Estimating conformal capacity using asymptotic matching

Abstract

Conformal capacity is a mathematical quantity relevant to a wide range of physical and mathematical problems and recently there has been a resurgence of interest in devising new methods for its computation. In this paper we show how ideas from matched asymptotics can be used to derive estimates for conformal capacity. The formulas derived here are explicit, and there is evidence that they provide excellent approximations to the exact capacity values even well outside the expected range of validity.

1. Introduction

The study of conformal invariants is of particular importance in the field of complex analysis (Ahlfors, 2010). Conformal capacity is one of the most essential such invariants and has been studied for many years (Pölya & Szego, 1953; Ransford, 1995). It has numerous applications in potential theory (Ransford, 1995), the study of quasiconformal mapping Gehring (1961) and in approximation theory (Belyi, 1997). As its name suggests, conformal capacity has its roots in the theory of electrical potentials (Baddoo & Trefethen, 2021). Mathematically, the conformal capacity of a domain $\mathcal{G}$ containing a subset $E$ is defined by the extremal value of the integral (Dubinin, 2014; Nasser et al., 2022)

where $\varphi (x,y)$ is a harmonic function with $\varphi (x,y)\ge 1$ for all $(x,y)\in E$ and $\varphi (x,y)\to 0$ as $(x,y)\to \partial \mathcal{G}$⁠. It is known that the extremal function $\varphi$ satisfies the following classical Dirichlet problem for Laplace’s equation (Ahlfors, 2010; Hakula et al., 2023):

with the following boundary conditions:

An elementary example is where $\mathcal{G}$ is taken as the unit disc with centre at the origin and $E$ is a concentric disc of radius $\rho$⁠, $0<\rho <1$⁠. The solution for the Dirichlet problem (1.2) in this geometry is given by $\varphi =\text{Re}[W(z)]$⁠, where the analytic function

is often called a complex potential for the problem. Conformal capacity can then be calculated from equation (1.1) using Green’s second identity (Ahlfors, 2010; Hakula et al., 2023) with the result

Since the capacity is conformally invariant, the calculation of capacity in doubly connected domains (where $\partial E$ forms an internal boundary component of a doubly connected domain) can be related to the problem of finding the conformal modulus $\rho$ of the conformal mapping to the target annular domain from a canonical concentric annulus, $\rho <|\zeta |<1$ say, with $|\zeta |=1$ mapping to $\partial \mathcal{G}$ and $|\zeta |=\rho$ mapping to $\partial E$⁠.

Another example where the capacity can be found in closed form is where $\mathcal{G}$ is the unit disc and $E$ is the slit $\{(x,y)|x\in [0,r],y=0\}$⁠, $0<r<1$⁠. This domain is called the ‘Grötzsch ring’ which has broad application in the theory of conformal mappings, in physics and in number theory (Anderson et al., 1992). The capacity in this case is given implicitly by the following formulas:

where the function $P(.,.)$ is a special function, closely related to the so-called prime function (Crowdy, 2020) of the annulus, defined by

The expression for the ring (1.6) can be derived because the explicit conformal map from an annulus $\rho <|\zeta |<1$ to this domain can be determined in closed form; see, for example, exercise 5.16 of Crowdy (2020), or pp. 293 of Nehari (2012). Interested readers can refer to Appendix  A for the derivation of formula (1.6). Incidentally, the prime function of the concentric annulus can be connected to the theory of elliptic functions (see chapter 14 of Crowdy (2020)) so that (1.6) can be related to an alternative expression (2.6) featured in Hakula et al. (2023). To be more precise, the capacity of the Grötzsch ring is given by the complete elliptic integral of the first kind as follows: (see the equation (3.11) of Anderson et al. (1992))

and hence $\rho$ in (1.6) has an explicit formula such that $\rho =\exp (-\mu (r))$⁠.

Formula (1.5) indicates how conformal capacity can be interpreted as a ‘conformal modulus’ of a domain and those, in turn, are related to important diagnostic quantities appearing in applications. Formula (1.5) can, for example, be understood physically as the ‘effective conductance’, or the total current into the annular conductor $\rho <|z|<1$ of uniform unit conductivity, when the inner boundary is set to unit voltage and the outer boundary is grounded (set to zero voltage). A number of other physical applications are discussed in detail by Papamichael (Papamichael, 1989) and Papamichael and Stylianopoulos (Papamichael & Stylianopoulos, 2010). For example, the effective transport coefficients of materials can be calculated by obtaining the conformal modulus of holey materials (Kramer et al., 2008). Acker (Acker, 1977) examined the heat loss of the cylindrical pipe in order to obtain the appropriate shape of the outer boundary of the pipe which minimizes the heat loss of the whole domain. Effective diffusion coefficients of diffusive material can be calculated by solving the 2D Laplace equation and integration of the flow over the flow area (Bell & Crank, 1974). These physical quantities have physical importance in each field, but mathematically they can be related to conformal capacity.

Despite its many applications it is rare to find explicit formulas for conformal capacity. A common tool used to compute the capacity is the theory of conformal mapping. Dalichau has described the various conformal mappings including Schwarz–Christoffel maps for simply connected symmetrical polygonal-shaped domains and calculated the capacities of these domains (Dalichau, 2020). For very general polygonal or polycircular domains, including multiply connected cases, the general theory has now been developed for the construction of such mappings (Crowdy, 2005; Crowdy & Fokas, 2007; Crowdy et al., 2012; Crowdy, 2020). This theory often reduces the problem to finding a finite set of accessory parameters (for polygons) or finding the solution of a differential equation (for polycircular-arc domains). This construction usually has to be done numerically, but the reduction to a finite set of unknowns offers significant advantage. On the other hand, one can use purely numerical approaches based on boundary integral formulations or their kin: for example, Nasser and Vuorinen (Nasser & Vuorinen, 2021) used the generalized Neumann kernel to calculate the capacity of a doubly connected domain. Baddoo and Trefethen (Baddoo & Trefethen, 2021) recently proposed an algorithm to evaluate capacity using rational function approximations. Indeed, many different numerical schemes for the computation of conformal capacity have now been proposed (Betsakos et al., 2023; Hakula et al., 2023; Liesen et al., 2023).

The purpose of this paper is to show how a quite different mathematical idea—the theory of matched asymptotics (Van Dyke, 1975)—can be used to a great advantage in providing estimates of conformal capacity. Perhaps surprisingly, we have not seen this idea applied to the computation of capacity which is all the more remarkable because the estimates it provides are, as will be shown here, very accurate even well beyond the expected range of validity. While the idea may not be familiar in the literature on the computation of capacity, asymptotic analysis is a powerful tool in the applied sciences and its principal tenets are well known (Van Dyke, 1975; Hinch, 1991). Tuck (Tuck, 1975) advocated its use in the calculation of so-called ‘blockage coefficients’ characterizing the net effect of occlusions obstructing ideal flows in channels as well as the ‘effective size’ of holes in a wall, and both concepts have much in common with conformal capacity. And, just as the notion of conformal capacity manifests itself in applications in many different guises, Crowdy (Crowdy, 2011) has shown how the idea of a blockage coefficient is analogous to the so-called hydrodynamic slip length used in surface engineering to quantify the frictional properties of superhydrophobic surfaces. In many ways, the present article adopts the spirit of Tuck’s approach to estimates of blockage coefficients and effective size but now for the estimation of conformal capacity.

To understand the central idea of the matching approach advocated here consider a small lens located at the centre of a unit disc shown in Fig. 1; this same geometry is featured in fig. 1 of Hakula et al. (2023). Let the unit disc be denoted by $\mathcal{G}$ and denote the lens by $E$⁠. The domain $\mathcal{G}\mathrm{\setminus }E$ is a doubly connected polycircular-arc domain (defined as a domain with boundaries made up of a union of circular arcs) and the general theory exists—see Crowdy & Fokas (2007); Crowdy et al. (2012); Crowdy (2020)—to compute the capacity of this domain using conformal mapping from a canonical concentric annulus; that calculation, while relatively straightforward (indeed, it will be carried out later), still requires numerical integration of a differential equation. However, using the matching approach to be described next, an explicit formula estimating the required capacity can be obtained using more elementary, albeit still non-trivial, considerations.

The main assumption is that the lens is small compared with the unit circle and also well separated from it. Then, using an electric circuit analogy for concreteness, an ‘outer’ observer viewing this set-up on the scale of the unit circle sees the small lens, set to unit voltage, effectively as a point current source at the origin and of (as yet) unknown strength $m$⁠. Resolution of the detailed geometry of the lens is not possible for this observer, as indicated in the middle of Fig. 1, and the corresponding complex potential for this outer observer is well approximated by

where the real part of this analytic function is the voltage potential, $\varphi$ say. This potential incorporates the condition that the outer boundary is grounded, $\varphi =0$ on $|z|=1$⁠. On the other hand, an ‘inner’ observer viewing the same configuration at the smaller scale of the lens does not notice the grounded outer boundary in the far distance; this is also indicated in Fig. 1. Hence, for the inner observer, the boundary value problem for the voltage potential $\varphi$ requires that $\varphi =1$ on $\partial E$ with the total current out of the lens equal to $m$⁠. The solution to this inner problem is more difficult to solve, but is readily done by utilizing the conformal map from the unit circle in a complex $\zeta$-plane to the outside of the lens with opening angle $2\theta$ in the $z$-plane as determined in Crowdy (2010) as

where $2a$ is the width of the lens and, for the matching approach to work, we assume $a\ll 1$⁠. Given that $\varphi =1$ on $\partial E$ the solution to the inner problem is

A constant in this expression has been chosen to ensure that the imaginary part of $W_{\text{inner}}(z)$ is zero to the right of the meniscus. It should be emphasized that the outer solution incorporates information on the grounded nature of the outer boundary, while the inner solution encodes the fact that the inner boundary has been set to unit voltage. The idea now is to match the outer and inner solutions at an intermediate length scale at which their validity is assumed to overlap: practically, this means that the limit of the inner solution as $z\to \mathrm{\infty }$ must be made to ‘match’ with the the outer solution as $z\to 0$⁠. As $z\to \mathrm{\infty }$⁠,

so that, in the same limit, the inner potential behaves as

Notice first that (1.9) and (1.13) have the same leading order asymptotics. Furthermore, the ‘matching’ of the constant terms in (1.9) and (1.13) determines a leading-order approximation to $m$ via the relation

The capacity of the domain is then approximated by

where $\partial /\partial n$ denotes the normal derivative outward to the boundary of $E$⁠. This simple explicit formula is found to furnish an excellent approximation even when the size of the inner polycircular lens becomes large. Figure 2 shows a comparison of the estimates given by this formula and a calculation of the capacity based on construction of a doubly connected polycircular-arc conformal mapping to be described in the next section. The capacity for this example was also calculated using very different numerical methods in Hakula et al. (2023).

This simple example demonstrates the power of the matching approach in providing useful estimates of conformal capacity. These are useful, for example, in obtaining initial guesses for iterative procedures to find more precise values based on solution of a conformal mapping accessory parameter problem. The remainder of this paper demonstrates, using a series of illustrative cases, the scope of these ideas.

2. The matching approach for doubly connected domains

The matching technique just described can also be applied to cases (A), (B) and (C) shown in Fig. 3. The choice of domain $\mathcal{G}$ in cases (A) and (B) is an infinite channel with height $2H$⁠. First we assume that the parameter $a$ characterizing the size of $E$ is small compared with $H$ so that the total flux out of the inner region can be seen as a total flux from a point source with strength $m$ at the centre of lens. The solution to the ‘outer problem’ of a single point source of strength $m$ in a channel with grounded walls can be found using elementary conformal mapping techniques (Ablowitz & Fokas, 2003):

where the Taylor expansion of the hyperbolic tangent is used to find the behaviour as $z\to 0$⁠. Since $E$ is a circular disc in this case, which is a special case of a lens, the inner solution is still (1.11) but now with $z=f(\zeta )=-a/\zeta$⁠:

The matching of constants in the inner and outer solution gives

Hence, the conformal capacity for case (A) is

Of course, case (A) is actually a special case of (B). For a general lens with opening angle $2\theta$⁠, a combination of the inner solution (1.13) and the outer solution (2.1) gives the matching condition

from which the capacity for case (B) is calculated as

The result (2.4) for case (A) is retrieved when $\theta =\pi /2$⁠.

For case (C), only the outer solution needs to be modified. The relevant complex potential for the outer solution is that for a point source with strength $m$ situated at the centre of a rectangle with height $2H$ and width $2L$⁠. It can be calculated using a simple exponential conformal mapping from the rectangle to a half annular region and the theory of the prime function associated with that annulus as described in Crowdy (2013, 2020). Indeed the outer solution is given by

where $P(\zeta ,\rho )$ is precisely the function defined earlier in (1.7); a derivation of this is given in Appendix  B. As $z\to 0$⁠, which corresponds to $\zeta \to -\text{i}\sqrt{r}$⁠, the outer solution has the local expansion

On matching (2.8) and (1.13) the capacity of this geometry is estimated by the non-trivial explicit formula

How accurate are these estimates? As it turns out, cases (A), (B) and (C) all involve doubly connected polycircular-arc domains (Crowdy & Fokas, 2007). These examples were chosen so that the accuracy of the estimates from the matching approach can be validated by computing the capacity using the conformal map of each doubly connected domain using the general theory described in Crowdy & Fokas (2007) (see also Crowdy (2020)). Crowdy and Fokas Crowdy & Fokas (2007) have shown that the conformal map $z=f(\zeta )$ from a concentric annulus in $\zeta$-plane to a doubly connected polycircular region in $z$-plane satisfies the ordinary differential equation

where $T(\zeta )$ is a so-called loxodromic function (Crowdy, 2020) that depends on the geometry of the domain; the curly brackets denote a Schwarzian derivative. The function $T(\zeta )$ depends on unknown accessory parameters as well as the modulus $\rho$ and these can be found (e.g. by a simple Newton method) using equations derived by solving the ordinary differential equation (2.10). More details on the construction of $T(\zeta )$ and finding the accessory parameters are given in Appendix  C.

To validate the accuracy of the conformal mapping method, we compare the capacities of the Grötzsch ring calculated by the conformal mapping approach with the explicit formula (1.8). The Grötzsch ring is a special case of a lens in the unit circle, i.e. the capacity is calculated by solving the ordinary differential equation (2.10) with $\theta =0$ and an automorphism of the unit disc given in (A.3).

Table 1 shows the comparison of the capacities of the Grötzsch ring with different values of the slit length $r$⁠. It can be seen that our conformal mapping approach produces approximately $1.0\times {10}^{-10}$ relative error. Thus it is reasonable to see the conformal mapping approach as the reference method throughout this paper.

First, the conformal mapping method was used to calculate the capacity of a lens in the unit circle as graphed in Fig. 2. The matching gives excellent estimates when the width of lens is less than $0.6$⁠, where the relative error is less than ${10}^{-2}$⁠.

Next, comparisons of the estimates given by the matching formula with the values given by the solution of the accessory parameter problem for cases (A), (B) and (C) are shown in Fig. 4. For case (C), the half of the opening angle $\theta$ is fixed to $\pi /4$⁠. In all cases, the matching procedure gives excellent estimates for the capacity even when $a$ grows comparable with unity.

It is important to mention that the capacity of the Grötzsch ring is also approximated with great accuracy by the proposed matching approach. When the length of the slit $r\ll 1$ in the Grötzsch ring, an outer observer sees the point sources with strength $m$ at the centre of the slit $[0,r]$⁠. Since $\text{Re}[W_{\text{outer}}(z)]=0$ on the unit circle, the outer solution has an explicit formulae:

Using the same conformal map (1.10) with $\theta =0$ and $z\to z-r/2$⁠, the inner solution around the slit is given by

Combining the outer solution (2.11) and the inner solution (2.12) gives the approximation for the capacity of the Grötzsch ring:

This result agrees with the asymptotics of $\mu (r)$ described on page 8 of Anderson et al. (1992), where it has been shown that $\mu (r)$ behaves like $\log (4/r)$ when $r$ tends to zero. The accuracy of the matching (2.13) is displayed in Table 1.

According to the equation (3.13) of Anderson et al. (1992), $\mu (r)$ is known to satisfy the inequalities:

where $r^{\prime }\equiv \sqrt{1-r^2}$⁠. Figure 5 shows the comparison of the capacity of the Grötzsch ring given by the matching approach (2.13) and (2.14). It is not surprising that ${\mu }_{\text{min}}$ and ${\mu }_{\text{max}}$ are more accurate than the matching approach because ${\mu }_{\text{min}}$ and ${\mu }_{\text{max}}$ include the expansions of higher orders than the matching approach. To be more precise, Taylor expansions of ${\mu }_{\text{min}}$ and ${\mu }_{\text{max}}$ around $r=0$ give

and

These expansions suggest that the matching approach gives the first two terms of the asymptotics in this case.

3. The matching approach for triply connected domains

The matching approach can be easily extended to multiply connected domains such as the two triply connected examples, cases (I) and (II), shown in Fig. 6. In both examples, $\mathcal{G}$ is again the unit disc.

First consider case (I) involving the triply connected domain $\mathcal{G}\mathrm{\setminus }E$⁠, $E\equiv \{E_1,E_2\}$⁠, denoted by $D_z$⁠, comprising the unit disc with two equal circular discs $E_1$ and $E_2$ excised. To align with the notation of Crowdy (2020) we will denote $\partial \mathcal{G}$ by $C_0$ and $\partial E_j$ by $C_j$ for $j=1,2$⁠. The centres of each disc are at $\delta$⁠, $-\delta$⁠, where $\delta \in \mathbb{R}$⁠, and both have radius $q\in \mathbb{R}$⁠, where $q$ and $\delta$ satisfies $0<q<\delta <1$⁠, $q+\delta <1$⁠. In order to find the capacity of the domain, consider the potential $\varphi$ in $D_z$ which satisfies

with the boundary conditions

As in Section 2, the complex potential $W(z)$⁠, $\varphi =\text{Re}[W(z)]$ will be sought.

In order to construct the outer solution, suppose that the radii $q\ll 1$ and that the two circles $C_1$ and $C_2$ are sufficiently separated from each other, i.e. $q\ll \delta$⁠, and neither are too close to $C_0$⁠. Hence an outer observer viewing at the scale of the unit circle sees the point sources with strength $m_1$ and $m_2$ at the centres of inner circles. Since $\text{Re}[W(z)]=0$ on $C_0$⁠, the outer solution is given by

As before, the strengths of two sources $m_1$ and $m_2$ are unknown.

There are now two ‘inner solutions’. The inner solution, valid near $C_1$⁠, defined as $W_{\text{inner}}^{(1)}$⁠, satisfies $\varphi =1$ on $\partial E$ is

where $m_1$ is the flux associated with this inclusion. This inner observer sees neither the outer boundary, nor the other circular boundary $C_2$⁠. Similarly, the inner solution around $C_2$ is

where $m_2$ is the flux associated with this inclusion. The observer associated with this inner solution sees neither the outer boundary, nor the other circular boundary $C_1$⁠.

The outer solution and the two inner solutions must be matched. As $z\to \delta$⁠, or as $z$ approaches $C_1$⁠, the outer solution (3.4) has the local expansion

which should match to $W_{\text{inner}}^{(1)}(z)$⁠. Similar arguments can be made as $z$ approaches $C_2$⁠. On matching, a linear system for $m_1$ and $m_2$ emerges:

By symmetry we expect $m_2=m_1$ and consequently,

The capacity for the domain $D_z=\mathcal{G}\mathrm{\setminus }E$⁠, $E\equiv \{E_1,E_2\}$ then follows from Green’s second identity as

For case (II), there are now two slits with length $2a$ centred at $(b,0)$ and $(-b,0)$ in the unit disc $\mathcal{G}$⁠. These slits are labelled as $L_1$ and $L_2$⁠. Suppose that $a\ll 1$ and $a\ll b$⁠. Using the same conformal map (1.10) with $\theta =0$⁠, the inner solution around $L_1$ is given by

Since the outer boundary is the unit circle, the same expression (3.7) for the outer solution can be used. On matching in this case the linear system for $m_1$ and $m_2$ is

By symmetry we expect $m_1=m_2$ and after solving the linear system (3.12), the capacity is estimated by

To test the accuracy of these estimates, the values of the capacity are computed using an alternative scheme. Following a general formulation described in Crowdy (2020), the potential problem (3.1) can be solved using linear combinations of two multi-valued analytic functions $v_1(z)$ and $v_2(z)$ relevant to the general function theory associated with multiply connected domains as described in Crowdy (2020). These functions satisfy

and

where ${\mathcal{P}}_{jk}^{-1}$ is a set of constants (Crowdy, 2020). Since the functions $v_1(z)$ and $v_2(z)$ can be readily evaluated using freely available codes (Applied and Computational Complex Analysis Group) we merely use these resources here and refer interested readers to Crowdy (2020) for more details.

For Case (I), the exact solution of the potential problem is given by

where the parameters ${\alpha }_1$ and ${\alpha }_2$ are determined by the boundary conditions (3.2). The following capacity is then given by the same technique as (3.10):

For Case (II), the potential is also given by formula (3.16), but now the preimage circular domain, $D_{\zeta }$ say, for which the two functions $v_1$ and $v_2$ are defined is different. The geometry of the circular domain $D_{\zeta }$ resembles that of $D_z$ in Case (I), but the two preimage circles $C_1$ and $C_2$ have different centres, at $(\tilde{\delta },0)$ and $(-\tilde{\delta },0)$ say, and both have a different radius, $\tilde{q}$ say (Crowdy, 2020). To find $\tilde{\delta }$ and $\tilde{q}$ it happens that the conformal mapping from the triply connected circular domain $D_{\zeta }$ to the domain $D_z$ of Case (II) can be found using the following sequence of conformal maps (Crowdy et al., 2012):

where $\omega (.,.)$ is the prime function associated with $D_{\zeta }$ (Crowdy, 2020). The parameters $\tilde{\delta }$ and $\tilde{q}$ are determined by the following conditions:

The prime function $\omega (.,.)$ can be readily evaluated using the aforementioned freely available codes and the two equations (3.19) readily solved using standard methods.

Figure 7 shows the comparison of the capacities calculated by the matching approach and those calculated using the method just described. As expected, the matching approach yields excellent approximations when the sizes of the inclusions are small.

4. Estimating other accessory parameters

The matching approach can provide estimates of conformal capacity which, in turn, can be viewed as determining a conformal modulus in a conformal mapping problem. In this final section, we show that the matching procedure can be extended to provide more general estimates of other accessory parameters as well.

The geometry to be considered happens to be the one relevant to a topical problem involving channel flows with superhydrophobic surfaces (Miyoshi et al., 2022). The period of the channel in the $x$-direction is $2L$⁠, the distance of displacement of the flat meniscus below the tips of the sidewalls is $H$ and the distance between the side wall gratings is 2G as shown in Fig. 8. The preimage circular domain $D_{\zeta }$ is taken to be the circular domain in a parametric complex $\zeta =\xi +\text{i}\eta$-plane interior to the unit circle, denoted by $C_0$⁠, but exterior to two circles $C_1$ and $C_2$ each of radius $q$ and having centres at $\pm \delta$⁠, where $\delta$ is purely imaginary. The conformal map from $D_{\zeta }$ to the groove region $D$ is known (Miyoshi et al., 2022) to be given by

where

and where $\omega (.,.)$ is the prime function associated with $D_{\zeta }$ (Crowdy, 2020). The circle $C_0$ in the $\zeta$-plane is mapped to the line $|y|\le G$ on the imaginary axis in the $z$-plane, and the inner circles $C_1$ and $C_2$ are mapped to the lines $x=\pm L$⁠, $|y|\le H+G$ of the periodic channel. The parameters $\delta$ and $q$ depend on the lengths $H$⁠, $G$ and $L$ and can easily be solved for given the functional form (4.1) of the conformal mapping. Once again, the prime function $\omega (.,.)$ is readily evaluated using freely available codes. While this procedure is straightforward, it is of interest to examine whether approximate estimates for $\delta$ and $q$ are forthcoming from a matching approach.

The following method leads to such estimates. Consider a potential problem for a harmonic $\varphi (\xi ,\eta )$ on $D_{\zeta }$⁠, where $\varphi =\alpha$ on $C_1$⁠, $\varphi =\beta$ on $C_2$ and $\varphi =0$ on $C_0$⁠, $\alpha ,\beta \in \mathbb{R}$⁠. The analytic extension of $\varphi$ is defined as $W(\zeta )=\varphi +\text{i}\chi$ as usual. Suppose that the flux $m_{\alpha }$ is associated with $C_1$ and $m_{\beta }$ is associated with $C_2$⁠. For the two ‘inner solutions’ near to each circle it is easy to argue that

where the notation should be obvious. Because the sources with strength $m_{\alpha }$ and $m_{\beta }$ lie at $\zeta =\delta$ and $\zeta =-\delta$⁠, respectively, the outer solution is

The matching of constant terms arising from local expansions of (4.3) and (4.4) in the usual way leads to a linear system of equations for $m_{\alpha }$ and $m_{\beta }$⁠:

The parameters $m_{\alpha }$ and $m_{\beta }$ therefore satisfy the following linear system:

We can now compute the quantity

This same quantity can be calculated using estimates based on matching in the target region itself. We assume that the distance between the side wall gratings $G$ is small compared with the height of the groove $H$⁠. Because of the conformal invariance of the boundary value problem, the potential $\Phi (z)\equiv \varphi ({\mathcal{Z}}^{-1}(z))$ satisfies

where the last flux condition comes from the symmetry of $D_{\zeta }$ about the $\eta$-axis. To solve this, consider

where $\hat{\Phi }$ satisfies

The flux $m$ of $\hat{\Phi }$ associated with the blue portion in Fig. 8 can now be estimated using the matching approach developed in this paper. Let the analytic extension of $\hat{\Phi }$ be $\hat{W}(z)$ so that $\hat{\Phi }=\text{Re}[\hat{W}(z)]$⁠. The inner solution for $\hat{W}(z)$ is given by

Modelling this flux as associated with a point source of strength $m$ at $z=0$ in the rectangle the outer solution is given using the conformal transformation as follows:

A derivation of this potential is given in Appendix  B. A local expansion around $\zeta =\sqrt{\rho }$ gives

Matching (4.11) and (4.13) implies the following expression for $m$⁠:

Now,

We now have two expressions for the same quantity: (4.7) and (4.16). On comparing the coefficients in front of ${\alpha }^2$ and $\alpha \beta$⁠, we arrive at a system of nonlinear equations for $\delta$ and $q$⁠:

Indeed, after some algebra, these can be solved to give

Figure 8 shows how well these formulas predict the values of $\delta$ and $q$⁠. Similar to the previous results, the matching approach can estimate these parameters when $G$ is small.

5. Discussion

By presenting a series of examples, and comparing with numerical calculations, this paper has demonstrated a practical procedure based on asymptotic matching of suitable ‘outer’ and ‘inner’ solutions to provide estimates of the capacity associated with multiply connected domains. The estimates show excellent agreement when there is a good separation of scales between the inner and outer regions, a feature on which the matching idea relies (Tuck, 1975; Van Dyke, 1975; Hinch, 1991). From the selection of examples explored here, it should be clear that the idea is very general and the approach can be applied to a wide variety of geometries. On a technical note, it is worth remarking that it is usual when using matched asymptotics to introduce a rescaled variable to distinguish the inner region from the outer region and this can be important when doing matching at higher orders in any asymptotic expansions. Here, however, this rescaling has not been introduced explicitly since the estimates for capacity derived here involve only the leading order asymptotics in each region. In principle, more accurate estimates can be obtained by higher order matching, and then the introduction of suitably scaled inner and outer variables is advised.

It has been seen that estimating conformal capacity is akin to estimating certain conformal moduli, which can be viewed as accessory parameters in a conformal mapping construction. Another very different method has recently been proposed for the estimation of accessory parameters based on the so-called isomonodromic tau function (Anselmo et al., 2018, 2020; Cunha et al., 2022). For example, in Anselmo et al. (2018); Cunha et al. (2022) this isomonodromy method was used to find an undetermined pervertex of polycircular arc domains with four vertices by finding the zero of a tau function (Anselmo et al., 2018; Cunha et al., 2022). It is interesting to compare the matching approach of this paper with predictions from this alternative theoretical scheme. Indeed, with regard to case (A) in Fig. 3, it is possible to make a direct comparison between them. Considering the upper half of the unit circle in a channel region, where the geometry becomes simply connected, the approximation for $\rho$ can be derived through the isomonodromy approach (Anselmo et al., 2018) as follows:

where

The derivation of equation (5.1) is explained in detail in Appendix  D. Figure 9 shows a comparison between the estimates from the matching approach and from the isomonodromy approach. Matching is more accurate when $a$ is small, while the isomonodromy approach is more accurate when $a$ is large. This is natural because the matching approach assumes that the internal circle is small compared with the height of the channel. The isomonodromy approach, however, assumes that the prevertex $t_0$ is nearly $0$ in order to approximate the tau function, which corresponds to $\rho$ getting large. When combined, these two approximations—each emerging from very different considerations—give excellent estimates across the range of parameters and investigating the relationship between the tau function and the matching approach advocated here is an interesting challenge for the future. In principle, it is possible to generalize the isomonodromy approach using tau functions to give good accuracy as $a\to 0$⁠.

Acknowledgments

The first author is grateful for the support of the Nakajima Foundation in Japan.

Conflict of interest

The authors report no conflicts of interest.

Data availability

The main outcome of this paper is to produce explicit formulas which can be readily evaluated by the reader with the help of freely available codes available at a cited Github repository.

References

A. Derivation of the capacity of Grötzsch ring in terms of the prime function

We derive the conformal map from the annulus in $\zeta$-plane to the unit disc exterior to the slit $x\in [0,r]$ in $z$-plane. The conformal mapping from the concentric annulus to the unit disc exterior to the slit $x\in [-r^{\prime },r^{\prime }]$ is given by Crowdy (2020)

where

Changing the slit region $x\in [-r^{\prime },r^{\prime }]$ to a different slit $x\in [0,r]$⁠, without changing the unit circle, just requires an automorphism of the unit disc:

Combining these two conformal maps gives the reported expression (1.6) for the capacity.

B. Derivations of (2.7) and (4.12)

Here we explain how to derive the potentials of (2.7) and (4.12). The potential (2.7) is a solution which satisfies $\text{Re}[W_{\text{outer}}(z)]=0$ on the boundary of the rectangle with a source $m$ at its center. Using the conformal map $\zeta =e^{-\pi (z+L+\text{i}H)/2H}$⁠, the rectangle is transformed into the lower half annular domain $D_{\zeta }^{-}$ with radius $r=e^{-\pi L/H}$⁠. The left figure of Fig. A1 shows the rectangle and lower half annular domain. The location of the point source with strength $m$ is now $\zeta =-\text{i}\sqrt{r}$⁠. The conformal invariance of the potential means that the function $X(\zeta )\equiv W_{\text{outer}}(z(\zeta ))$ has a simple source term with a strength $m$ at $\zeta =-\text{i}\sqrt{r}$ and satisfies

On the real axis, i.e. $\bar{\zeta }=\zeta$⁠, the second condition of (B.1) means $\overline{X(\zeta )}=\bar{X}(\zeta )=-X(\zeta )$⁠. The Schwarz reflection principle means that the function $X(\zeta )$ satisfies $\bar{X}(\zeta )\equiv \overline{X(\stackrel{―}{\zeta })}=-X(\zeta )$ on $D_{\zeta }$⁠. This property indicates that $X(\zeta )$ has a sink at $\zeta =\text{i}\sqrt{r}$⁠. On $\zeta \in C_j^{+}$⁠, $\bar{\zeta }\in C_j^{-}$ for $j=1,2$⁠, so

Thus, $X(\zeta )$ has a source at $\zeta =-\text{i}\sqrt{r}$ and a sink at $\zeta =\text{i}\sqrt{r}$ and satisfies

Now we define

Because $P(\zeta ,r)$ has a simple zero at $\zeta =1$⁠, $X(\zeta )$ has a source $\zeta =-\text{i}\sqrt{r}$ and a sink at $\zeta =\text{i}\sqrt{r}$⁠. On $\zeta \in C_0$⁠, i.e. $\bar{\zeta }={\zeta }^{-1}$⁠, we have

where we used the properties of the prime function

These identities are easy to derive directly from the infinite product definition (1.7) or see Crowdy (2020) for a more general perspective. Hence, $\text{Re}[X(\zeta )]=0$ on $\zeta \in C_0$⁠. It is also easy to check that $\text{Re}[X(\zeta )]=0$ on $C_1$ and $\bar{X}(\zeta )=-X(\zeta )$⁠. Thus the expression (2.7) follows.

The potential (4.12) is a solution which satisfies $\hat{\Phi }\equiv \text{Re}[{\hat{W}}_{\text{outer}}(z)]=0$ on the left and right sides of a rectangle and ${\displaystyle \frac{\partial \hat{\Phi }}{\partial y}}=0$ on the top and bottom sides of a rectangle with a source $m$ at the centre of the rectangle. By the Cauchy–Riemann equations, the boundary condition on the top and bottom sides becomes that $\text{Im}[{\hat{W}}_{\text{outer}}(z)]$ is constant on the left and right sides. Using the conformal map $\hat{\zeta }=e^{-\pi (z+L)/(H+G)}$⁠, the rectangle region in the $z$-plane is transformed into the whole annular domain in the $\hat{\zeta }=\xi +\text{i}\eta$-plane. The figure on the right in Fig. A1 shows the rectangle and annular domain. The radius of the inner circle is $\rho =e^{-2\pi L/(H+G)}$⁠. The left and right sides of the rectangle are mapped to the boundary of the annulus, and the top and bottom sides of the rectangle are mapped to the same portion $\xi \in [-1,-\rho ]$⁠. The location of the point source is now $\hat{\zeta }=\sqrt{\rho }$⁠. The function $Y(\hat{\zeta })\equiv {\hat{W}}_{\text{outer}}(z(\hat{\zeta }))$ has a simple source term with a strength $m$ at $\hat{\zeta }=\sqrt{r}$ and satisfies

where $c_1,c_2\in \mathbb{R}$⁠. Note that the last condition of (B.8) comes from the fact that the centre line of the rectangle $\{(x,y)|-L<x<L,y=0\}$ is mapped to $\{(\xi ,\eta )|\rho <\xi <1,\eta =0\}$⁠, and by symmetry

Then, the Cauchy–Riemann equation means that $\text{Im}[Y(\zeta )]$ is constant on $\rho <\xi <1,\eta =0$⁠.

Now we define

The function $Y(\hat{\zeta })$ has a source at $\hat{\zeta }=\sqrt{\rho }$⁠. On $\hat{\zeta }\in C_0$ i.e. $\bar{\hat{\zeta }}={\hat{\zeta }}^{-1}$⁠, we have

where we used the properties (B.7). Hence $\text{Re}[Y(\hat{\zeta })]=0$ on $\hat{\zeta }\in C_0$⁠. It is also easy to show that $\text{Re}[Y(\hat{\zeta })]=0$ on $\hat{\zeta }\in C_1$⁠. Thus the expression (B.10) follows.

C. Conformal mapping of doubly connected polycircular-arc domains

For the numerical calculation of doubly connected polycircular-arc domains, we need to solve the third-order differential equation (Crowdy & Fokas, 2007; Crowdy, 2020) for the conformal map $z=Z(\zeta )$ given by

where the loxodromic function $T(\zeta )$ and initial conditions on the differential equation depend on each geometry (Crowdy, 2020). For the sake of simplicity, we first define two important functions

We note that $K(\zeta ,\rho )$ has a single pole at $\zeta =1$⁠, and $L(\zeta ,\rho )$ has a double pole at $\zeta =1$⁠. The turning angle (Crowdy & Fokas, 2007; Crowdy, 2020) of the meniscus $\alpha$ is related to the opening angle of the lens $\theta$ via

For completeness, we now list the functional form of the loxodromic function $T(\zeta )$ for each case.

1. (Lens in a circle): Because of the symmetry, the prevertices of the edges of the lens are set to be at $\zeta =\pm \rho$⁠. Then we can write

where $C$ is a real parameter to be found.

2. Case (A) (Circle in a channel): We let $\zeta =\pm 1$ be transplanted to $x\to \pm \mathrm{\infty }$⁠. Because of the symmetry, we can set

where $C\in \mathbb{R}$ is a parameter to be found.

3. Case (B) (Lens in a channel): By the combination of the case (i) and case (A), we can derive

where $C\in \mathbb{R}$ is a parameter to be found.

4. Case (C) (Lens in a rectangle): Because of the symmetry, the point $\beta$ on the unit circle is taken to be transplanted to the top right corner of the rectangle. Then we can derive

where $\gamma ,C\in \mathbb{R}$ and

More details on how to construct these functions can be found in Crowdy (2020).

D. Derivation of (5.1)

Here we explain how to derive (5.1). Considering the zero of the isomonodromy tau function, Anselmo et. al. and Anselmo et al. (2018) found the approximation of the unknown prevertex $t_0\in \mathbb{R}$ associated with the conformal map $z=F_1(t)$ to the semi-circular obstacle in an infinite channel in the $z$-plane from the upper half plane in the complex $t$-plane. Note that $F_1(0)=-\mathrm{\infty }$⁠, $F_1(t_0)=-a$⁠, $F_1(1)=a$ and $F_1(\mathrm{\infty })=\mathrm{\infty }$ as shown in Fig. D2(i). According to example (C) in section 5 of Anselmo et al. (2018), the approximation for the parameter $t_0$ is given by

Because this expression comes from the lower orders of the expansion of the tau function around $t=0$⁠, this expression is accurate when $t_0$ is small, which corresponds to the case where $a/H$ is large. Now we associate $t_0$ with the inner radius $\rho$⁠. We define a conformal map

This function $F_2(\zeta )$ maps $\zeta =1$ to infinity, $\zeta =-1$ to the origin, $\zeta =\rho$ to $t=1$⁠, and $\zeta =-\rho$ to $t=t_0$⁠, respectively. This gives

This is the same as equation (5.1).

Author notes