## Publications

172 results found

Green C, Snipes M, Ward L,
et al., 2022, Harmonic-measure distribution functions for a class of multiply connected symmetrical slit domains, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 478, Pages: 1-20, ISSN: 1364-5021

The harmonic-measure distribution function, or h-function, of a planar domain Ω⊂C with respect to a basepoint z0∈Ω is a signature that profiles the behaviour in Ω of a Brownian particle starting from z0. Explicit calculation of h-functions for a wide array of simply connected domains using conformal mapping techniques has allowed many rich connections to be made between the geometry of the domain and the behaviour of its h-function. Until now, almost all h-function computations have been confined to simply connected domains. In this work, we apply the theory of the Schottky–Klein prime function to explicitly compute the h-function of the doubly connected slit domain C∖([−1/2,−1/6]∪[1/6,1/2]). In view of the connection between the middle-thirds Cantor set and highly multiply connected symmetric slit domains, we then extend our methodology to explicitly construct the h-functions associated with symmetric slit domains of arbitrary even connectivity. To highlight both the versatility and generality of our results, we graph the h-functions associated with quadruply and octuply connected slit domains.

Crowdy D, 2022, Equilibrium tilt of slippery elliptical rods in creeping simple shear, *Journal of Fluid Mechanics*, Vol: 931, Pages: 1-11, ISSN: 0022-1120

It is shown that shape anisotropy and intrinsic surface slip lead to equilibrium tilt of slippery particles in a creeping simple shear flow, even for nearly shape-isotropic particles with a cross-section that is close to circular provided the Navier-slip length is sufficiently large. We study a rigid particle with an elliptical cross-section, and of infinite extent in the vorticity direction, in simple shear. A Navier-slip boundary condition is imposed on its surface. When a Navier-slip length parameter λ is infinite, an analytical solution is derived for the Stokes flow around a particle tilting in equilibrium at an angle (1/2)cos−1((1−k)/(1+k)) to the flow direction where 0≤k≤1 is the ratio of the semi-minor to semi-major axes of its elliptical cross-section. A regular perturbation analysis about this analytical solution is then performed for small values of 1/λ and a numerical continuation method implemented for larger values. It is found that an equilibrium continues to exist for any anisotropic particle k<1 provided λ≥λcrit(k) where λcrit(k) is a critical Navier-slip length parameter determined here. As the case k→1 of a circular cross-section is approached, it is found that λcrit(k)→∞ , so the range of Navier-slip lengths allowing equilibrium tilt shrinks as shape anistropy is lost. Novel theoretical connections with equilibria for constant-pressure gas bubbles with surface tension are also pointed out.

Crowdy D, Nelson R, Carneiro da Cunha B,
et al., 2021, Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case, *Journal of Physics A: Mathematical and Theoretical*, Vol: 55, ISSN: 1751-8113

The prevertices of the conformal map between a generic, n-vertex, simply connected, polycircular arc domain and the upper half plane are determined by ﬁnding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of 5 vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko & Lisovyy [Comm. Math. Phys., 363, 2018)] is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.

Crowdy D, 2021, Viscous Marangoni flow driven by insoluble surfactant and the complex Burgers equation, *SIAM Journal on Applied Mathematics*, Vol: 81, Pages: 2526-2546, ISSN: 0036-1399

A new mathematical connection is established between a class of two dimensional viscous Marangoni flows driven by insoluble surfactant and the complex Burgers equation. It is shown that the Marangoni-driven dynamics of a bath of viscous fluid at zero Reynolds and capillary number, and with a linear equation of state, is described by the evolution of a lower-analytic function with positive imaginary part on the real line satisfying the complex Burgers equation. Surface diffusion of surfactant plays the role of viscosity in the more familiar real-valued Burgers equation arising in gas dynamics. Using this mathematical connection it is shown that, at arbitrary surface P ́eclet number, the Marangoni dynamics is linearizable, and integrable, via a transformation of Cole-Hopf type. A new class of time-evolving exact solutions is identified for the Marangoni-induced fluid motion at any finite surface P ́eclet number. These are shown to be given by a class of evolvingN- pole solutions which differ from, and generalize, known pole dynamics solutions to the real Burgers equation. Analogous meromorphic solutions describing spatially singly-periodic Marangoni flows are also reported. For infinite surface P ́eclet number it is shown how a generalized method of characteristics leads to an implicit form of the general solution. For a special choice of initial condition it is demonstrated that this implicit solution can be made explicit and, from it, the formation at finite time of an instantaneous weak singularity is observed. Together these new solutions afford a mathematical view of the effect of surface diffusion on Marangoni flows via the evolution of complex singularities in a non-physical region of the complex plane. The observations open up valuable new mathematical connections between viscous Marangoni flows and the theory of caloric functions, Calogero-Moser systems, random matrices and Dyson diffusi

Hauge J, Crowdy D, 2021, A new approach to the complex Helmholtz equation with applications to diffusion wave fields, impedance spectroscopy and unsteady Stokes flow, *IMA Journal of Applied Mathematics*, Vol: 86, Pages: 1287-1326, ISSN: 0272-4960

A new transform pair representing solutions to the complex Helmholtz equation in a convex twodimensional polygon is derived using the theory of Bessel’s functions and Green’s second identity. Thederivation is a direct extension of that given by Crowdy [IMA J. Appl. Math, 80, (2015)] for “FourierMellin transform” pairs associated with Laplace’s equation in various domain geometries. It is shownhow the new transform pair fits into the collection of ideas known as the Fokas transform where the keystep in solving any given boundary value problem is the analysis of a global relation. Here we contextualize those global relations from the point of view of “reciprocal theorems” which are familiar tools inthe study of the effective properties of physical systems. A survey of the many uses of this new transform approach to the complex Helmholtz equation in applications is given. This includes calculationof effective impedance in electrochemical impedance spectroscopy and in other spectroscopy methodsin diffusion wave field theory, application to the 3w method for measuring thermal conductivity and tounsteady Stokes flow. A theoretical connection between this analysis of the global relations and Lorentzreciprocity in mathematical physics is also pointed out.

Baddoo PJ, Oza AU, Moore NJ,
et al., 2021, Generalization of waving-plate theory to multiple interacting swimmers, *Communications on Pure and Applied Mathematics*, ISSN: 0010-3640

Early research in aerodynamics and biological propulsion was dramatically advanced by the analytical solutions of Theodorsen, von K ́arm ́an, Wu and others. While these classical solutions apply only to isolated swimmers, the flow interactions between multiple swimmers are relevant to many practical applications, including the schooling and flocking of animal collectives. In this work, we derive a class of solutions that describe the hydrodynamic interactions between an arbitrary number of swimmers in a two-dimensional inviscid fluid. Our approach is rooted in multiply-connected complex analysis and exploits several recent results. Specifically, the transcendental (Schottky–Klein) prime function serves as the basic building block to construct the appropriate conformal maps and leading-edge-suction functions, which allows us to solve the modified Schwarz problem that arises. As such, our solutions generalize classical thin aerofoil theory, specifically Wu’s waving-plate analysis, to the case of multiple swimmers.For the case of a pair of interacting swimmers, we develop an efficient numerical implementation that allows rapid computations of the forces on each swimmer. We investigate flow-mediated equilibria and find excellent agreement between our new solutions and previously reported experimental results. Our solutions recover and unify disparate results in the literature, thereby opening the door for future studies into the interactions between multiple swimmers.

Crowdy D, 2021, Exact solutions for the formation of stagnant caps of insoluble surfactant on a planar free surface, *Journal of Engineering Mathematics*, Vol: 133, ISSN: 0022-0833

A class of exact solutions is presented describing the time evolutionof insoluble surfactant to a stagnant-cap equilibrium on the surface of deepwater in the Stokes flow regime at zero capillary number and infinite surfaceP´eclet number. This is done by demonstrating, in a two-dimensional modelsetting, the relevance of the forced complex Burgers equation to this problemwhen a linear equation of state relates the surface tension to the surfactantdensity. A complex-variable version of the method of characteristics can thenbe deployed to find an implicit representation of the general solution. A specialclass of initial conditions is considered for which the associated solutions canbe given explicitly. The new exact solutions, which include both spreading andcompactifying scenarios, provide analytical insight into the unsteady formation of stagnant caps of insoluble surfactant. It is also shown that first-orderreaction kinetics modelling sublimation or evaporation of the insoluble surfactant to the upper gas phase can be incorporated into the framework; this leadsto a forced complex Burgers equation with linear damping. Generalized exactsolutions to the latter equation at infinite surface P´eclet number are also foundand used to study how reaction effects destroy the surfactant cap equilibrium.

Crowdy D, 2021, Slip length formulas for longitudinal shear flow over a superhydrophobic grating with partially filled cavities, *Journal of Fluid Mechanics*, Vol: 925, Pages: 1-11, ISSN: 0022-1120

Explicit formulas are given for the hydrodynamic slip lengths associated with longitudinal shear flow over a superhydrophobic grating where the menisci have partially invaded the cavities and are only weakly curved. For flat menisci that have depinned from the top of the grating and have displaced downwards into the cavities, the axial velocity is determined analytically and the slip length extracted from it. This solution is then combined with an integral identity to determine the first-order correction to the slip length when the displaced menisci bow weakly into the cavity. It is argued that the new formulas provide useful upper bounds for quantifying slip in microchannel flows involving partially filled cavities. The new solutions are natural extensions of prior results due to Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372) for shear flow over mixed no-slip/no-shear surfaces and due to Bechert & Bartenwerfer (J. Fluid Mech., vol. 206, 1989, pp. 105–129) for shear flow over blade-shaped riblets.

Crowdy D, MIYOSHI H, Nelson R, 2021, The prime function, the Fay trisecant identity, andthe van der Pauw method. On some conjectures on the resistivity of a holey conductor, *Computational Methods and Function Theory - Springer*, Vol: 21, Pages: 707-736, ISSN: 1617-9447

The van der Pauw method is a well-known experimental techniquein the applied sciences for measuring physical quantities such as the electricalconductivity or the Hall coefficient of a given sample. Its popularity isattributable to its flexibility: the same method works for planar samples ofany shape provided they are simply connected. Mathematically, the method isbased on the cross-ratio identity. Much recent work has been done by appliedscientists attempting to extend the van der Pauw method to samples withholes (“holey samples”). In this article we show the relevance of two newfunction theoretic ingredients to this area of application: the prime functionassociated with the Schottky double of a multiply connected planar domainand the Fay trisecant identity involving that prime function. We focus hereon the single-hole (doubly connected, or genus one) case. Using these newtheoretical ingredients we are able to prove several mathematical conjecturesput forward in the applied science literature.

Crowdy D, Krishnamurthy V, Constantin A,
et al., 2021, Liouville chains: new hybrid vortex equilibria of the 2D Euler equation, *Journal of Fluid Mechanics*, Vol: 921, ISSN: 0022-1120

A large class of new exact solutions to the steady, incompressible Euler equation on the plane is presented. These hybrid solutions consist of a set of stationary point vortices embedded in a background sea of Liouville-type vorticity that is exponentially related to the stream function. The input to the construction is a “pure” point vortex equilibrium in a background irrotational flow. Pure point vortex equilibria also appear as a parameter A in the hybrid solutions approaches the limits A → 0, ∞. While A → 0 reproduces the input equilibrium, A → ∞ produces a new pure point vortex equilibrium. We refer to the family of hybrid equilibria continuously parametrised by A as a “Liouville link”. In some cases, the emergent point vortex equilibrium as A → ∞ can itself be the input for a second family of hybrid equilibria linking, in a limit, to yet another pure point vortex equilibrium. In this way, Liouville links together form a “Liouville chain”. We discuss several examples of Liouville chains and demonstrate that they can have a finite or an infinite number of links. We show here that the class of hybrid solutions found by Crowdy (2003) and by Krishnamurthy et al. (2019) form the first two links in one such infinite chain. We also show that the stationary point vortex equilibria recently studied by Krishnamurthy et al. (2020) can be interpreted as the limits of a Liouville link. Our results point to a rich theoretical structure underlying this class of equilibria of the 2D Euler equation.

Crowdy D, 2021, Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant, *Physical Review Fluids*, Vol: 6, ISSN: 2469-990X

An analytical solution is derived for the flow generated by a self-propelling two-dimensional Marangoni boat driven by reactive insoluble surfactant on a deep layer of fluid of viscosity μ at zero Reynolds number, capillary number, and surface Péclet number. In the model, surfactant emitted from the edges of the boat causes a surface tension disparity across the boat. Once emitted, the surfactant diffuses along the interface and sublimates to the upper gas phase. A linear equation of state relates the surface tension to the surfactant concentration. The propulsion speed of the boat is shown to be U0=Δσ(2πμ)−1e√DaK0(√Da) where Da is a Damköhler number measuring the reaction rate of the surfactant to its surface diffusion, Δσ is the surface tension disparity between the front and rear of the boat, and K0 is the order-zero modified Bessel function. Explicit expressions for the stream function associated with the Stokes flow beneath the boat are found facilitating ready examination of the Marangoni-induced streamlines. An integral formula, derived using the reciprocal theorem, is also given for the propulsion speed of the boat in response to a more general Marangoni stress distribution.

Crowdy D, Nelson R, Krishnamurthy V, 2021, "H-states'': exact solutions for a rotating hollow vortex, *Journal of Fluid Mechanics*, Vol: 913, Pages: R5-1-R5-11, ISSN: 0022-1120

Exact solutions are found for an N-fold rotationally symmetric, steadily rotating hollow vortex where a continuous real parameter governs its deformation from a circular shape and N≥2 is an integer. The vortex shape is found as part of the solution. Following the designation ‘V-states’ assigned to steadily rotating vortex patches (Deem & Zabusky, Phys. Rev. Lett., vol. 40, 1978, pp. 859–862) we call the analogous rotating hollow vortices ‘H-states’. Unlike V-states where all but the N=2 solution – the Kirchhoff ellipse – must be found numerically, it is shown that all H-state solutions can be written down in closed form. Surface tension is not present on the boundaries of the rotating H-states but the latter are shown to be intimately related to solutions for a non-rotating hollow vortex with surface tension on its boundary (Crowdy, Phys. Fluids, vol. 11, 1999a, pp. 2836–2845). It is also shown how the results here relate to recent work on constant-vorticity water waves (Hur & Wheeler, J. Fluid Mech., vol. 896, 2020, R1) where a connection to classical capillary waves (Crapper, J. Fluid Mech., vol. 2, 1957, pp. 532–540) is made.

Chen M, Stokes Y, Crowdy D,
et al., 2021, Investigation of oversized channels in tubular fibre drawing, *Optical Materials Express*, Vol: 11, Pages: 905-912, ISSN: 2159-3930

In a previous study, we compared experiments on drawing of axisymmetric tubular optical fibres to a mathematical model of this process. The model and experiments generally agreed closely. However, for some preforms and operational conditions, the internal channel of the drawn fibre was larger than predicted by the model. We have further investigated this phenomenon of an oversized channel with to determine the mechanism behind the size discrepancy. In particular we have explored the possibility of channel expansion similar to ‘self-pressurisation’ in fibres drawn from preforms that have been first sealed to the atmosphere, as previously described by Voyce et al. [J. Lightwave Technol. 27, 871 (2009) [CrossRef] ]. For this, two pieces from each of two preforms with different inner to outer diameter ratios were drawn to fibre, one open to the atmosphere and the other with a sealed end. In addition, we have sectioned a cooled neck-down region from a previous experiment, for which the fibre had an oversized channel compared to the model prediction, and measured the cross-sectional slices. We here compare this new experimental data with the predictions of the previously derived model for drawing of an unsealed preform and a new model, developed herein, for drawing of a sealed tube. We establish that the observed oversized channels are not consistent with the self-pressurisation model for the sealed tube.

MIYOSHI H, Crowdy D, Nelson R, 2021, Fay meets van der Pauw: the trisecant identity and the resistivity of holey samples, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, ISSN: 1364-5021

The van der Pauw method is commonly used in the applied sciences to find the resistivity of a simply connected, two-dimensional conducting laminate. Given the usefulness of this “4-point probe” method there has been much recent interest in trying to extend it to holey, that is, multiply connected, samples. This paper introduces two new mathematical tools to this area of investigation – the prime function on the Schottky double of a planar domain and the Fay trisecant identity – and uses them to show how the van der Pauw method can be extended to find the resistivity of a sample with a hole. We show that an integrated form of the Fay trisecant identity provides valuable information concerning the appearance of “envelopes” observed in the case of holey samples by previous authors. We find explicit formulas for these envelopes, as well as an approximate formula relating two pairs of resistance measurements to the sample resistivity that is expected to be valid when the hole is sufficiently small and not too close to the outer boundary. We describe how these new mathematical tools have enabled us to prove certain conjectures recently made in the engineering literature.

Nelson R, Krishnamurthy V, Crowdy D, 2021, The corotating hollow vortex pair: steady merger and break-up via a topological singularity, *Journal of Fluid Mechanics*, Vol: 907, ISSN: 0022-1120

The shapes of two steadily rotating, equal circulation, two-dimensional hollow vortices are determined and their properties examined. By means of a numerical scheme that accounts for the doubly connected nature of the fluid domain, it is shown that a one-parameter family of solutions exists that is a continuation of a corotating point vortex pair. Withb= 2 set as the distance between the vortex centroids we find that each vortex reaches a maximum possible area of 0.796 corresponding toa/b= 0.260 where a is a measure of the vortex core radius proposed by Meunieret al[Phys. Fluids,14, (2002)]. Results are compared to those of a previous study by Saffman & Szeto [Phys. Fluids,23, (1980)] in which two corotating patches of uniform vorticity are considered in place of the hollow vortices studied here. The general behaviour of the two systems is seen to be similar but some differences are highlighted, especially when the vortices become close to touching due to the accumulation of vorticity in thin extended fingers emanating from each of the vortices. The numerical scheme captures the family of equilibria very close to a critical configuration where these fingers tend to touch at the centre of rotation corresponding to a/b≈0.283. By a simple adaptation of the numerical scheme to compute 2-fold rotationally symmetric equilibria for a single rotating hollow vortex we then show that its limiting configuration is one where a thin waist forms leading to two separate parts of its single boundary drawing close together. We give evidence that the limit of this single vortex configuration coincides with the limit of the two-vortex configuration. The limiting configuration itself turns out not to be physically admissible leading to what we refer to as a topological singularity since no physical quantities blow up, indeed they appear to be continuous as the limiting state is approached from the two topologically distinct directions.

Crowdy D, 2021, Superhydrophobic annular pipes: a theoretical study, *Journal of Fluid Mechanics*, Vol: 906, Pages: A15-1-A15-33, ISSN: 0022-1120

Analytical solutions are presented for longitudinal flow along a superhydrophobic annularpipe where one wall, either the inner or outer, is a fully no-slip boundary while the otheris a no-slip wall decorated by a rotationally symmetric pattern of no-shear longitudinalstripes. Formulas are given for the effective slip length associated with laminar flow alongthe superhydrophobic pipe and the friction properties are characterized. It is shownhow these new solutions generalize two solutions to mixed no-slip/no-shear boundaryvalue problems due to Philip [J. Appl. Math. Phys., 23, (1972)] for flow in a singlewalled superhydrophobic pipe and a superhydrophobic channel. This is done by providingalternative representations of Philip’s two solutions, including a useful new formula forthe effective slip length for his channel flow solution. For a superhydrophobic annularpipe with inner-wall no-shear patterning there is an optimal pipe bore for enhancinghydrodynamic slip for a given pattern of no-shear stripes. These optimal pipes havea ratio of inner-outer pipe radii in the approximate range 0.5–0.6 and depending onlyweakly on the geometry of the surface patterning. Boundary point singularities are foundto be deleterious to the slip suggesting that, in designing slippery pipes, maximizing thesize of uninterrupted no-shear regions is preferable to covering the same surface area witha larger number of smaller no-shear zones. The results add to a list of analytical solutionsto mixed boundary value problems relevant to modelling superhydrophobic surfaces

Constantin A, Crowdy D, Krishnamurthy V,
et al., 2021, Stuart-type polar vortices on a rotating sphere, *Discrete and Continuous Dynamical Systems Series A*, Vol: 41, Pages: 201-215, ISSN: 1078-0947

Stuart vortices are among the few known smooth explicit solu-tions of the planar Euler equations with a nonlinear vorticity, and they can beadapted to model inviscid flow on the surface of a fixed sphere. By means ofaperturbativeapproachweshowthatthemethodusedtoinvestigateStuartvortices on a fixed sphere provides insight into the dynamics of the large-scalezonal flows on a rotating sphere that model the background flow of polar vor-tices. Our approach takes advantage of the fact that while a sphere is spinningaround its polar axis, every point on the sphere has the same angular velocitybut its tangential velocity is proportional to the distance from the polar axisof rotation, so that points move fastest at the Equator and slower as we gotowards the poles, both of which remain fixed.

Crowdy D, 2020, Collective viscous propulsion of a two-dimensional flotilla of Marangoni boats, *Physical Review Fluids*, Vol: 5, Pages: 124004 – 1-124004 – 17, ISSN: 2469-990X

A closed-form solution is presented for the collective Marangoni-induced motion of a two-dimensional periodic array, or “flotilla”, of Marangoni boats on deep water at zero Reynolds, capillary and surface P´eclet numbers. The physical set-up is identical to the model of Marangoni propulsion proposed by Lauga & Davis [J. Fluid Mech., 705, (2012)] but accounts now for interaction effects between boats, and in a simpler two-dimensional setting. The boats are modelled as identical thin floating strips each self-actuated by a trailing edge surfactant source that lowers the surface tension there according to a linear equation of state. The collective Marangoni propulsion speed of a flotilla of boats is found to be (2πµδ) −1∆σ log sec(πδ/2) where δ is the meniscus coverage fraction, µ is the subphase fluid viscosity and ∆σ is the surfactant-induced surface tension disparity across each boat. The theoretical result exemplifies the mechanism for collective rectilinear motion due to Marangoni convection caused by the diffusion of insoluble surfactant.Keywords: Marangoni boat, camphor boat, viscous propulsion, active particle.

Yariv E, Crowdy D, 2020, Phoretic self-propulsion of Janus discs in the fast-reaction limit, *Physical Review Fluids*, Vol: 5, ISSN: 2469-990X

Due to the net interfacial consumption of solute, the two-dimensional problem of phoretic swimming is ill posed in the standard description of diffusive transport, where the solute concentration satisfies Laplace's equation. It becomes well posed when solute advection is accounted for. We consider here the case of weak advection, where solute transport is analyzed using matched asymptotic expansions in two separate asymptotic regions, a near-field region in the vicinity of the swimmer and a far-field region where solute advection enters the dominant balance. We carry out the analysis for a standard Janus configuration, where half of the particle boundary is active and the other half is inert. Our main focus lies in the limit of fast reaction, which leads to a mixed boundary-value problem in the near field. That problem is solved using conformal mapping techniques. Our asymptotic scheme furnishes an implicit equation for the particle velocity s in the direction of the active portion of its boundary, 2s(8ln8D|s|a−γ)=bc∞/a, wherein a is the particle radius, D the solute diffusivity, c∞ its far-field concentration, b the diffusio-osmotic slip coefficient, and γ the Euler-Mascheroni constant. The nonlinear dependence of s upon bc∞ is a signature of the nonvanishing effect of solute advection.

Anselmo da Silva T, Carneiro da Cunha B, Nelson R,
et al., 2020, Schwarz-Christoffel accessory parameter for quadrilaterals via isomonodromy, *Journal of Physics A: Mathematical and Theoretical*, Vol: 53, ISSN: 1751-8113

We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz-Christoffel type: in particular, the mapping from the upper half plane to a 4-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters -- the pre-image of the polygonal vertices -- via a special ``zero curvature limit'' in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.

Krishnamurthy VS, Wheeler MH, Crowdy DG,
et al., 2020, A transformation between stationary point vortex equilibria: Transformation point vortex equilibria, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 476, ISSN: 1364-5021

A new transformation between stationary point vortex equilibria in the unbounded plane is presented. Given a point vortex equilibrium involving only vortices with negative circulation normalized to -1 and vortices with positive circulations that are either integers or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant. When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations, each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler-Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko (Loutsenko 2004 J. Phys. A: Math. Gen. 37, 1309-1321 (doi:10.1088/0305-4470/37/4/017)). For the latter polynomials, the existence of such a transformation appears to be new. The new transformation, therefore, unifies a wide range of disparate results in the literature on point vortex equilibria.

Krishnamurthy V, Wheeler M, Crowdy D,
et al., 2020, A transformation between stationary point vortex equilibria, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 476, Pages: 1-21, ISSN: 1364-5021

A new transformation between stationary point vortex equilibria in the unbounded plane is presented.Given a point vortex equilibrium involving only vortices with negative circulation normalized to−1 and vortices with positive circulations that are either integers, or half-integers, the transformation produces a new equilibrium with a free complex parameter that appears as an integration constant.When iterated the transformation can produce infinite hierarchies of equilibria, or finite sequences that terminate after a finite number of iterations,each iteration generating equilibria with increasing numbers of point vortices and free parameters. In particular, starting from an isolated point vortex as a seed equilibrium, we recover two known infinite hierarchies of equilibria corresponding to the Adler–Moser polynomials and a class of polynomials found, using very different methods, by Loutsenko[J. Phys. A: Math. Gen. 37, (2004)]. For the latter polynomials the existence of such a transformation appears to be new. The new transformation therefore unifies a wide range of disparate results in the literature on point vortex equilibria.

Kirk T, Karamanis G, Crowdy D,
et al., 2020, Thermocapillary stress and meniscus curvature effects on slip lengths in ridged microchannels, *Journal of Fluid Mechanics*, Vol: 894, ISSN: 0022-1120

Pressure-driven flow in the presence of heat transfer through a microchannel patterned with parallel ridges is considered. The coupled effects of curvature and thermocapillary stress along the menisci are captured. Streamwise and transverse thermocapillary stresses along menisci cause the flow to be three-dimensional, but when the Reynolds number based on the transverse flow is small the streamwise and transverse flows decouple. In this limit, we solve the streamwise flow problem, i.e. that in the direction parallel to the ridges, using a suite of asymptotic limits and techniques – each previously shown to have wide ranges of validity thereby extending results by Hodes et al. (J. Fluid Mech., vol. 814, 2017, pp. 301–324) for a flat meniscus. First, we take the small-ridge-period limit, and then we account for the curvature of the menisci with two further complementary limits: (i) small meniscus curvature using boundary perturbation; (ii) arbitrary meniscus curvature but for small slip (or cavity) fractions using conformal mapping and the Poisson integral formula. Heating and cooling the liquid always degrade and enhance (apparent) slip, respectively, but their effect is greatest for large meniscus protrusions, with positive protrusion (into the liquid) being the most sensitive. For strong enough heating the solutions become complex, suggesting instability, with large positive protrusions transitioning first.

Yariv E, Crowdy D, 2020, Longitudinal thermocapillary flow over a dense bubble mattress, *SIAM Journal on Applied Mathematics*, Vol: 80, Pages: 1-19, ISSN: 0036-1399

A common form of superhydrophobic surface is made out of a periodically groovedsolid substrate, wherein cylindrical bubbles are trapped in a Cassie state. When a macroscopic tem-perature gradient is externally applied, Marangoni forces generate thermocapillary flow of character-istic magnitudeU=−aGσT/μ, in which 2ais the groove width,Gthe applied-gradient magnitude,μthe liquid viscosity, andσTthe derivative of interfacial-tension coefficient with respect to the temper-ature. We consider the case of a gradient which is applied parallel to the grooves. Assuming a highlyconducting solid substrate, we seek to calculate the longitudinal velocity component, anti-parallelto the applied gradient, and in particular its “slip” value, attained at large distances away from thesurface. Normalized byU, this value depends only upon the bubble protrusion angle and the solidfraction . In this paper we consider the small solid-fraction limit 1 and focus upon the case of90◦protrusion angle, for which this limit is known to be highly singular in the comparable problemof shear-driven flow [Schnitzer, Phys. Rev. Fluids, 1 (2016), 052101]. Using matched asymptoticexpansions, we find that the dimensionless slip velocity is given byπ2/√8 + ln +I−ln(8π2) +o(1).The first two terms in this expansion follow from a lubrication-type analysis of the narrow gap regionseparating two neighboring bubbles. The subsequentO(1) terms follow from asymptotic matchingwith the bubble-scale region, where the bubbles appear to be touching. The solution in that regionis obtained using conformal mapping techniques, with the constantIgiven as an explicit integral,evaluated numerically to be 2.27.

Crowdy D, Luca E, 2019, Analytical solutions for two-dimensional singly periodic Stokes flow singularity arrays near walls, *Journal of Engineering Mathematics*, Vol: 119, Pages: 199-215, ISSN: 0022-0833

New analytical representations of the Stokes flows due to periodic arrays of point singularities in a two-dimensional no-slip channel and in the half-planenear a no-slip wall are derived. The analysis makes use of a conformal mappingfrom a concentric annulus (or a disc) to a rectangle and a complex variable formulation of Stokes flow to derive the solutions. The form of the solutions is amenable tofast and accurate numerical computation without the need for Ewald summationor other fast summation techniques.

Baddoo P, Crowdy D, 2019, Periodic Schwarz-Christoffel mappings with multiple boundaries per period, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 475, ISSN: 1364-5021

We present an extension to the theory of SchwarzChristoffel (S-C) mappings by permitting the targetdomain to be a single period window of a periodicconfiguration having multiple polygonal (straightline) boundaries per period. Taking the arrangementsto be periodic in the x direction in an (x, y) plane,three cases are considered; these differ in whether theperiod window extends off to infinity as y → ±∞,or extends off to infinity in only one direction(y → +∞ or y → −∞), or is bounded. The preimagedomain is taken to be a multiply connected circulardomain. The new S-C mapping formulas are shownto be expressible in terms of the Schottky-Kleinprime function associated with the circular preimagedomains. As usual for an S-C map, the formulasare explicit but depend on a finite set of accessoryparameters. The solution of this parameter problemis discussed in detail, and illustrative examples arepresented to highlight the essentially constructivenature of the results.

Yariv E, Crowdy D, 2019, Thermocapillary flow between grooved superhydrophobic surfaces: transverse temperature gradients, *Journal of Fluid Mechanics*, Vol: 871, Pages: 775-798, ISSN: 0022-1120

We consider the thermocapillary motion of a liquid layer which is bounded between two superhydrophobic surfaces, each made up of a periodic array of highly conducting solid slats, with flat bubbles trapped in the grooves between them. Following the recent analysis of the longitudinal problem (Yariv, J. Fluid Mech., vol. 855, 2018, pp. 574–594), we address here the transverse problem, where the macroscopic temperature gradient that drives the flow is applied perpendicular to the grooves, with the goal of calculating the volumetric flux between the two surfaces. We focus upon the situation where the slats separating the grooves are long relative to the groove-array period, for which case the temperature in the solid portions of the superhydrophobic plane is piecewise uniform. This scenario, which was investigated numerically by Baier et al. (Phys. Rev. E, vol. 82 (3), 2010, 037301), allows for a surprising analogy between the harmonic conjugate of the temperature field in the present problem and the unidirectional velocity in a comparable longitudinal pressure-driven flow problem over an interchanged boundary. The main body of the paper is concerned with the limit of deep channels, where the problem reduces to the calculation of the heat transport and flow about a single surface and the associated ‘slip’ velocity at large distance from that surface. Making use of Lorentz’s reciprocity, we obtain that velocity as a simple quadrature, providing the analogue to the expression obtained by Baier et al. (2010) in the comparable longitudinal problem. The rest of the paper is devoted to the diametric limit of shallow channels, which is analysed using a Hele-Shaw approximation, and the singular limit of small solid fractions, where we find a logarithmic scaling of the flux with the solid fraction. The latter two limits do not commute.

Crowdy D, Krishnamurthy V, Wheeler M,
et al., 2019, Steady point vortex pair in a field of Stuart-type vorticity, *Journal of Fluid Mechanics*, Vol: 874, Pages: R1-1-R1-11, ISSN: 0022-1120

A new family of exact solutions to the two-dimensional steady incompressible Eulerequation is presented. The solutions provide a class of hybrid equilibria comprisingtwo point vortices of unit circulation – a point vortex pair – embedded in a smooth seaof non-zero vorticity of ‘Stuart-type’ so that the vorticity ω and the stream functionψ are related by ω = aebψ − δ(x − x0) − δ(x + x0), where a and b are constants. Wealso examine limits of these new Stuart-embedded point vortex equilibria where theStuart-type vorticity becomes localized into additional point vortices. One such limitresults in a two-real-parameter family of smoothly deformable point vortex equilibriain an otherwise irrotational flow. The new class of hybrid equilibria can be viewedas continuously interpolating between the limiting pure point vortex equilibria. Atthe same time the new solutions continuously extrapolate a similar class of hybridequilibria identified by Crowdy (Phys. Fluids, vol. 15, 2003, pp. 3710–3717).

Mayer M, Hodes M, Kirk T,
et al., 2019, Effect of surface curvature on contact resistance between cylinders, *Journal of Heat Transfer*, Vol: 141, ISSN: 0022-1481

Due to the microscopic roughness of contacting materials, an additional thermal resistance arises from the constriction and spreading of heat near contact spots. Predictive models for contact resistance typically consider abutting semi-infinite cylinders subjected to an adiabatic boundary condition along their outer radius. At the nominal plane of contact, an isothermal and circular contact spot is surrounded by an adiabatic annulus and the far-field boundary condition is one of constant heat flux. However, cylinders with flat bases do not mimic the geometry of contacts. To remedy this, we perturb the geometry of the problem such that, in cross section, the circular contact is surrounded by an adiabatic arc. When the curvature of this arc is small, we employ a series solution for the leading-order (flat base) problem. Then, Green's second identity is used to compute the increase in spreading resistance in a single cylinder, and thus the contact resistance for abutting ones, without fully resolving the temperature field. Complementary numerical results for contact resistance span the full range of contact fraction and protrusion angle of the arc. The results suggest as much as a 10–15% increase in contact resistance for realistic contact fraction and asperity slopes. When the protrusion angle is negative, the decrease in spreading resistance for a single cylinder is also provided.

Luca E, Crowdy DG, 2018, A transform method for the biharmonic equation in multiply connected circular domains, *IMA Journal of Applied Mathematics*, Vol: 83, Pages: 942-976, ISSN: 0272-4960

A new transform approach for solving mixed boundary value problems for the biharmonic equation in simply and multiply connected circular domains is presented. This work is a sequel to Crowdy (2015, IMA J. Appl. Math., 80, 1902–1931) where new transform techniques were developed for boundary value problems for Laplace’s equation in circular domains. A circular domain is defined to be a domain, which can be simply or multiply connected, having boundaries that are a union of circular arc segments. The method provides a flexible approach to finding quasi-analytical solutions to a wide range of problems in fluid dynamics and plane elasticity. Three example problems involving slow viscous flows are solved in detail to illustrate how to apply the method; these concern flow towards a semicircular ridge, a translating and rotating cylinder near a wall as well as in a channel geometry.

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