## Publications

184 results found

Miyoshi H, Rodriguez-Broadbent H, Crowdy D, 2024, Numerical validation of analytical formulas for channel flows over liquid-infused surfaces, *Journal of Engineering Mathematics*, Vol: 144, ISSN: 0022-0833

This paper provides numerical validation of some new explicit, asymptotically exact, analytical formulas describing channel flows over liquid-infused surfaces, an important class of surfaces of current interest in surface engineering. The new asymptotic formulas, reproduced here, were derived in a recent companion paper by the authors. The numerical validation is done by presenting a novel computational method for calculating longitudinal flow in a periodic channel involving finite-length closed liquid-filled grooves with a flat two-fluid interface, a challenging problem given the two-fluid nature of the flow. The formulas are asymptotically exact for wide channels where the grooves on the lower wall of the channel are well separated; the numerical method devised here, however, is subject to no such restrictions. Significantly, it is shown here that the asymptotic formulas remain good global approximants for the flow over a wide range of flow geometries, including those well outside the asymptotic parameter range for which they were derived. It is found that the formulas are more reliable for liquid-infused surfaces than for superhydrophobic surfaces.

Baddoo PJ, Moore NJ, Oza AU,
et al., 2023, Generalization of waving-plate theory to multiple interacting swimmers, *Communications on Pure and Applied Mathematics*, Vol: 76, Pages: 3811-3851, 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, Mayer M, Hodes M, 2023, Asymmetric thermocapillarity-based pump: concept and exactly solved model, *Physical Review Fluids*, Vol: 8, ISSN: 2469-990X

A mechanism for a microfluidic pump that leverages alternating adverse and favorable thermocapillary stresses along menisci in a (periodically) fully developed transverse flow in a microchannel is exemplified. The transverse ridges are the interdigitated teeth of cold and hot (isothermal) “combs” and free surface menisci span the interstitial regions between them. The teeth are asymmetrically positioned so that the widths of adjacent menisci differ. This architecture is essentially that of the theoretical pump proposed by Adjari [Phys. Rev. E 61, R45(R) (2000)] but exploits thermocapillarity rather than electro-osmotic slip to drive unidirectional pumping. A theoretical model of the multiphysics pumping mechanism is given that is solved in closed form. Two explicit formulas for the pumping speed are provided. One is derived from the exact solution to the full problem; the other follows from the reciprocal theorem for Stokes flow combined with an exact solution to a distinct problem resolving apparent slip over superhydrophobic surfaces [D. G. Crowdy, Phys. Fluids 23, 072001 (2011)]. A conceptual design of the pump is also outlined; this involves no moving parts, requires no external driving pressure, and pumps a continuous stream of liquid through a microchannel, as opposed to a series of discrete droplets. Since there is only a periodic component of the pressure field the microchannel could be made arbitrarily long and the menisci, which would be essentially flat, are more robust than for conventional pressure-driven flow.

Crowdy D, Curran A, Papageorgiou D, 2023, Fast reaction of soluble surfactant can remobilize a stagnant cap, *Journal of Fluid Mechanics*, Vol: 969, Pages: 1-29, ISSN: 0022-1120

Analytical solutions are derived showing that a stagnant cap of surfactant at the interface between two viscous fluids caused by a linear extensional flow can be remobilized by fast kinetic exchange of surfactant with one of the fluids. Using a complex variable formulation of this multiphysics problem at zero capillary number, zero Reynolds number and zero bulk Péclet number, and assuming a linear equation of state, it is shown that the system is governed by a forced complex Burgers equation at arbitrary surface Péclet number. Consequently, this nonlinear system is shown to be linearizable using a complex analogue of the Cole–Hopf transformation. Steady equilibria of the system at any finite value of the surface Péclet number are found explicitly in terms of parabolic cylinder functions. While surface diffusion is naturally expected to mollify sharp gradients associated with stagnant caps and to remobilize the interface, this work gives an analytical demonstration of the less intuitive result that fast kinetic exchange has a similar effect. Indeed, the analytical approach here imposes no limit on the surface Péclet number, which can be taken to be infinitely large so that surface diffusion is completely absent. Mathematically, the solution structure is then very rich allowing a theoretical investigation of this extreme case where it is seen that fast surfactant exchange with the bulk can alone remobilize a stagnant cap. Remarkably, it is also possible to track explicitly the time evolution of the system to these remobilized equilibria by finding time-evolving exact solutions.

Crowdy D, Keeler J, 2023, Exact solutions for submerged von Kármán point vortex streets cotravelling with a wave on a linear shear current, *Journal of Fluid Mechanics*, Vol: 969, Pages: 1-26, ISSN: 0022-1120

New exact solutions are presented to the problem of steadily travelling water waves with vorticity wherein a submerged von Kármán point vortex street cotravels with a wave on a linear shear current. Surface tension and gravity are ignored. The work generalizes an earlier study by Crowdy & Nelson (Phys. Fluids, vol. 22, 2010, 096601) who found analytical solutions for a single point vortex row cotravelling with a water wave in a linear shear current. The main theoretical tool is the Schwarz function of the wave, and the work builds on a novel framework set out recently by Crowdy (J. Fluid Mech., vol. 954, 2022, A47). Conformal mapping theory is used to construct Schwarz functions with the requisite properties and to parametrize the waveform. A two-parameter family of solutions is found by solving a pair of nonlinear algebraic equations. This system of equations has intriguing properties: indeed, it is degenerate, which radically reduces the number of possible solutions, although the space of physically admissible equilibria is still found to be rich and diverse. For inline vortex streets, where the two vortex rows are aligned vertically, there is generally a single physically admissible solution. However, for staggered streets, where the two vortex rows are offset horizontally, certain parameter regimes produce multiple solutions. An important outcome of the work is that while only degenerate von Kármán point vortex streets can exist in an unbounded simple shear current, a broad array of such equilibria is possible in a shear current beneath a cotravelling wave on a free surface.

Miyoshi H, Crowdy D, 2023, Generalized Schwarz integral formulas for multiply connected domains, *SIAM Journal on Applied Mathematics*, Vol: 83, Pages: 966-984, ISSN: 0036-1399

A generalization of the Schwarz integral formula in the class of multiply connected circular domains is constructed. A classical Schwarz integral formula retrieves, up to an imaginary constant, an analytic function in a domain given its real part on the boundary of a domain; Poisson integral formulas are well-known examples for simply connected domains. The generalized integral formulas derived here retrieve an analytic function given more general linear combinations of its real and imaginary parts on each boundary component of a multiply connected domain. Those linear combinations can be different on each boundary component. The chief mathematical tool is the prime function of a multiply connected circular domain. A Schwarz integral formula for such domains, retrieving an analytic function given its real part on all boundary components, was derived in terms of the prime function by Crowdy [Complex Variables and Elliptic Equations, 53(3), 221-236, (2008)]. The present paper combines those formulas with use of radial slit conformal mappings, also expressible in terms of the prime function, to produce integral expressions for analytic functions where more general linear combinations of their real and imaginary parts are specified on the boundary components of a multiply connected domain. We refer to the resulting expressions as generalized Schwarz integral formulas. Their usefulness and versatility are showcased by application to three topical problems: finding the potential around periodic interdigitated electrodes, solving the free boundary problem for hollow vortex wakes behind a bluff body, and determining the two-phase flow over a so-called liquid-infused surface.

Miyoshi H, Crowdy D, 2023, Estimating conformal capacity using asymptotic matching, *IMA Journal of Applied Mathematics*, Vol: 88, Pages: 472-497, ISSN: 0272-4960

Conformal capacity is a mathematical quantity relevant to a wide range of physical and mathematicalproblems and there has been a recent 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 conformalcapacity. The formulas derived here are explicit, and evidence is given that they provide excellentapproximations to the exact capacity values even well outside the expected range of validity.

Crowdy D, Rodriguez-Broadbent H, 2023, Superhydrophobic surfaces with recirculating interfacial flow due to surfactants are 'effectively' immobilised, *Journal of Fluid Mechanics*, Vol: 956, Pages: 1-11, ISSN: 0022-1120

At high surface Péclet numbers, it is common to associate the presence of surfactants with surface immobilization, where a free surface becomes indistinguishable from a no-slip surface. A different mechanism has recently been proposed for longitudinal shear flow along a unidirectional trench (Baier & Hardt, J. Fluid Mech., vol. 949, 2022, A34) wherein, at high Marangoni numbers, the meniscus spanning the finite-length trench becomes a constant-shear-stress surface due to contamination by incompressible surfactant. That model predicts recirculating interfacial flows on the meniscus, a phenomenon that has been observed experimentally (Song et al., Phys. Rev. Fluids, vol. 3, issue 3, 2018, 033303). By finding an explicit solution to the constant-shear-stress model at all protrusion angles and calculating the effective slip length for a dilute mattress of such surfactant-laden trenches, we show that those effective slip lengths are almost indistinguishable from those for a surface whose menisci have the same deflection but have been completely immobilized (i.e. they are no-slip surfaces). This means that, despite the presence of non-trivial recirculating vortical flows on the menisci, the aggregate slip characteristics of such surfaces are that they have been effectively immobilized. This surprising result underscores the need for caution in comparing theory with experiments based on effective slip properties alone.

Crowdy D, 2023, Exact solutions for steadily travelling water waves with submerged point vortices, *Journal of Fluid Mechanics*, Vol: 954, Pages: 1-35, ISSN: 0022-1120

This paper presents a novel theoretical framework, based on the concept of the Schwarz function of a wave, for understanding water waves with vorticity in the absence of gravity and capillarity. The framework leads naturally to a taxonomy of three subcases, herein referred to as cases 1, 2 and 3, into which fall three existing studies of water waves incorporating uniform vorticity and submerged point vortices. This provides a theoretical unification of several seemingly unrelated results in the literature. It also provides a route to finding new exact solutions with this paper focussing on new solutions falling within the case 2 category. Among several presented here are a submerged point vortex pair cotravelling with a solitary deep-water wave, von Kármán point vortex streets cotravelling with a periodic deep-water wave and a point vortex row cotravelling with a wave in water of finite depth. Some other more exotic waveforms are also constructed. All these new solutions generalize those of Crowdy & Roenby (Fluid Dyn. Res., vol. 46, 2014) who found steady waves in deep water cotravelling with a submerged point vortex row for which the free surface shapes turn out to coincide with those of pure capillary waves on deep water found by Crapper (J. Fluid Mech., vol. 2, 1957). The new exact solutions are likely to provide a useful basis for asymptotic or numerical studies when additional effects such as gravity and capillarity are incorporated.

Crowdy D, Rodriguez-Broadbent H, 2022, Superhydrophobicity can enhance convective heat transfer in pressure-driven pipe flow, *Quarterly Journal of Mechanics and Applied Mathematics*, Vol: 75, Pages: 315-346, ISSN: 0033-5614

Theoretical evidence is given that it is possible for superhydrophobicity to enhance steady laminar convective heat transfer in pressure-driven flow along a circular pipe or tube with constant heat flux. Superhydrophobicity here refers to the presence of adiabatic no-shear zones in an otherwise solid no-slip boundary. Adding such adiabatic no-shear zones reduces not only hydrodynamic friction, leading to greater fluid volume fluxes for a given pressure gradient, but also reduces the solid surface area through which heat enters the fluid. This leads to a delicate trade-off between competing mechanisms so that the net effect on convective heat transfer along the pipe, as typically measured by a Nusselt number, is not obvious. Existing evidence in the literature suggests that superhydrophobicity always decreases the Nusselt number, and therefore compromises the net heat transfer. In this theoretical study, we confirm this to be generally true but, significantly, we identify a situation where the opposite occurs and the Nusselt number increases thereby enhancing convective heat transfer along the pipe.

Mayer M, Crowdy D, 2022, Superhydrophobic surface immobilization by insoluble surfactant, *Journal of Fluid Mechanics*, Vol: 949, Pages: 1-31, ISSN: 0022-1120

The effects of insoluble surfactants, satisfying a Langmuir equation of state, on transverse Stokes flow over a superhydrophobic surface of unidirectional grooves with flat menisci are examined. The phenomenon of surface immobilization, whereby surfactants cause part or all of the Cassie-state menisci to become effectively no-slip zones thereby degrading the slip properties of the surface, is of primary interest. The study employs a combined analytical and numerical approach allowing an exploration of surfactant effects over the full range of surface P ́eclet numbers, Marangoni numbers and surfactant loads. For small surface P ́eclet and Marangoni numbers, perturbation theory is used to gain basic insights into the physical mechanisms at work. That analysis also provides checks on a robust numerical scheme, built around a complex variable formulation of the problem, used to compute solutions across a wide range of non-dimensional parameter values. Two distinct mechanisms are found to be responsible for surface immobilization. In the first, most commonly seen at higher surface P ́eclet numbers, a stagnant cap forms because swept surfactant immobilizes a section of the meniscus before any portion of the meniscus reaches maximum packing. Such a cap exists even for a linear equation of state and grows in length with increasing Marangoni number. A second immobilization mechanism is associated with the non-linear equation of state: an immobilized region forms because the surfactant concentration reaches its maximum value near the downstream edge of the meniscus. Immobilization of the latter form can occur at much lower surface P ́eclet numbers, even if the Marangoni number is small, as long as there is sufficient surfactant inthe system. The study enhances understanding of how insoluble surfactants can degrade slip over superhydrophobic surfaces.

MIYOSHI H, Rodriguez-Broadbent H, Curran A,
et al., 2022, Longitudinal flow in superhydrophobic channels with partially invaded grooves, *Journal of Engineering Mathematics*, Vol: 137, ISSN: 0022-0833

Analytical expressions are derived for the longitudinal flow in a super-hydrophobic microchannel where flat menisci in the Cassie state havepartially invaded the grooves between no-slip blades. Using these solu-tions, the effective slip lengths are computed and compared with recentanalytical results for unbounded shear flow over the same class ofsurfaces. Expressions for the first-order corrections to these effectiveslip lengths when the menisci are weakly curved are also derived.A mathematical connection to superhydrophobic channel flows wherethe flat menisci are still pinned to the tops of the pillars is alsomade, resulting in novel analytical expressions for those solutions too.

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, 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

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.

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.

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.

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