Publications
184 results found
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.
Stokes YM, Crowdy DG, Ebendorff-Heidepriem H, et al., 2019, Can We Fabricate That Fibre?, IUTAM Symposium on Recent Advances in Moving Boundary Problems in Mechanics, Publisher: SPRINGER INTERNATIONAL PUBLISHING AG, Pages: 1-13, ISSN: 1875-3507
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.
Crowdy DG, Nelson R, Anselmo T, et al., 2018, Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 474, ISSN: 1364-5021
We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.
Crowdy DG, Luca E, 2018, Fast evaluation of the fundamental singularities of two-dimensional doubly periodic Stokes flow, Journal of Engineering Mathematics, Vol: 111, Pages: 95-110, ISSN: 0022-0833
Analytical representations of the flows associated with doubly periodic arrangements of point singularities of two-dimensional Stokes flow are derived. The analysis makes use of a conformal mapping from a concentric annulus to a rectangle. A natural mathematical object known as the Schottky–Klein prime function is used to derive the solutions avoiding the theory of elliptic functions. The new expressions have the advantage of being immediately amenable to fast evaluation without the need for Ewald or other fast summation methods or acceleration techniques for lattice sums.
Smith DJ, Pedley TJ, Crowdy DG, 2018, Special issue in honour of Professor John Blake FIMA CMath, IMA JOURNAL OF APPLIED MATHEMATICS, Vol: 83, Pages: 553-555, ISSN: 0272-4960
Nelson R, Zuo L, Weijermars R, et al., 2018, Applying improved analytical methods for modelling flood displacement fronts in bounded reservoirs (Quitman field, east Texas), Journal of Petroleum Science and Engineering, Vol: 166, Pages: 1018-1041, ISSN: 1873-4715
We apply an improved potential flow model based on conformal mapping to study the sweep pattern of waterfloods in bounded reservoirs. Solutions for streamlines and flood advancement obtained with the new model are validated using an independent but more intricate numerical streamline simulation method. Subsequently, the use of the benchmarked model is demonstrated in a review of the flow patterns in the Quitman field, a tabular reservoir comprising an 18 ft thick payzone of Harris sand interfingering with shales of the Eagle Ford formation. The reservoir is bound by impervious faults modelled by the method of images in earlier studies of the 1970's. In the present study, Riemann's mapping theorem in combination with the Schottky-Klein prime function is applied to find complex potentials that describe the flow in the bounded reservoir. Such an approach can model waterflooding in marginal fields like the Quitman oil field more accurately than previous potential flow methods and can visualize the sweep pattern and compute time-of-flight contours in a simpler and faster fashion than numerical streamline simulators.
Crowdy D, Hodes M, Kirk T, 2018, Spreading and contact resistance formulae capturing boundary curvature and contact distribution effects, Journal of Heat Transfer, Vol: 140, ISSN: 0022-1481
There is a substantial and growing body of literature which solves Laplace's equation governing the velocity field for a linear-shear flow of liquid in the unwetted (Cassie) state over a superhydrophobic surface. Usually, no-slip and shear-free boundary conditions are applied at liquid–solid interfaces and liquid–gas ones (menisci), respectively. When the menisci are curved, the liquid is said to flow over a “bubble mattress.” We show that the dimensionless apparent hydrodynamic slip length available from studies of such surfaces is equivalent to (i) the dimensionless spreading resistance for a flat, isothermal heat source flanked by arc-shaped adiabatic boundaries and (ii) the dimensionless thermal contact resistance between symmetric mating surfaces with flat contacts flanked by arc-shaped adiabatic boundaries. This is important because real surfaces are rough rather than smooth. Furthermore, we demonstrate that this observation provides a significant source of new and explicit results on spreading and contact resistances. Significantly, the results presented accommodate arbitrary solid-to-solid contact fraction and arc geometry in the contact resistance problem for the first time. We also provide formulae for the case when each period window includes a finite number of no-slip (or isothermal) and shear free (or adiabatic) regions and extend them to the case when the latter are weakly curved. Finally, we discuss other areas of mathematical physics to which our results are directly relevant.
Crowdy DG, Krishnamurthy VS, 2017, The effect of core size on the speed of compressible hollow vortex streets, Journal of Fluid Mechanics, Vol: 836, Pages: 797-827, ISSN: 0022-1120
The effect of weak compressibility on the speed of steadily translating staggered vortex streets of hollow vortices in isentropic subsonic flow is studied. A small-Mach-number perturbation expansion about the incompressible solutions for staggered streets of hollow vortices found recently by Crowdy & Green (Phys. Fluids, 2011, vol. 23, 126602) is carried out; the latter solutions provide a desingularization of the classical point vortex streets of von Kármán. The first-order compressible flow correction is calculated. We employ a novel scheme based on a complex variable formulation of the compressible flow equations (the Imai–Lamla method) combined with conformal mapping theory to track the vortex shape in this free boundary problem. The analysis to find the perturbed streamfunction and compressible vortex shapes is greatly facilitated by exploiting a calculus based on use of the Schottky–Klein prime function of a conformally equivalent parametric annulus. It is found that, for a vortex street of specified aspect ratio comprising vortices of specified circulation, the vortex core size is a key determinant of whether compressibility increases or decreases the steady propagation speed (relative to the incompressible street with the same parameters) and that both eventualities are possible. We focus attention on streets with aspect ratios around 0.28, which is close to the neutrally stable case for incompressible flow, and find that a critical vortex core size exists at which compressibility does not affect the speed of the street at first order in the (squared) Mach number. Streets comprising vortices with core size below the critical value speed up due to compressibility; larger vortices slow down.
Crowdy D, 2017, Effect of shear thinning on superhydrophobic slip: Perturbative corrections to the effective slip length, Physical Review Fluids, Vol: 2, ISSN: 2469-990X
Analytical expressions are derived for the first-order correction to the effective slip length of a weakly shear-thinning Carreau-Yasuda fluid in both longitudinal and transverse semi-infinite shear flow over a unidirectional superhydrophobic surface of flat no-shear slots. The formulas, which are derived using suitably generalized forms of the standard reciprocal theorem for Stokes flow, are given by explicit integrals which require only numerical quadrature for their evaluation. For both longitudinal and transverse flow we find that for a given no-shear fraction of the superhydrophobic surface and a given power-law index characterizing the Carreau-Yasuda fluid, there is a critical imposed strain rate of the shear at which the enhancement of effective slip is maximal. The theoretical results are qualitatively consistent with recent numerical work by other authors for the transverse case.
Crowdy DG, Krishnamurthy VS, 2017, Speed of a von Kármán point vortex street in a weakly compressible fluid, Physical Review Fluids, Vol: 2, ISSN: 2469-990X
Analytical expressions are obtained for the change in speed of translation of the von Kármán point vortex streets of given aspect ratios due to the effects of weak compressibility in subsonic flow of an isentropic fluid. We also clarify the nature of the force-free condition on a weakly compressible point vortex in equilibrium. For staggered streets, it is found that the speed of a compressible point vortex street can both increase and decrease relative to its incompressible counterpart of the same aspect ratio. Compressibility increases the speeds of streets with aspect ratios less than the critical value of 0.38187, at which no change in speed occurs to first order in the (squared) Mach number. In particular, the compressible counterpart to the neutrally stable incompressible point vortex street of aspect ratio 0.28056 is found to propagate with increased speed. Streets with aspect ratios larger than 0.38187 slow down under the effects of compressibility, with the slowdown becoming maximal at a street aspect ratio of 0.52630. On the other hand, the speed of unstaggered streets always increases, with the first-order correction in speed relative to its incompressible value increasing maximally at aspect ratios around κ=0.36216.
Crowdy DG, 2017, Slip length for transverse shear flow over a periodic array of weakly curved menisci, Physics of Fluids, Vol: 29, ISSN: 1070-6631
By exploiting the reciprocal theorem of Stokes flow, we find an explicit expression for the first order slip length correction, for small protrusion angles, and for transverse shear over a periodic array of curved menisci. The result is the transverse flow analogue of the longitudinal flow result of Sbragaglia and Prosperetti [“A note on the effective slip properties for microchannel flows with ultrahydrophobic surfaces,” Phys. Fluids 19, 043603 (2007)]. For small protrusion angles, it also generalizes the dilute-limit result of Davis and Lauga [“Geometric transition in friction for flow over a bubble mattress,” Phys. Fluids 21, 011701 (2009)] to arbitrary no-shear fractions. While the leading order slip lengths for transverse and longitudinal flow over flat no-shear slots are well-known to differ by a factor of 2, the first order slip length corrections for weakly protruding menisci in each flow are found to be identical.
Crowdy DG, 2017, Effective slip lengths for immobilized superhydrophobic surfaces, Journal of Fluid Mechanics, Vol: 825, ISSN: 1469-7645
Analytical solutions are found for both longitudinal and transverse shear flow, at zero Reynolds number, over immobilized superhydrophobic surfaces comprising a periodic array of near-circular menisci penetrating into a no-slip surface and where the menisci are no longer shear-free but are taken to be no-slip zones. Explicit formulae for the associated longitudinal and transverse effective slip lengths are derived; these are then compared with analogous results for superhydrophobic surfaces of the same characteristic geometry but where the menisci are shear-free. The new formulae give results that are consistent with recent experimental observations that have prompted suggestions that menisci that are assumed to be free of shear have in fact been immobilized. Significantly, for transverse shear flow, it is found that at critical downward meniscus protrusion angles of around , for many surface geometries, it is impossible to distinguish, purely from the effective slip length, between a no-shear and a no-slip boundary condition. We also find that immobilized menisci bowing into the grooves at supercritical angles just below can be almost twice as slippery to transverse shear as no-shear menisci. The results are relevant to recent discussion as to whether surface immobilization, due to contamination by surfactants or other physical mechanisms, is compromising drag reduction properties expected from an assumed no-shear condition.
Crowdy DG, 2017, Perturbation analysis of subphase gas and meniscus curvature effects for longitudinal flows over superhydrophobic surfaces, Journal of Fluid Mechanics, Vol: 822, Pages: 307-326, ISSN: 0022-1120
Integral expressions for the first-order correction to the effective slip length for longitudinal flows over a unidirectional superhydrophobic surface with rectangular grooves are determined under the assumptions that the meniscus curvature is small and the viscosity contrast between the groove-trapped subphase gas and the working fluid is significant. Both pressure-driven channel flows and semi-infinite shear flows are considered. Reciprocity ideas, based on use of Green’s second identity, provide the integral expressions with integrands dependent on known flat-meniscus solutions found by Philip (Z. Angew. Math. Phys., vol. 23, 1972, pp. 353–372). The results extend earlier work by Sbragaglia & Prosperetti (Phys. Fluids, vol. 19, 2007, 043603) on how weak meniscus curvature affects hydrodynamic slip. In particular, we derive a new integral expression for the first-order slip length correction due to weak meniscus curvature.
Crowdy DG, Brzezicki S, 2017, Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol: 473, ISSN: 0080-4630
An analytical method to find the flow generated bythe basic singularities of Stokes flow in a wedge ofarbitrary angle is presented. Specifically, we solve abiharmonic equation for the streamfunction of theflow generated by a point stresslet singularity andsatisfying no-slip boundary conditions on the twowalls of the wedge. The method, which is readilyadapted to any other singularity type, takes fullaccount of any transcendental singularities arisingat the corner of the wedge. The approach is alsoapplicable to problems of plane strain/stress of anelastic solid where the biharmonic equation alsogoverns the Airy stress function.
Ishimoto K, Crowdy DG, 2017, Dynamics of a treadmilling microswimmer near a no-slip wall in simple shear, Journal of Fluid Mechanics, Vol: 821, Pages: 647-667, ISSN: 1469-7645
Induction of flow is commonly used to control the migration of a microswimmer in a confined system such as a microchannel. The motion of a swimmer, in general, is governed by nonlinear equations due to non-trivial hydrodynamic interactions between the flow and the swimmer near a wall. This paper derives analytical expressions for the equations of motion governing a circular treadmilling swimmer in simple shear near a no-slip wall by combining the reciprocal theorem for Stokes flow with an exact solution for the dragging problem of a cylinder near a wall. We demonstrate that the reduced dynamical system possesses a Hamiltonian structure, which we use to show that the swimmer cannot migrate stably at a constant distance from a wall but only exhibit periodic oscillatory motion along the wall, or to escape from it. A treadmilling swimmer with the lowest two treadmilling modes is investigated in detail by means of a bifurcation analysis of the reduced dynamical system. It is found that the swimming direction of oscillatory motion is clarified by the sign of the Hamiltonian in the absence of flow, and that the induction of the flow suppresses upstream migration but aligns swimmer orientations in downstream migration. These results could inform strategies for the transport and control of micro-organisms and micromachines.
Chen MJ, Stokes YM, Buchak P, et al., 2016, Asymptotic modelling of a six-hole MOF, Journal of Lightwave Technology, Vol: 34, Pages: 5651-5656, ISSN: 1558-2213
We model the drawing of a six-hole microstructured optical fibre with a combination of asymptotic techniques and a new efficient numerical method, and compare this to a previous set of experiments and finite element simulations. The new approach accurately models the deformation of the inner channels and predicts cross-sectional fibre geometries that are a better match to the experiments than the fibres, more computationally expensive simulation technique.
Crowdy DG, 2016, Finite gap Jacobi matrices and the Schottky-Kleinprime function, Computational Methods and Function Theory, Vol: 17, Pages: 319-341, ISSN: 1617-9447
A covering map formalism for studying the spectral curves assocaitedwith finite gap Jacobi matrices is presented. We advocate a constructivefunction theoretic framework based on use of the Schottky-Klein prime function.The single gap, or genus-one, case is studied in explicit detail.
Crowdy DG, Kropf E, Green CC, et al., 2016, The Schottky-Klein prime function: a theoretical and computational tool for applications, IMA Journal of Applied Mathematics, Vol: 81, Pages: 589-628, ISSN: 1464-3634
This article surveys the important role, in a variety of applied mathematical contexts, played by the so-called Schottky–Klein (S–K) prime function. While it is a classical special function, introduced by 19th century investigators, its theoretical significance for applications has only been realized in the last decade or so, especially with respect to solving problems defined in multiply connected, or ‘holey’, domains. It is shown here that, in terms of it, many well-known results pertaining only to the simply connected case (no holes) can be generalized, in a natural way, to the multiply connected case, thereby contextualizing those well-known results within a more general framework of much broader applicability. Given the wide-ranging usefulness of the S–K prime function it is important to be able to compute it efficiently. Here we introduce both a new theoretical formulation for its computation, as well as two distinct numerical methods to implement the construction. The combination of these theoretical and computational developments renders the S–K prime function a powerful new tool in applied mathematics.
Champneys AR, Crowdy D, Papageorgiou D, 2016, Some highlights from 50 years of the IMA Journal of Applied Mathematics, IMA JOURNAL OF APPLIED MATHEMATICS, Vol: 81, Pages: 393-408, ISSN: 0272-4960
Crowdy DG, 2016, Flipping and scooping of curved 2D rigid fibers in simple shear: the Jeffery equations, Physics of Fluids, Vol: 28, ISSN: 1089-7666
The dynamical system governing the motion of a curved rigid two-dimensionalcircular-arc fiber in simple shear is derived in analytical form. This is achieved byfinding the solution for the associated low-Reynolds-number flow around such a fiberusing the methods of complex analysis. Solutions of the dynamical system displaythe “flipping” and “scooping” recently observed in computational studies of threedimensionalfibers using linked rigid rod and bead-shell models [Wang et al, Phys.Fluids, 24, (2012)]. To complete the Jeffery-type equations for a curved fiber in alinear flow field we also derive its evolution equations in an extensional flow. It isexpected that the equations derived here also govern the motion of slender, curved,three-dimensional rigid fibers when they evolve purely in the plane of shear or strain.
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