally, although only simply and doubly connected geometries have been treated in detail in this paper, there appears to be no obstruction to generalization of the main result to domains of arbitrary finite connectivity, that is, to nematic liquid crystal domains of connectivity $M + 1$ for any $M \geq 0$ . In that case, conformal mapping from a circular domain $D_{ζ}$ with $M$ circular holes is relevant. All the function theoretic machinery based on the prime function appropriate to such circular domains exists [13] and it is likely that a rich constructive framework extending the work herein can be built for nematic liquid crystals much the same as has already been done in point vortex dynamics [19,24,25]. This is the subject of ongoing work by the authors.

Data accessibility

Data available from the Zenodo Repository [30].

Declaration of AI use

We have not used AI-assisted technologies in creating this article.

Authors’ contributions

H.M.: conceptualization, formal analysis, funding acquisition, methodology, writing— original draft; D.C.: conceptualization, formal analysis, project administration, supervision, writing— review and editing; H.M.: conceptualization, funding acquisition, project administration, resources, writing—review and editing; T.N.: conceptualization, funding acquisition, methodology, project administration, writing—review and editing.

All authors gave final approval for publication and agreed to be held accountable for the work performed therein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

The first author was supported by a Postdoctoral Research Fellowship from the Japan Society for the Promotion of Science (JSPS-24KJ0041). The second author was partially funded by EPSRC Grant EP/V062298/1. The third author was partially supported by the Japan Society for the Promotion of Science KAKENHI (grant numbers JP23H00086 and JP23H04406).

Appendix A. Derivation of local force condition for arbitrary simply connected domains

We associate (4.5) with a conformal mapping $z = g (ζ)$ for arbitrarily shaped domains. The Taylor expansion of $g (ζ)$ around $ζ = ζ_{k}$ gives

\begin{array}{lrr} \frac{1}{ζ - ζ_{k}} = \frac{g^{'} (ζ_{k})}{z - z_{k}} + \frac{g^{''} (ζ_{k})}{2 g^{'} (ζ_{k})} + O (z - z_{k}) . \end{array}

(A 1)

The derivative of the complex potential $f_{Ω} (z)$ defined in (2.9) with respect to $ζ$ is given by

\begin{array}{ll} \frac{\partial f_{Ω}}{\partial ζ} & = - \frac{i q_{k}}{ζ - ζ_{k}} + q_{k} \frac{\partial R (ζ, ζ_{k})}{\partial ζ} + \sum_{l \neq k}^{N} q_{l} \frac{\partial G (ζ, ζ_{l})}{\partial ζ} - i \frac{g^{″} (ζ)}{g^{'} (ζ)} \\ = - i q_{k} (\frac{g^{'} (ζ_{k})}{z - z_{k}} + \frac{g^{″} (ζ_{k})}{2 g^{'} (ζ_{k})}) + q_{k} \frac{\partial R (ζ, ζ_{k})}{\partial ζ} + \sum_{l \neq k}^{N} q_{l} \frac{\partial G (ζ, ζ_{l})}{\partial ζ} - i \frac{g^{″} (ζ)}{g^{'} (ζ)} + O (z - z_{k}) . \end{array}

(A 2)

By the chain rule and the expansion of the conformal map $g (ζ)$ , we have

\begin{array}{ll} \frac{\partial f_{Ω}}{\partial ζ} = \frac{\partial f_{Ω}}{\partial z} \frac{d z}{d ζ} = \frac{\partial f_{Ω}}{\partial z} (g^{'} (ζ_{k}) + (z - z_{k}) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + \dots) . \end{array}

(A 3)

Using (A 2) and (A 3), the derivative of $f_{Ω}$ with respect to $z$ is given by

\begin{array}{ll} \frac{\partial f_{Ω}}{\partial z} = \frac{\partial f_{Ω}}{\partial ζ} {(g^{'} (ζ_{k}) + (z - z_{k}) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})})}^{- 1} + O (z - z_{k}) = - \frac{i q_{k}}{z - z_{k}} + {\hat{V}}_{k} + O (z - z_{k}), \end{array}

(A 4)

where ${\hat{V}}_{k}$ is calculated using the following Taylor expansion:

\begin{array}{ll} {\hat{V}}_{k} & = \frac{1}{g^{'} (ζ_{k})} (i (\frac{q_{k}}{2} - 1) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + q_{k} {\frac{\partial R (ζ, ζ_{k})}{\partial ζ} |}_{ζ = ζ_{k}} + \sum_{l \neq k}^{N} q_{l} {\frac{\partial G (ζ, ζ_{l})}{\partial ζ} |}_{ζ = ζ_{k}}) \\ = \frac{1}{g^{'} (ζ_{k})} (i (\frac{q_{k}}{2} - 1) \frac{g^{″} (ζ_{k})}{g^{'} (ζ_{k})} + {\frac{\partial {\hat{f}}_{k}}{\partial ζ} |}_{ζ = ζ_{k}}), \end{array}

(A 5)

where ${\hat{f}}_{k} (ζ)$ is defined in (4.1). To derive the first line in (A 5), the reciprocity conditions (3.10) and (3.11) are used. Hence, the final expression (4.6) is obtained.

Appendix B. Proof of energy formula for doubly connected domains

Here, we prove the energy formula for annular regions (5.12). By direct calculation, it can be shown that the above formula can be written using ${\hat{f}}_{k} (ζ)$ defined in (4.1):

\begin{array}{ll} F_{d} & = - (1 + \frac{Q_{n}^{2}}{(\log ρ)^{2}}) \log ρ + \sum_{k = 1}^{N} q_{k} I m [{\hat{f}}_{k} (ζ_{k})] - \sum_{k = 1}^{N} I m [(i + \frac{Q_{n}}{\log ρ}) q_{k} \log ζ_{k}] . \end{array}

(B 1)

Following [14], the energy can be written as a contour integral along the branch cuts as follows:

\begin{array}{ll} \begin{aligned} F = \frac{K}{2} \int_{D_{ζ} ∖ D (ϵ)} | \nabla ϕ |^{2} d S = \frac{K}{2} \cdot \frac{1}{2 i} \oint_{\partial (D_{ζ} ∖ D (ϵ))} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ, \end{aligned} \end{array}

(B 2)

where $f_{D_{ζ}} (ζ)$ is defined in (5.1). The path of the contour integral is shown in figure 7. The boundary of the unit disc is denoted as $C_{0}$ and that of the inner disc is denoted as $C_{1}$ . The integral paths $l_{k}^{\pm}$ , $k = 1, \dots, N$ and $l_{ρ}^{\pm}$ are chosen so that the paths do not cross the branch cuts of the logarithmic singularities at $ζ = ζ_{k}$ , $k = 1, \dots, N$ . We also choose branch cuts from the inner disc denoted as $l_{ρ}^{\pm}$ so that they do not cross the other branch cuts and $\partial D_{k} (ϵ)$ . We denote the intersection between $l_{k}^{\pm}$ and $C_{0}$ as $a_{k}^{\pm}$ and that between $l_{k}^{\pm}$ and $\partial D_{k} (ϵ)$ as $b_{k}^{\pm}$ . We also denote the intersection between $l_{ρ}^{\pm}$ and $C_{0}$ as $- 1^{\pm}$ and that between $l_{ρ}^{\pm}$ and $C_{1}$ as $- ρ^{\pm}$ . The imaginary parts of $\log (a_{k}^{\pm} - ζ_{k})$ and $\log (b_{k}^{\pm} - ζ_{k})$ are denoted as $α_{k}^{\pm}$ and $β_{k}^{\pm}$ , respectively. The study by Miyoshi et al. [14] explains the procedure for simply connected domains in detail.

Paths of contour integral. — Figure 7. Paths of contour integral in $D_{ζ} \ D (ϵ)$ (left) and $g^{- 1} (Ω \ D (ϵ)) = D_{ζ} \ \tilde{D} (ϵ)$ (right). (Left) The disc with inner radius $ϵ ≪ 1$ should be eliminated to calculate the free energy defined by (3.1). The path of integrals are chosen such that they do not cross any other branch cuts. The inner disc $D_{2} (ϵ)$ should be modified when a conformal map $g (ζ)$ is considered as shown in the right figure.

Line integral along $l_{ρ}^{\pm}$ :

When $ζ^{\pm}$ is on $l_{ρ}^{\pm}$ , the function $f_{D_{ζ}}$ satisfies

\begin{array}{ll} f_{D_{ζ}} (ζ^{+}) - f_{D_{ζ}} (ζ^{-}) = 2 i π (- i + \frac{Q_{n}}{\log ρ}) . \end{array}

(B 3)

Using relation (B 3), the line integral along $l_{ρ}^{\pm}$ can be evaluated as follows:

\begin{array}{ll} \frac{1}{2 i} \int_{l_{ρ}^{+}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ - \frac{1}{2 i} \int_{l_{ρ}^{-}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ = - π (i + \frac{Q_{n}}{\log ρ}) [f_{D_{ζ}} (- ρ^{+}) - f_{D_{ζ}} (- 1^{+})] . \end{array}

(B 4)

Line integral along $l_{k}^{\pm}$

We use the same technique as that described in [14]. On the path $ζ^{\pm} \in l_{k}^{\pm}$ , the function $f_{D_{ζ}} (ζ)$ satisfies

\begin{array}{ll} f_{D_{ζ}} (ζ^{+}) - f_{D_{ζ}} (ζ^{-}) = 2 π q_{k} . \end{array}

(B 5)

Using relation (B.5) and $\log (b_{k}^{-} - ζ_{k}) = \log ϵ + i β_{k}^{-}$ , the line integral along $l_{k}^{\pm}$ is given by

\begin{array}{ll} \frac{1}{2 i} \int_{l_{k}^{+}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ - \frac{1}{2 i} \int_{l_{k}^{-}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ = - i π q_{k} \int_{l_{k}^{-}} \frac{\partial f_{D_{ζ}}}{\partial ζ} d ζ \\ = - π q_{k}^{2} \log ϵ - i π q_{k} {\hat{f}}_{k} (ζ_{k}) - i π q_{k}^{2} β_{k}^{-} + i π q_{k} f_{D_{ζ}} (a_{k}^{-}) + O (ϵ), \end{array}

(B 6)

where ${\hat{f}}_{k} (ζ)$ is defined by (4.1).

Line integral along $\partial D_{k} (ϵ)$

We use ${\hat{f}}_{k} (ζ)$ and the derivative of the function. Using the parameterization around $\partial D_{k}$ as $ζ = ζ_{k} + ϵ e^{i θ}$ , $θ \in [β_{k}^{-}, β_{k}^{+}]$ , we have

\begin{array}{ll} \frac{1}{2 i} \oint_{\partial D_{k} (ϵ)} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ & = \frac{1}{2} \int_{β_{k}^{-}}^{β_{k}^{+}} [- \frac{i q_{k}}{ϵ e^{i θ}} + \frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ_{k} + ϵ e^{i θ})] \bar{[- i q_{k} \log ϵ e^{i θ} + {\hat{f}}_{k} (ζ_{k} + ϵ e^{i θ})]} ϵ e^{i θ} d θ \\ = - π q_{k}^{2} \log ϵ + i π q_{k} \bar{{\hat{f}}_{k} (ζ_{k})} + i π q_{k}^{2} β_{k}^{-} + i π^{2} q_{k}^{2} + O (ϵ \log ϵ) . \end{array}

(B 7)

Line integral along $C_{1}$

For $ζ = ρ e^{i θ} \in C_{1}$ , $θ \in [- π, + π)$ , the use of (5.9) gives

\begin{array}{ll} R e [f_{D_{ζ}} (ζ)] = R e [\sum_{k = 1}^{N} q_{k} G (ζ, ζ_{k})] + θ + Q_{n} - \frac{π}{2} = θ + n π - \frac{π}{2} . \end{array}

(B 8)

The line integral along $C_{1}$ becomes

\begin{array}{ll} - \frac{1}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{f_{D_{ζ}}} d ζ = - \frac{1}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} (2 θ + π (2 n - 1) - f_{D_{ζ}}) d ζ, \end{array}

(B 9)

where a minus sign is placed in front of the integral because the integral along $C_{1}$ is taken in the clockwise direction. Using integration by parts, the first term of (B 9) is

\begin{array}{lrr} - \frac{2}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} θ d ζ = 2 i π f_{D_{ζ}} (- ρ^{+}) + 2 π^{2} (- i + \frac{Q_{n}}{\log ρ}) - i \int_{- π}^{π} f_{D_{ζ}} (ρ e^{i θ}) d θ, \end{array}

(B 10)

where we used (B 3). To calculate the final term of (B 10), we note that

\begin{array}{ll} - i \int_{- π}^{π} f_{D_{ζ}} (ρ e^{i θ}) d θ & = - i \int_{- π}^{π} [\sum_{k = 1}^{N} q_{k} G (ζ, ζ_{k}) + (- i + \frac{Q_{n}}{\log ρ}) \log (ρ e^{i θ})] d θ + i π^{2} \\ = - 2 π \sum_{k = 1}^{N} q_{k} \log ζ_{k} - 2 i π \log ρ (- i + \frac{Q_{n}}{\log ρ}) + i π^{2}, \end{array}

(B 11)

where we used (5.4) and

\begin{array}{lrr} \int_{- π}^{π} \log P (ρ e^{i θ} α, ρ) d θ = \int_{- π}^{π} \log P (ρ e^{i θ} / β, ρ) d θ = 0, \end{array}

(B 12)

for any complex parameters $α$ , $β$ strictly inside $D_{ζ}$ . This can be proven by the standard residue theorem of logarithmic functions; that is

\begin{array}{lrr} \oint_{C_{0}} \frac{\log (1 - t ζ)}{ζ} d ζ = 0, | t | < 1 . \end{array}

(B 13)

The second term of (B 9) becomes

\begin{array}{lrr} - π (2 n - 1) \frac{1}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} d ζ = - π^{2} (2 n - 1) (- i + \frac{Q_{n}}{\log ρ}), \end{array}

(B 14)

where we used (B 3). We use (B 3) again to show that for the third term of (B 9),

\begin{array}{lrr} + \frac{1}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} f_{D_{ζ}} d ζ = \frac{1}{4 i} [f_{D_{ζ}}^{2}]_{ζ = - ρ^{-}}^{ζ = - ρ^{+}} = π (- i + \frac{Q_{n}}{\log ρ}) [f_{D_{ζ}} (- ρ^{+}) - π i (- i + \frac{Q_{n}}{\log ρ})] . \end{array}

(B 15)

By combining (B 10), (B 14) and (B 15), the line integral around $C_{1}$ is given by

\begin{array}{lrlr} - \frac{1}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} f_{D_{ζ}} d ζ & = π (i + \frac{Q_{n}}{\log ρ}) f_{D_{ζ}} (- ρ^{+}) - 2 π \sum_{k = 1}^{N} q_{k} \log ζ_{k} \\ + π (- i + \frac{Q_{n}}{\log ρ}) [2 π (1 - n) - i π \frac{Q_{n}}{\log ρ} - 2 i \log ρ] + i π^{2} . \end{array}

(B 16)

Line integral along $C_{0}$

We note that

\begin{array}{lrr} \frac{1}{2 i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} f_{D_{ζ}} d ζ = \frac{1}{2 i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} (2 ϕ (θ) - π - f_{D_{ζ}}) d ζ, \end{array}

(B 17)

where $ϕ (θ) \equiv θ + χ (θ)$ , where $χ (θ)$ is a step function that has a jump at $θ = α_{k}^{\pm}$ by $- 2 π q_{k}$ . The visualization of $χ (θ)$ is shown in fig. 1 in [14]. The first term of (B 17) becomes

\begin{array}{lrlr} \frac{1}{i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} ϕ d ζ & = - i \sum_{k = 1}^{N} [f_{D_{ζ}} (e^{i θ}) (θ + χ (θ))]_{θ = α_{k}^{-}}^{θ = α_{k}^{+}} - i [f_{D_{ζ}} (e^{i θ}) (θ + χ (θ))]_{θ = π^{-}}^{θ = π^{+}} + i \int_{- π}^{π} f_{D_{ζ}} (e^{i θ}) d θ \\ = - 2 i π \sum_{k = 1}^{N} q_{k} (f_{D_{ζ}} (a_{k}^{-}) + α_{k}^{-} + χ (α_{k}^{-}) + 2 π q_{k}) - 2 i π f_{D_{ζ}} (- 1^{+}) \\ - 2 π^{2} (- i + \frac{Q_{n}}{\log ρ}) + i \int_{- π}^{π} f_{D_{ζ}} (e^{i θ}) d θ \\ = - 2 i π \sum_{k = 1}^{N} q_{k} (f_{D_{ζ}} (a_{k}^{-}) + π q_{k}) - 2 i π f_{D_{ζ}} (- 1^{+}) - 2 π^{2} \frac{Q_{n}}{\log ρ} + i π^{2}, \end{array}

(B 18)

where we used

\begin{array}{lrr} i \int_{- π}^{π} f_{D_{ζ}} (e^{i θ}) d θ = i \int_{- π}^{π} Re [f_{D_{ζ}} (e^{i θ})] d θ = 2 i π q_{k} \sum_{k = 1}^{N} (α_{k}^{-} + χ (α_{k}^{-}) + π q_{k}) - i π^{2}, \end{array}

(B 19)

where from the first equality, we used the functional property of $\hat{P} (., .)$ and $P (., .)$ . The relation comes from

\begin{array}{lrlr} \int_{- π}^{π} (\log | P (e^{i θ} / ζ_{k}, ρ) | + \log | - ζ_{k} |) d θ = \int_{- π}^{π} (\log | e^{i θ} - ζ_{k} | + \log | \hat{P} (e^{i θ} / ζ_{k}, ρ) |) d θ = 0, \end{array}

(B 20)

\begin{array}{lrlr} \int_{- π}^{π} \log | P (e^{i θ} ζ_{k}, ρ) | d θ = \int_{- π}^{π} (\log | 1 - e^{i θ} ζ_{k} | + \log | \hat{P} (e^{i θ} ζ_{k}, ρ) |) d θ = 0, \end{array}

(B 21)

where we used the residue theorem (B 13). Using the residue theorem at $ζ = ζ_{k}$ and $ζ = 0$ , the second term in (B 17) becomes

\begin{array}{lrr} - \frac{π}{2 i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} d ζ = - π^{2} (- i + \frac{Q_{n}}{\log ρ}), \end{array}

(B 22)

where we used $\sum_{k = 1}^{N} q_{k} = 0$ . The third term of (B 17) becomes

\begin{array}{lrlr} - \frac{1}{2 i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} f_{D_{ζ}} d ζ = - \frac{1}{4 i} \sum_{k = 1}^{N} [f_{D_{ζ}}^{2}]_{ζ = a_{k}^{-}}^{ζ = a_{k}^{+}} - \frac{1}{4 i} [f_{D_{ζ}}^{2}]_{ζ = - 1^{-}}^{ζ = - 1^{+}} \\ = i π \sum_{k = 1}^{N} q_{k} (f_{D_{ζ}} (a_{k}^{-}) + π q_{k}) - π (- i + \frac{Q_{n}}{\log ρ}) (f_{D_{ζ}} (- 1^{+}) - i π (- i + \frac{Q_{n}}{\log ρ})) . \end{array}

(B 23)

By combining (B 18), (B 22) and (B 23), the line integral along $C_{0}$ is given by

\begin{array}{lrlr} \frac{1}{2 i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} f_{D_{ζ}} d ζ & = - i π \sum_{k = 1}^{N} q_{k} (f_{D_{ζ}} (a_{k}^{-}) + π q_{k}) \\ - π (i + \frac{Q_{n}}{\log ρ}) f_{D_{ζ}} (- 1^{+}) + i π^{2} \frac{Q_{n}}{\log ρ} (i + \frac{Q_{n}}{\log ρ}) + i π^{2} . \end{array}

(B 24)

Summation

The summation of (i)–(v) gives

\begin{array}{lrlr} \frac{1}{2 i} \oint_{\partial (D_{ζ} \ D (ϵ))} \frac{\partial f_{D_{ζ}}}{\partial ζ} & f_{D_{ζ}} d ζ = - 2 π \sum_{k = 1}^{N} q_{k}^{2} \log ϵ + 2 π \sum_{k = 1}^{N} q_{k} Im [{\hat{f}}_{k} (ζ_{k})] - 2 π \sum_{k = 1}^{N} q_{k} \log ζ_{k} \\ - 2 π (1 + \frac{Q_{n}^{2}}{(\log ρ)^{2}}) \log ρ - 2 π (- i + \frac{Q_{n}}{\log ρ}) \sum_{k = 1}^{N} Im [q_{k} \log ζ_{k}] . \end{array}

(B 25)

By direct calculation, the formula (B 25) except the first term is the same as (B 1) multiplied by $2 π$ .

For general domains, we consider the conformal map $z = g (ζ)$ from annular region $D_{ζ}$ to the domain $Ω$ . We define the mapped region ${\tilde{D}}_{k} (ϵ) \equiv g^{- 1} (D_{k} (ϵ))$ . Since the Frank free energy is conformally invariant, the free energy is calculated in the annular region. The integral path is shown on the right side of figure 7. However, two simple modifications are needed; that is, the radius of inner disc ${\tilde{D}}_{k} (ϵ)$ becomes $ϵ / | g^{'} (ζ_{k}) | + O (ϵ^{2})$ and the complex function becomes (5.13). By using the expression in (5.13), the contour integral for the energy is given by

\begin{array}{lrlr} F & = \frac{K}{2} \cdot \frac{1}{2 i} \oint_{\partial (D_{ζ} \ \tilde{D} (ϵ))} \frac{\partial f_{Ω}}{\partial ζ} f_{Ω} d ζ = \frac{K}{2} \cdot \frac{1}{2 i} \oint_{\partial (D_{ζ} \ \tilde{D} (ϵ))} (\frac{\partial f_{D_{ζ}}}{\partial ζ} + \frac{\partial h}{\partial ζ}) (f_{D_{ζ}} + h) d ζ \\ = \frac{K}{2} \cdot \frac{1}{2 i} \oint_{\partial (D_{ζ} \ \tilde{D} (ϵ))} (\frac{\partial f_{D_{ζ}}}{\partial ζ} f_{D_{ζ}} + \frac{\partial f_{D_{ζ}}}{\partial ζ} h + \frac{\partial h}{\partial ζ} f_{D_{ζ}} + \frac{\partial h}{\partial ζ} h) d ζ, \end{array}

(B 26)

where $h (ζ) \equiv - i \log g^{'} (ζ)$ . The first term of (B 26) divided by $K / 2$ is the same as (B 25), except the radius $ϵ$ now becomes $ϵ / | g^{'} (ζ_{k}) |$ as the inner disc ${\tilde{D}}_{k} (ϵ)$ is excluded. The final term of (B 26) is

\begin{array}{lrr} \frac{K}{2} \cdot \frac{1}{2 i} \oint_{\partial (D_{ζ} \ \tilde{D} (ϵ))} \frac{\partial h}{\partial ζ} h d ζ = \frac{K}{2} [\int_{D_{ζ}} {| \frac{g^{''}}{g^{'}} |}^{2} d S + O (ϵ^{2})] . \end{array}

(B 27)

To prove (5.14), some simple modifications to (5.12) are needed. Due to the single-valueness of the integrand along $l_{k}^{\pm}$ and $l_{ρ}^{\pm}$ , the second term of (B 26) is

\begin{array}{lrr} \int_{l_{k}^{+}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ - \int_{l_{k}^{-}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ = \int_{l_{ρ}^{+}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ - \int_{l_{ρ}^{-}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ = 0 . \end{array}

(B 28)

Also,

\begin{array}{lrlr} - \frac{1}{2 i} \oint_{\partial {\tilde{D}}_{k} (ϵ)} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ & = - \frac{1}{2} \int_{β_{k}^{-}}^{β_{k}^{+}} [- \frac{i q_{k}}{\tilde{ϵ} e^{i θ}} + \frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ_{k} + ϵ e^{i θ} + ζ_{k})] h (\tilde{ϵ} e^{i θ} + ζ_{k}) \tilde{ϵ} e^{i θ} d θ + O (ϵ \log ϵ) \\ = π i q_{k} h (ζ_{k}) + O (ϵ \log ϵ) = - π q_{k} \log g^{'} (ζ_{k}) + O (ϵ \log ϵ), \end{array}

(B 29)

and

\begin{array}{lrr} \frac{1}{2 i} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ - \frac{1}{2 i} \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ = - π \sum_{k = 1}^{N} q_{k} \log g^{'} (ζ_{k}), \end{array}

(B 30)

where we used the residue theorem, that is

\begin{aligned} \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{h} d ζ - \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} \bar{h} d ζ & = \bar{\sum_{k = 1}^{N} \int_{a_{k}^{-}}^{a_{k + 1}^{+}} \frac{\partial \bar{f_{D_{ζ}}}}{\partial θ} h d θ - \int_{- π}^{π} \frac{\partial \bar{f_{D_{ζ}}}}{\partial θ} h d θ} \\ = \bar{2 \int_{- π}^{π} h (e^{i θ}) d θ - 2 \int_{- π}^{π} h (ρ e^{i θ}) d θ - \oint_{C_{0}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ + \oint_{C_{1}} \frac{\partial f_{D_{ζ}}}{\partial ζ} h d ζ}, \end{aligned}

where we used $\partial f_{D_{ζ}} / \partial θ + \partial f_{D_{ζ}} / \partial θ = 2$ on $C_{0}$ and $C_{1}$ except $ζ = a_{k}^{\pm}$ , $k = 1, \dots, N$ and $ζ = - 1^{\pm}$ .

For the third term of (B 26), we note that

\begin{array}{lrr} \frac{1}{2 i} \int_{l_{ρ}^{+}} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ - \frac{1}{2 i} \int_{l_{ρ}^{-}} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ = - π (i + \frac{Q_{n}}{\log ρ}) (h (- ρ) - h (- 1)), \end{array}

(B 31)

where we used (B 3) and

\begin{array}{lrlr} \frac{1}{2 i} \int_{l_{k}^{+}} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ - \frac{1}{2 i} \int_{l_{k}^{-}} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ & = - i π q_{k} [h (b_{k}^{-}) - h (a_{k}^{-})] = - i π q_{k} [h (ζ_{k}) - h (a_{k}^{-})] + O (ϵ), \end{array}

and

\begin{array}{lrlr} - \frac{1}{2 i} \oint_{\partial {\tilde{D}}_{k} (ϵ)} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ & = O (ϵ \log ϵ) . \end{array}

(B 32)

The line integral along $C_{0}$ gives

\begin{array}{lrlr} \frac{1}{2 i} \oint_{C_{0}} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ = \frac{1}{2 i} \oint_{C_{0}} \frac{\partial h}{\partial ζ} (2 ϕ - π - f_{D_{ζ}}) d ζ \\ = - i π \sum_{k = 1}^{N} q_{k} h (a_{k}^{-}) - π (i + \frac{Q_{n}}{\log ρ}) h (- 1) + i \int_{- π}^{π} h (e^{i θ}) d θ - \frac{1}{2 i} \oint_{C_{0}} (π \frac{\partial h}{\partial ζ} - h \frac{\partial f_{D_{ζ}}}{\partial ζ}) d ζ, \end{array}

and the line integral along $C_{1}$ is

\begin{array}{lrr} - \frac{1}{2 i} \oint_{C_{1}} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ = π (i + \frac{Q_{n}}{\log ρ}) h (- ρ) - i \int_{- π}^{π} h (ρ e^{i θ}) d θ + \frac{1}{2 i} \oint_{C_{1}} (π (2 n - 1) \frac{\partial h}{\partial ζ} - h \frac{\partial f_{D_{ζ}}}{\partial ζ}) d ζ . \end{array}

It is noted that

\begin{array}{lrr} i \int_{- π}^{π} h (e^{i θ}) d θ - i \int_{- π}^{π} h (ρ e^{i θ}) d θ = \oint_{C_{0}} \frac{h (ζ)}{ζ} d ζ - \int_{C_{1}} \frac{h (ζ)}{ζ} d ζ = 0, \end{array}

(B 33)

and due to the fact that the inner and outer boundaries are not self-intersecting,

\begin{array}{lrr} \oint_{C_{0}} \frac{\partial h}{\partial ζ} d ζ = - i \oint_{C_{0}} \frac{g^{''}}{g^{'}} d ζ = \oint_{C_{1}} \frac{\partial h}{\partial ζ} d ζ = - i \oint_{C_{1}} \frac{g^{''}}{g^{'}} d ζ = 0 . \end{array}

(B 34)

Hence, the third term of (B 26) becomes

\begin{array}{lrr} \frac{1}{2 i} \oint_{\partial (D_{ζ} \ \tilde{D} (ϵ))} \frac{\partial h}{\partial ζ} f_{D_{ζ}} d ζ = - 2 π \sum_{k = 1}^{N} q_{k} \log g^{'} (ζ_{k}), \end{array}

(B 35)

where we used the residue theorem to calculate

\begin{array}{lrr} \frac{1}{2 i} \oint_{C_{0}} h \frac{\partial f_{D_{ζ}}}{\partial ζ} d ζ - \frac{1}{2 i} \oint_{C_{1}} h \frac{\partial f_{D_{ζ}}}{\partial ζ} d ζ = - π \sum_{k = 1}^{N} q_{k} \log g^{'} (ζ_{k}) . \end{array}

(B 36)

Combining these, the formula (5.14) is derived.

Appendix C. Local force conditions for a concentric annulus

Using the energy formula, this section proves that the formula derived using the local force condition is equivalent to the condition for the local extrema of topological defects in annular regions. The derivative of $F$ given in (5.12) with respect to $ζ_{k}$ gives

\begin{array}{lrr} \frac{\partial F}{\partial ζ_{k}} = π q_{k} K [q_{k} \frac{\partial}{\partial ζ_{k}} Im [R (ζ_{k}, ζ_{k})] + 2 \sum_{l \neq k}^{N} q_{l} \frac{\partial}{\partial ζ_{k}} Im [G (ζ_{k}, ζ_{l})] - (1 + i \frac{Q_{n}}{\log ρ}) \frac{1}{ζ_{k}}], \end{array}

(C 1)

where we used the reciprocal properties of $G (ζ, ζ_{k})$ and $R (ζ, ζ_{k})$ given in (5.7) and (5.8), respectively, and the following properties:

\begin{array}{lrr} \frac{\partial Q_{n}}{\partial ζ_{k}} = \frac{i q_{k}}{2 ζ_{k}}, \frac{\partial}{\partial ζ_{k}} (Re [\log ζ_{k}]) = \frac{1}{2 ζ_{k}} . \end{array}

(C 2)

Hence, the derivative of the energy in (C 1) becomes

\begin{array}{lrr} \frac{\partial F}{\partial ζ_{k}} = - i π q_{k} K [q_{k} {\frac{\partial R}{\partial ζ_{k}} (ζ, ζ_{k}) |}_{ζ = ζ_{k}} + \sum_{l \neq k}^{N} q_{l} {\frac{\partial G}{\partial ζ} (ζ, ζ_{l}) |}_{ζ = ζ_{k}} + (- i + \frac{Q_{n}}{\log ρ}) \frac{1}{ζ_{k}}] . \end{array}

(C 3)

Similar to §5, we now associate the derivative of the energy (C 1) with the local force conditions. The derivative of ${\hat{f}}_{k} (ζ)$ evaluated at $ζ = ζ_{k}$ is

\begin{array}{lrlr} {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}} & = q_{k} {\frac{\partial R}{\partial ζ} (ζ, ζ_{k}) |}_{ζ = ζ_{k}} + \sum_{l \neq k}^{N} q_{l} {\frac{\partial G}{\partial ζ} (ζ, ζ_{l}) |}_{ζ = ζ_{k}} + (- i + \frac{Q_{n}}{\log ρ}) \frac{1}{ζ_{k}} . \end{array}

(C 4)

Hence, the derivative of the free energy is written as

\begin{array}{lrr} \frac{\partial F}{\partial ζ_{k}} = - i π q_{k} K {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}} . \end{array}

(C 5)

Appendix D. Local force conditions for arbitrary doubly connected domains

Using the same technique as that used for simply connected domains in §3, the derivative of $f_{Ω} (z)$ defined in (5.13) is given using the chain rule for $z = g (ζ)$ as follows:

\begin{array}{lrr} \frac{\partial f_{Ω}}{\partial z} = - \frac{i q_{k}}{z - z_{k}} + {\hat{V}}_{k} + O (ζ - ζ_{k}), \end{array}

(D 1)

where ${\hat{V}}_{k}$ is given by the same expression (A 5) with different $f_{D_{ζ}} (ζ)$ defined in (5.1). By direct calculation of the derivative of ${\hat{f}}_{Ω, k} (z) \equiv f_{Ω} (z) + i q_{k} \log (z - z_{k})$ , it can be shown that

\begin{array}{lrr} \frac{\partial F}{\partial z_{k}} = - i π q_{k} K {\frac{\partial {\hat{f}}_{Ω, k}}{\partial z} (z) |}_{z = z_{k}} . \end{array}

(D 2)

Therefore, to find the equilibria for topological defects in doubly connected domains whose mapping from a concentric annulus is known, it is sufficient to find ${ζ_{k}}$ satisfying the local conditions

\begin{array}{lrr} g^{'} (ζ_{k}) {\hat{V}}_{k} = - i π q_{k} K (i (\frac{q_{k}}{2} - 1) \frac{g^{''} (ζ_{k})}{g^{'} (ζ_{k})} + {\frac{\partial {\hat{f}}_{k}}{\partial ζ} (ζ) |}_{ζ = ζ_{k}}) = 0, k = 1, \dots, N . \end{array}

(D 3)

Footnotes

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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences cover image

February 2025

Volume 481Issue 2308

Article Information

DOI:https://doi.org/10.1098/rspa.2024.0569
Published by:Royal Society
Online ISSN:1471-2946

History:

Manuscript received31/07/2024
Manuscript accepted13/12/2024
Published online19/02/2025

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Equivalent local force conditions minimizing the Frank free energy for topological defect equilibria in nematic liquid crystals

Abstract

1. Introduction

2. Topological defects and the Frank free energy

3. Frank free energy formula in simply connected domains

4. Local force conditions for nematic liquid crystals

(a) Local force conditions in the unit disc

(b) Local force conditions for arbitrary simply connected domains

(c) Example 1: equilibria in a unit disc

(d) Example 2: equilibria in a quadrature domain

5. Local force conditions for annular regions

(a) Example 3: equilibria of pairs of topological defects in annular regions

6. Final remarks

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Declaration of AI use

Authors’ contributions

Conflict of interest declaration

Funding

Appendix A. Derivation of local force condition for arbitrary simply connected domains

Appendix B. Proof of energy formula for doubly connected domains

Appendix C. Local force conditions for a concentric annulus

Appendix D. Local force conditions for arbitrary doubly connected domains

Footnotes

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