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  • 1
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    The wake of a two-dimensional ship in the low-speed limit: results for multi-cornered hulls (2014)
    Cambridge University Press
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    Publication Date: 2014-02-17
    Description: In the Dagan & Tulin (J. Fluid Mech., vol. 51, 1972, pp. 529-543) model of ship waves, a blunt ship moving at low speeds can be modelled as a two-dimensional semiinfinite body. A central question for these reduced models is whether a particular ship design can minimize, or indeed eliminate, the wave resistance. In the previous part of our work (Trinh et al., J. Fluid Mech., vol. 685, 2011, pp. 413-439), we demonstrated why a single corner can never be made waveless. In this accompanying paper, we continue our investigations with the study of more general piecewise-linear, or multicornered ships. By using exponential asymptotics, we demonstrate how the production of waves can be directly ascertained by the positions and angles of the corners. In particular, this theory answers the question raised by Farrow & Tuck (J. Austral. Math. Soc. B, vol. 36, 1995, pp. 424-437) as to why certain bulbous-like obstructions can minimize the production of waves. General results for wavelessness are given for a class of hulls, and numerical computations of the nonlinear ship-wave problem are used to confirm analytical predictions. Finally, we discuss open questions regarding hulls without corners and more general three-dimensional bluff bodies. © Cambridge University Press 2014.
    Print ISSN: 0022-1120
    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 2
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    On the distinguished limits of the Navier slip model of the moving contact line problem (2015)
    Cambridge University Press
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    Publication Date: 2015-04-28
    Description: When a droplet spreads on a solid substrate, it is unclear what the correct boundary conditions are to impose at the moving contact line. The classical no-slip condition is generally acknowledged to lead to a non-integrable singularity at the moving contact line, which a slip condition, associated with a small slip parameter, λ, serves to alleviate. In this paper, we discuss what occurs as the slip parameter, λ, tends to zero. In particular, we explain how the zero-slip limit should be discussed in consideration of two distinguished limits: one where time is held constant, t=O(1), and one where time tends to infinity at the rate t=O(|log λ|). The crucial result is that in the case where time is held constant, the λ → 0 limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if λ → 0 and t → ∞, then contact line slippage is a leading-order singular effect. © © 2015 Cambridge University PressÂ.
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    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 3
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    New singularities for Stokes waves (2016)
    Cambridge University Press
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    Publication Date: 2016-05-31
    Description: In 1880, Stokes famously demonstrated that the singularity that occurs at the crest of the steepest possible water wave in infinite depth must correspond to a corner of . Here, the complex velocity scales like where is the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity moves into the complex plane, and is of order (Grant J. Fluid Mech., vol. 59, 1973, pp. 257-262). Grant conjectured that as the highest wave is approached, other singularities must coalesce at the crest so as to cancel the square-root behaviour. Despite recent advances, the complete singularity structure of the Stokes wave is still not well understood. In this work, we develop numerical methods for constructing the Riemann surface that represents the extension of the water wave into the complex plane. We show that a countably infinite number of distinct singularities exist on other branches of the solution, and that these singularities coalesce as Stokes' highest wave is approached. © 2016 Cambridge University Press.
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  • 4
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    Complex singularities near the intersection of a free surface and wall. Part 1. Vertical jets and rising bubbles (2018)
    Cambridge University Press
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    Publication Date: 2018-10-04
    Description: It is known that in steady-state potential flows, the separation of a gravity-driven free surface from a solid exhibits a number of peculiar characteristics. For example, it can be shown that the fluid must separate from the body so as to form one of three possible in-fluid angles: (i) , (ii) or (iii) an angle such that the surface is locally perpendicular to the direction of gravity. These necessary separation conditions were notably remarked upon by Dagan & Tulin (J. Fluid Mech., vol. 51 (3), 1972, pp. 529-543) in the context of ship hydrodynamics, but they are of crucial importance in many potential-flow applications. It is not particularly well understood why there is such a drastic change in the local separation behaviours when the global flow is altered. The question that motivates this work is the following: outside of a formal balance-of-terms argument, why must cases (i)-(iii) occur and furthermore, what are the connections between them? In this work, we seek to explain the transitions between the three cases in terms of the singularity structure of the associated solutions once they are extended into the complex plane. A numerical scheme is presented for the analytic continuation of a vertical jet (or alternatively a rising bubble). It will be shown that the transition between the three cases can be predicted by observing the coalescence of singularities as the speed of the jet is modified. A scaling law is derived for the coalescence rate of singularities. © 2018 Cambridge University Press.
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    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 5
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    Do waveless ships exist? Results for single-cornered hulls (2011)
    Cambridge University Press
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    Publication Date: 2011-10-06
    Description: Consider low-speed potential flow past a ship modelled as a semi-infinite two-dimensional body with constant draught. Is it possible to design the hull in such a way as to eliminate the waves produced downstream of the ship? In 1977, Vanden-Broeck & Tuck had conjectured that a single-cornered piecewise-linear hull will always generate a wake; in this paper, we show how recently developed tools in exponential asymptotics can be used to confirm this conjecture. In particular, we show how the formation of waves near a ship is a necessary consequence of singularities in the ship's geometry (or its analytic continuation). Comprehensive numerical computations confirm the analytical predictions. © 2011 Cambridge University Press.
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    Electronic ISSN: 1469-7645
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  • 6
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    On reduced models for gravity waves generated by moving bodies (2017)
    Cambridge University Press
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    Publication Date: 2017-01-26
    Description: In 1983, Tulin published a report proposing a framework for reducing the equations for gravity waves generated by moving bodies into a single nonlinear differential equation solvable in closed form (Proceedings of the 14th Symposium on Naval Hydrodynamics, 1983, pp. 19-51). Several new and puzzling issues were highlighted by Tulin, notably the existence of weak and strong wave-making regimes, and the paradoxical fact that the theory seemed to be applicable to flows at low speeds, 'but not too low speeds'. These important issues were left unanswered, and despite the novelty of the ideas, Tulin's report fell into relative obscurity. Now, 30 years later, we will revive Tulin's observations, and explain how an asymptotically consistent framework allows us to address these concerns. Most notably, we demonstrate, using the asymptotic method of steepest descents, how the production of free-surface waves can be related to the arrangement of integration contours connected to the shape of the moving body. This approach provides a new and powerful methodology for the study of geometrically nonlinear wave-body interactions. © 2017 Cambridge University Press.
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    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 7
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    A pinned or free-floating rigid plate on a thin viscous film (2014)
    Cambridge University Press
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    Publication Date: 2014-11-11
    Description: A pinned or free-floating rigid plate lying on the free surface of a thin film of viscous fluid, which itself lies on top of a horizontal substrate that is moving to the right at a constant speed is considered. The focus of the present work is to describe how the competing effects of the speed of the substrate, surface tension, viscosity, and, in the case of a pinned plate, the prescribed pressure in the reservoir of fluid at its upstream end, determine the possible equilibrium positions of the plate, the free surface, and the flow within the film. The present problems are of interest both in their own right as paradigms for a range of fluid-structure interaction problems in which viscosity and surface tension both play an important role, and as a first step towards the study of elastic effects. © 2014 Cambridge University Press.
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    Electronic ISSN: 1469-7645
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  • 8
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    New gravity–capillary waves at low speeds. Part 2. Nonlinear geometries (2013)
    Cambridge University Press
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    Publication Date: 2013-04-29
    Description: When traditional linearized theory is used to study gravity-capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions are waveless to every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. In Part 1 (Trinh & Chapman, J. Fluid Mech., vol. 724, 2013b, pp. 367-391), we showed how exponential asymptotics could be used to study the problem when the size of the obstruction is first linearized. In this paper, we extend the analysis to the nonlinear problem, thus allowing the full geometry to be considered at leading order. When applied to the classic problem of flow over a step, our analysis reveals the existence of six classes of gravity-capillary waves, two of which share a connection with the usual linearized solutions first discovered by Rayleigh. The new solutions arise due to the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects. ©2013 Cambridge University PressA.
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 9
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    New gravity–capillary waves at low speeds. Part 1. Linear geometries (2013)
    Cambridge University Press
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    Publication Date: 2013-04-29
    Description: When traditional linearized theory is used to study gravity-capillary waves produced by flow past an obstruction, the geometry of the object is assumed to be small in one or several of its dimensions. In order to preserve the nonlinear nature of the obstruction, asymptotic expansions in the low-Froude-number or low-Bond-number limits can be derived, but here, the solutions invariably predict a waveless surface at every order. This is because the waves are in fact, exponentially small, and thus beyond-all-orders of regular asymptotics; their formation is a consequence of the divergence of the asymptotic series and the associated Stokes Phenomenon. By applying techniques in exponential asymptotics to this problem, we have discovered the existence of new classes of gravity-capillary waves, from which the usual linear solutions form but a special case. In this paper, we present the initial theory for deriving these waves through a study of gravity-capillary flow over a linearized step. This will be done using two approaches: in the first, we derive the surface waves using the standard method of Fourier transforms; in the second, we derive the same result using exponential asymptotics. Ultimately, these two methods give the same result, but conceptually, they offer different insights into the study of the low-Froude-number, low-Bond-number problem. ©2013 Cambridge University PressA.
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    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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  • 10
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    Bending of elastic fibres in viscous flows: the influence of confinement (2013)
    Cambridge University Press
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    Publication Date: 2013-02-27
    Description: We present a mathematical model and corresponding series of microfluidic experiments examining the flow of a viscous fluid past an elastic fibre in a three-dimensional channel. The fibre's axis lies perpendicular to the direction of flow and its base is clamped to one wall of the channel; the sidewalls of the channel are close to the fibre, confining the flow. Experiments show that there is a linear relationship between deflection and flow rate for highly confined fibres at low flow rates, which inspires an asymptotic treatment of the problem in this regime. The three-dimensional problem is reduced to a two-dimensional model, consisting of Hele-Shaw flow past a barrier, with boundary conditions at the barrier that allow for the effects of flexibility and three-dimensional leakage. The analysis yields insight into the competing effects of flexion and leakage, and an analytical solution is derived for the leading-order pressure field corresponding to a slit that partially blocks a two-dimensional channel. The predictions of our model show favourable agreement with experimental results, allowing measurement of the fibre's elasticity and the flow rate in the channel. © 2013 Cambridge University Press.
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    Electronic ISSN: 1469-7645
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics , Physics
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