Publications
184 results found
Crowdy DG, Buchak PM, 2016, Surface-tension-driven Stokes flow: a numerical method based on conformal geometry, Journal of Computational Physics, Vol: 317, Pages: 347-361, ISSN: 1090-2716
A novel numerical scheme is presented for solving the problem of two dimensional Stokes flows with free boundarieswhose evolution is driven by surface tension. The formulation is based on a complex variable formulation of Stokes flowand use of conformal mapping to track the free boundaries. The method is motivated by applications to modelling thefabrication process for microstructured optical fibres (MOFs), also known as “holey fibres”, and is therefore tailored forthe computation of multiple interacting free boundaries. We give evidence of the efficacy of the method and discuss itsperformance.
Crowdy DG, 2016, Analytical formulas for longitudinal slip lengths over unidirectional superhydrophobic surfaces with curved menisci, Journal of Fluid Mechanics, Vol: 791, ISSN: 0022-1120
This paper reports new analytical formulas for the longitudinal slip lengths for simpleshear over a superhydrophobic surface, or bubble mattress, comprising a periodic arrayof unidirectional circular menisci, or bubbles, protruding into, or out of, the fluid. Theaccuracy of the formulas is tested against results from full numerical simulations; theyare found to give small relative errors even at large no-shear fractions. In the dilute limitthe formulas reduce to an earlier result by the author [Phys. Fluids, 22, 121703, (2011)].They also extend analytical results of Sbragaglia & Prosperetti [Phys Fluids, 19, 043603,(2007)] beyond a small protrusion angle limit.
Crowdy D, 2016, Geometry-Fitted Fourier-Mellin Transform Pairs, 10th International ISAAC Congress, Publisher: SPRINGER, Pages: 37-53, ISSN: 2194-1009
Chen MJ, Stokes YM, Buchak P, et al., 2015, Drawing tubular fibres: experiments versus mathematical modelling, Optical Materials Express, Vol: 6, Pages: 166-180, ISSN: 2159-3930
A series of six experiments drawing tubular fibres are comparedto some recent mathematical modelling of this fabrication process. Theimportance of fibre tension in determining the internal geometry of thefibre is demonstrated, confirming a key prediction of the models. There isevidence of self-pressurisation of the internal channel, where an additionalpressure is induced in the internal channel as the fibre is drawn, and thedependence of the magnitude of this pressure on fibre tension is discussed.Additionally, there is evidence that the difference between the glass and furnacetemperatures is proportional to the furnace temperature and dependenton the preform geometry.
Crowdy DG, 2015, A constructive method for plane-wave representations of special functions, Journal of Mathematical Analysis and Applications, Vol: 436, Pages: 149-167, ISSN: 0022-247X
A general constructive scheme for the derivation of plane-wave representationsof special functions is proposed. Illustrative examples of the construction aregiven. As one case study, new integral representations of the elliptic Weierstrass℘ function are derived; these complement, and generalize, similar new planewaveintegral representations of the same function recently found by Dienstfrey& Huang [J. Math. Anal. Appl., 316, 142–160, (2006)] using other techniques.Our approach is inspired by recent developments in the so-called Fokas transformfor the solution of boundary value problems for partial differential equations.Keywords: plane wave representation, elliptic function, transform method,special functions
Crowdy DG, 2015, Uniform flow past a periodic array of cylinders, European Journal of Mechanics B - Fluids, Vol: 56, Pages: 120-129, ISSN: 0997-7546
The problem of uniform flow past a linear periodic array, or grating, of circular cylinders is solved by a new transform technique recently devised by the author. The solutions are expressed as an integral representation dependent on a set of coefficients which are the solutions of an explicit and well-conditioned linear system that is easily solved. We also give new explicit approximation formulas for the so-called blockage coefficients that are accurate well beyond the dilute limit relevant to cylinders that are small relative to their spacing. Finally, we describe an alternative approach to the flow problems through conformal mapping theory and produce a transform representation of a useful conformal mapping from the disc-in-channel geometry to a concentric annulus. The latter result is expected to be useful in many other applications.
Crowdy DG, 2015, A transform method for Laplace's equation in multiply connected circular domains, IMA Journal of Applied Mathematics, Vol: 80, Pages: 1902-1931, ISSN: 0272-4960
A new general transform method for the solution of mixed boundary value problems for Laplace’s equationin multiply connected circular domains is formulated. Circular domains are those with boundariesthat are a union of circular arcs and/or straight line segments. Constructive schemes based on the methodare showcased by application to some concrete problems in fluid mechanics. In particular, a boundaryvalue problem of topical interest in the study of superhydrophobic surfaces is solved and compared withan existing numerical solution.
Crowdy DG, 2015, Effective slip lengths for longitudinal shear flow over partial-slip circular bubble mattresses., Fluid Dynamics Research, Vol: 47, ISSN: 1873-7005
The problem of longitudinal shear flow over a circular bubble mattress withpartial slip and protrusion angle 90o is solved in a quasi-analytical fashion by anovel transform scheme recently devised by the author. The general approachcan be readily adapted to other mixed boundary value problems. From theanalysis explicit approximations for the effective slip lengths are found as afunction of the Navier-slip parameter and the area fraction of the surfacecovered by protrusions. These new approximation formulas for the slip lengthsboth unify and extend those based on empirical polynomial fits to numericaldata given recently by Ng and Wang (2011 Fluid Dyn. Res. 43 065504).
Chen MJ, Stokes YM, Buchak P, et al., 2015, Microstructured optical fibre drawing with active channel pressurisation, Journal of Fluid Mechanics, Vol: 783, Pages: 137-165, ISSN: 1469-7645
The use of channel pressurisation in drawing microstructured optical fibres (MOFs) potentially allows for fine control of the internal structure of the fibre. By applying extra pressure inside the channels it is possible to counteract the effect of surface tension which would otherwise act to close the channels in the fibre as it is drawn. This paper extends the modelling approach of Stokes et al. (J. Fluid Mech., vol. 755, 2014, pp. 176–203) to include channel pressurisation. This approach treats the problem as two submodels for the flow, one in the axial direction along the fibre and another in the plane perpendicular to that direction. In the absence of channel pressurisation these models decoupled and were solved independently; we show that they become fully coupled when the internal channels are pressurised. The fundamental case of a fibre with an annular cross-section (containing one central channel) will be examined in detail. In doing this we consider both a forward problem to determine the shape of fibre from a known preform and an inverse problem to design a preform such that when drawn it will give a desired fibre geometry. Criteria on the pressure corresponding to fibre explosion and closure of the channel will be given that represent an improvement over similar criteria in the literature. A comparison between our model and a recent experiment is presented to demonstrate the effectiveness of the modelling approach. We make use of some recent work by Buchak et al. (J. Fluid Mech., vol. 778, 2015, pp. 5–38) to examine more complicated fibre geometries, where the cross-sectional shape of the internal channels is assumed to be elliptical and multiple channels are present. The examples presented here demonstrate the versatility of our modelling approach, where the subtleties of the interaction between surface tension and pressurisation can be revealed even for complex patterns of cross-sectional channels.
Crowdy DG, 2015, Fourier-Mellin transforms for circular domains, Computational Methods and Function Theory, Vol: 15, Pages: 655-687, ISSN: 1617-9447
Generalized Fourier–Mellin transforms for analytic functions defined insimply connected circular domains are derived. Circular domains are taken to be thosewith boundaries that are a finite union of circular arcs, including straight line edges. Theresults are an extension to circular domains of the generalized Fourier transforms forconvex polygons (having only straight line edges) derived by Fokas and Kapaev (IMAJ Appl Math 68:355–408, 2003). First, a new, elementary derivation of the latter resultfor polygons is given based on Cauchy’s integral formula and a spectral representationof the Cauchy kernel. This rederivation extends in a natural way to the case of circulardomains once an adapted spectral representation of the Cauchy kernel is established.Domains with boundaries that are a combination of circular arc and straight lineedges can be treated similarly. The newly derived transforms are generalizations ofthe classical Fourier and Mellin transforms to general circular domains. It is shownby example how they can be used to solve boundary value problems for Laplace’sequation in such domains. The notions of spectral matrix and fundamental contour,which arise naturally in the formulation, are also introduced.
Buchak P, Crowdy DG, Stokes YM, et al., 2015, Elliptical pore regularisation of the inverse problem for microstructured optical fibre fabrication, JOURNAL OF FLUID MECHANICS, Vol: 778, ISSN: 0022-1120
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- Citations: 17
Crowdy DG, 2015, Stress fields around two pores in an elastic body: exact quadrature domain solutions, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 471, ISSN: 1471-2946
Analytical solutions are given for the stress fields,in both compression and far-field shear, in a twodimensionalelastic body containg two interactingnon-circular pores. The two complex potentialsgoverning the solutions are found by using aconformal mapping from a preimage annulus withthose potentials expressed in terms of the SchottkyKleinprime function for the annulus. Solutionsfor a three-parameter family of elastic bodies withtwo equal symmetric pores are presented and thecompressibility of a special family of pore pairs isstudied in detail. The methodology extends to twounequal pores.The importance for boundary value problems ofplane elasticity of a special class of planar domainsknown as quadrature domains is also elucidated. Thisobservation provides the route to generalization ofthe mathematical approach here to finding analyticalsolutions for the stress fields in bodies containing anyfinite number of pores.
Crowdy DG, Tanveer S, Delillo T, 2015, Hybrid basis scheme for computing electrostatic fields exterior to close-to-touching discs, IMA Journal of Numerical Analysis, Vol: 36, Pages: 743-769, ISSN: 0272-4979
This paper presents a simple and effective new numerical scheme for the computation of electrostatic fields exterior to a collection of close-to-touching discs. The method is presented in detail for the two-disc case. The key idea is to represent the required complex potential using a hybrid set of basis functions comprising the usual Fourier–Laurent expansion about each circle centre complemented by a subsidiary expansion in a variable associated with conformal mapping of the physical domain to a concentric annulus domain. We also rigorously prove that there is a representation of the solution in the hybrid basis with a faster decay rate of coefficients than is obtained by using a nonhybrid basis, thereby providing a rationalization for the success of the method. The numerical scheme is easy to implement and adaptable to the case of multiple close-to-touching cylinders.
Crowdy D, 2015, Exact solutions for the static dewetting of two-dimensional charged conducting droplets on a substrate, PHYSICS OF FLUIDS, Vol: 27, ISSN: 1070-6631
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- Citations: 4
Davis AMJ, Crowdy DG, 2015, Matched asymptotics for a spherical low-Reynolds-number treadmilling swimmer near a rigid wall, IMA JOURNAL OF APPLIED MATHEMATICS, Vol: 80, Pages: 634-650, ISSN: 0272-4960
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- Citations: 5
Vasconcelos GL, Marshall JS, Crowdy DG, 2015, Secondary Schottky-Klein prime functions associated with multiply connected planar domains, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 471, ISSN: 1364-5021
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- Citations: 9
Crowdy DG, Luca E, 2014, Solving Wiener-Hopf problems without kernel factorization, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 470, ISSN: 1364-5021
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- Citations: 13
Stokes YM, Buchak P, Crowdy DG, et al., 2014, Drawing of micro-structured fibres: circular and non-circular tubes, JOURNAL OF FLUID MECHANICS, Vol: 755, ISSN: 0022-1120
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- Citations: 31
Crowdy DG, Roenby J, 2014, Hollow vortices, capillary water waves and double quadrature domains, FLUID DYNAMICS RESEARCH, Vol: 46, ISSN: 0169-5983
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- Citations: 8
Crowdy DG, 2014, Vortex patch equilibria of the Euler equation and random normal matrices, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, Vol: 47, ISSN: 1751-8113
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- Citations: 2
Crowdy DG, 2013, Wall effects on self-diffusiophoretic Janus particles: a theoretical study, JOURNAL OF FLUID MECHANICS, Vol: 735, Pages: 473-498, ISSN: 0022-1120
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- Citations: 49
Crowdy DG, Davis AMJ, 2013, Stokes flow singularities in a two-dimensional channel: a novel transform approach with application to microswimming, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 469, Pages: 1-14, ISSN: 1364-5021
A transform method for determining the flow generated by the singularities of Stokes flow in a two-dimensional channel is presented. The analysis is based on a general approach to biharmonic boundary value problems in a simply connected polygon formulated by Crowdy & Fokas in this journal. The method differs from a traditional Fourier transform approach in entailing a simultaneous spectral analysis in the independent variables both along and across the channel. As an example application, we find the evolution equations for a circular treadmilling microswimmer in the channel correct to third order in the swimmer radius. Significantly, the new transform method is extendible to the analysis of Stokes flows in more complicated polygonal microchannel geometries.
Crowdy DG, 2013, Surfactant-induced stagnant zones in the Jeong-Moffatt free surface Stokes flow problem, PHYSICS OF FLUIDS, Vol: 25, ISSN: 1070-6631
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- Citations: 11
Crowdy DG, 2013, Exact solutions for cylindrical 'slip-stick' Janus swimmers in Stokes flow, JOURNAL OF FLUID MECHANICS, Vol: 719, ISSN: 0022-1120
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- Citations: 11
Davis AMJ, Crowdy DG, 2013, Matched asymptotics for a treadmilling low-Reynolds-number swimmer near a wall, QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS, Vol: 66, Pages: 53-73, ISSN: 0033-5614
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- Citations: 8
Crowdy D, 2013, Analytical formulae for source and sink flows in multiply connected domains, THEORETICAL AND COMPUTATIONAL FLUID DYNAMICS, Vol: 27, Pages: 1-19, ISSN: 0935-4964
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- Citations: 14
Crowdy DG, Smith SGL, Freilich DV, 2013, Translating hollow vortex pairs, EUROPEAN JOURNAL OF MECHANICS B-FLUIDS, Vol: 37, Pages: 180-186, ISSN: 0997-7546
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- Citations: 14
Davis AMJ, Crowdy DG, 2012, Stresslet asymptotics for a treadmilling swimmer near a two-dimensional corner: hydrodynamic bound states, PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, Vol: 468, Pages: 3765-3783, ISSN: 1364-5021
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- Citations: 8
Smith SGL, Crowdy DG, 2012, Structure and stability of hollow vortex equilibria, JOURNAL OF FLUID MECHANICS, Vol: 691, Pages: 178-200, ISSN: 0022-1120
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- Citations: 14
Crowdy D, 2012, CONFORMAL SLIT MAPS IN APPLIED MATHEMATICS, ANZIAM JOURNAL, Vol: 53, Pages: 171-189, ISSN: 1446-1811
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- Citations: 28
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