arXiv subjects

Philippe H. Trinh

Publications and source records attributed to Philippe H. Trinh.

34 records · Page 2Linked to original sources

arXiv2019

Exponential asymptotics for the eigenvalues in the broken PT-symmetric region

Stemming from the seminal work of Bender & Boettcher in 1998 (Phys. Rev. Lett. vol. 80 pp. 5243-5246), there has been great interest in the study of PT-symmetric models of quantum mechanics, where the primary focus is with the study of non-Hermitian Hamiltonians that nevertheless produce countably infinite sets of real-valued eigenvalues. One of the fundamental models of such a system is governed by the Hamiltonian $H = \hat{p}^2 + x^2(ix)^{\varepsilon}$. In their work, Bender & Boettcher proposed a WKB methodology for the prediction of the discrete eigenvalues in the so-called unbroken region of $\varepsilon > 0$. However, the authors noted that this methodology fails to predict those 'broken' eigenvalues for $\varepsilon < 0$. Here, we shall explain why the traditional WKB methodology fails, and we shall demonstrate how eigenvalues for all relevant values of $\varepsilon$ can be predicted using techniques in exponential asymptotics. These predictions provide excellent agreement to exact numerical results over nearly the entire range of values. Moreover, such techniques can be extended to a much wider range PT-symmetric problems.

math-ph↗

arXiv2019

Shear-induced instabilities of flows through submerged vegetation

We consider the instabilities of flows through a submerged canopy, and show how the full governing equations of the fluid-structure interactions can be reduced to a compact framework that captures many key features of vegetative flow. By modelling the canopy as a collection of homogeneous elastic beams, we predict the steady configuration of the plants in response to a unidirectional flow. This treatment couples the beam equations in the canopy to the fluid momentum equations. Our linear stability analysis suggests new insights into the development of instabilities at the surface of the vegetative region. In particular, we show that shear at the top of the canopy is a dominant factor in determining the onset of instabilities known as monami. Based on numerical and asymptotic analysis of the generalised eigenvalue problem, the system is shown to be stable if the canopy is sufficiently sparse or if the plants are sufficiently flexible.

physics.flu-dyn↗

arXiv2016

The influence of van der Waals interactions on a bubble moving in a tube

We consider the unsteady thin-film dynamics of a long bubble of negligible viscosity that advances at a uniform speed in a cylindrical capillary tube. The bubble displaces a viscous, nonwetting fluid, creating a thin film between its interface and the tube walls. The film is considered thin enough that intermolecular forces in the form of van der Waals attractions are significant and thin-film rupture is possible. In the case of negligible intermolecular forces, a steady-state solution exits where a film of uniform thickness is deposited in the annular region between the bubble interface and the tube walls. However, once intermolecular interactions are important, the interface is perturbed out of its steady state and either (i) the perturbation grows sufficiently before reaching the rear meniscus of the bubble such that rupture occurs; or (ii) the perturbation remains small due to weak intermolecular forces until it leaves the bubble interface through the rear meniscus. We obtain, both numerically and asymptotically, the time-scale over which rupture occurs, and thus, we find a critical capillary number, depending on the bubble length and the strength of the intermolecular forces, below which the film is predicted to rupture.

physics.flu-dyn↗

arXiv2016

New singularities for Stokes waves

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 $120^\circ$. Here, the complex velocity scales like $f^{1/3}$ where $f$ is the complex potential. Later in 1973, Grant showed that for any wave away from the steepest configuration, the singularity $f = f^*$ moves into the complex plane, and is of order $(f-f^*)^{1/2}$ (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 exists on other branches of the solution, and that these singularities coalesce as Stokes' highest wave is approached.

physics.flu-dyn↗

arXiv2015

A topological study of gravity waves generated by moving bodies using the method of steepest descents

The standard analytical approach for studying gravity free-surface waves generated by a moving body often relies upon a linearization of the physical geometry, where the body is considered asymptotically small in one or several of its dimensions. In this paper, a methodology that avoids any such geometrical simplification is presented for the case of flows at low speeds. The approach is made possible through a reduction of the water-wave equations to a complex-valued integral equation that can be studied using the method of steepest descents. The main result is a theory that establishes a correspondence between a given physical flow geometry, with the topology of the Riemann surface formed by the steepest descent paths. Then, when a geometrical feature of the body is modified, a corresponding change to the Riemann surface is observed, and the resultant effects to the water waves can be derived. This visual procedure is demonstrated for the case of two-dimensional free-surface flow past a surface-piercing ship and over an angled step in a channel.

physics.flu-dyn↗

arXiv2015

On reduced models for gravity waves generated by moving bodies

In 1982, Marshall P. 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 [Proc. 14th Symp. on Naval Hydrodynamics, 1982, 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 thirty years later, we will revive Tulin's observations, and explain how an asymptotically consistent framework allows us to address these concerns. Most notably, we will explain, 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 an intuitive and visual procedure for studying nonlinear wave-body interactions.

physics.flu-dyn↗

arXiv2015

The dynamics of localized spot patterns for reaction-diffusion systems on the sphere

In the singularly perturbed limit corresponding to a large diffusivity ratio between two components in a reaction-diffusion (RD) system, quasi-equilibrium spot patterns are often admitted, producing a solution that concentrates at a discrete set of points in the domain. In this paper, we derive and study the differential algebraic equation (DAE) that characterizes the slow dynamics for such spot patterns for the Brusselator RD model on the surface of a sphere. Asymptotic and numerical solutions are presented for the system governing the spot strengths, and we describe the complex bifurcation structure and demonstrate the occurrence of imperfection sensitivity due to higher order effects. Localized spot patterns can undergo a fast time instability and we derive the conditions for this phenomena, which depend on the spatial configuration of the spots and the parameters in the system. In the absence of these instabilities, our numerical solutions of the DAE system for $N = 2$ to $N = 8$ spots suggest a large basin of attraction to a small set of possible steady-state configurations. We discuss the connections between our results and the study of point vortices on the sphere, as well as the problem of determining a set of elliptic Fekete points, which correspond to globally minimizing the discrete logarithmic energy for $N$ points on the sphere.

nlin.PS↗

arXiv2014

An elastic plate on a thin viscous film

We consider the steady-state analysis of a pinned elastic plate lying on the free surface of a thin viscous fluid, forced by the motion of a bottom substrate moving at constant speed. A mathematical model incorporating elasticity, viscosity, surface tension, and pressure forces is derived, and consists of a third-order Landau-Levich equation for the thin film, and a fifth-order beam equation for the plate. A numerical and asymptotic analysis is presented in the relevant limits of the elasticity and Capillary numbers. We demonstrate the emergence of boundary-layer effects near the ends of the plate, which are likely to be a generic phenomenon for singularly perturbed elastocapillary problems.

physics.flu-dyn↗

arXiv2014

Exponential Asymptotics and Stokes Line Smoothing for Generalized Solitary Waves

In a companion paper, Grimshaw (Asymptotic Methods in Fluid Mechanics, 2010, pp. 71-120) has demonstrated how techniques of Borel summation can be used to elucidate the exponentially small terms that lie hidden beyond all orders of a divergent asymptotic expansion. Here, we provide an alternative derivation of the generalized solitary waves of the fifth-order Korteweg-de Vries equation. We will first optimally truncate the asymptotic series, and then smooth the Stokes line. Our method provides an explicit view of the switching-on mechanism, and thus increased understanding of the Stokes Phenomenon.

math.CA↗

arXiv2014

Exponential asymptotics and problems with coalescing singularities

Problems in exponential asymptotics are typically characterized by divergence of the associated asymptotic expansion in the form of a factorial divided by a power. In this paper, we demonstrate that in certain classes of problems that involve coalescing singularities, a more general type of exponential-over-power divergence must be used. As a model example, we study the water waves produced by flow past an obstruction such as a surface-piercing ship. In the low speed or low Froude limit, the resultant water waves are exponentially small, and their formation is attributed to the singularities in the geometry of the obstruction. However, in cases where the singularities are closely spaced, the usual asymptotic theory fails. We present both a general asymptotic framework for handling such problems of coalescing singularities, and provide numerical and asymptotic results for particular examples.

math.CA↗

arXiv2014

On the distinguished limits of the Navier slip model of the moving contact line problem

When a droplet spreads on a solid substrate, it is unclear what are the correct boundary conditions 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, for 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:: 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 $λ\to 0$ limit converges to the slip-free equation, and contact line slippage occurs as a regular perturbative effect. However, if $λ\to 0$ and $t \to \infty$, then contact line slippage is a leading-order singular effect.

physics.flu-dyn↗

arXiv2014

A pinned or free-floating plate on a thin viscous film. Part 1: A rigid plate

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.

physics.class-ph↗

arXiv2014

Curvature suppresses the Rayleigh-Taylor instability

The dynamics of a thin liquid film on the underside of a curved cylindrical substrate is studied. The evolution of the liquid layer is investigated as the film thickness and the radius of curvature of the substrate are varied. A dimensionless parameter (a modified Bond number) that incorporates both geometric parameters, gravity, and surface tension is identified, and allows the observations to be classified according to three different flow regimes: stable films, films with transient growth of perturbations followed by decay, and unstable films. Experiments and theory confirm that, below a critical value of the Bond number, curvature of the substrate suppresses the Rayleigh-Taylor instability.

physics.flu-dyn↗

arXiv2013

New gravity-capillary waves at low speeds. Part 1: Linear geometries

When traditional linearised 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 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 linearised step; this will be done using two approaches: in the first, we derive the surface waves using the 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, low-Bond number problem.

math-ph↗

arXiv2013

New gravity-capillary waves at low speeds. Part 2: Nonlinear geometries

When traditional linearised 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 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, we showed how exponential asymptotics could be used to study the problem when the size of the obstruction is first linearised. 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, from which two share a connection with the usual linearised solutions first discovered by Lord Rayleigh. The new solutions arise due to the availability of multiple singularities in the geometry, coupled with the interplay of gravitational and cohesive effects.

math-ph↗

arXiv2013

Waveless ships in the low speed limit: Results for multi-cornered hulls

In the low-speed limit, a blunt ship modeled as two-dimensional semi-infinite body with a single corner can never be made waveless. This was the conclusion of the previous part of our work in Trinh et al. (2011), which focused on the Dagan & Tulin (1972) model of ship waves in the low speed limit. In this accompanying paper, we continue our investigations with the study of more general, piecewise-linear, or multi-cornered ships. The low-speed or low-Froude limit, coupled with techniques in exponential asymptotics allows us to derive explicit formulae relating the geometry of the hull to the form of the waves. Configurations with closely spaced corners present a non-trivial extension of the theory, and we present the general methodology for their study. Lastly, numerical computations of the nonlinear ship-wave problem are presented in order to confirm the analytical predictions.

math-ph↗