arXiv subjects

Lennon Ó Náraigh

Publications and source records attributed to Lennon Ó Náraigh.

At least 19 recordsLinked to original sources

arXiv2025

An integrated theoretical, experimental, and numerical study of small-amplitude water waves

We introduce an inexpensive experimental setup for analyzing free-surface water waves: a $1\,\mathrm{m}$-long tabletop flume made from perspex, driven by a variable-frequency piston wavemaker built from Lego. Using mobile-phone video capture, we collect experimental data and compare it with predictions from linear gravity-capillary wave theory and with multiphase simulations performed in OpenFOAM. We find excellent quantitative agreement across all three approaches. Our setup may be valuable for students with a background in Mathematical Modelling who lack hands-on laboratory experience. To explore this, we report on a survey of students who completed an integrated theoretical, experimental, and computational project. While students found the experience enhanced their learning of Fluid Mechanics, they also noted the need for better support in setting up and running CFD simulations.

physics.flu-dyn↗

arXiv2025

Bounds on the Spreading Radius in Droplet Impact: The Two-Dimensional Case

We consider the problem of a cylindrical (quasi-two-dimensional) droplet impacting on a hard surface. Cylindrical droplet impact can be engineered in the laboratory, and a theoretical model of the system can also be used to shed light on various complex experiments involving the impact of liquid sheets. We formulate a rim-lamella model for the droplet-impact problem. Using Gronwall's Inequality, we establish theoretical bounds for the maximum spreading radius $\mathcal{R}_{max}$ in droplet impact, specifically $k_1 \mathrm{Re}^{1/3}-k_2(1-\cos\vartheta_{adv})^{1/2}(\mathrm{Re}/\mathrm{We})^{1/2}\leq \mathcal{R}_{max}/R_0\leq k_1\mathrm{Re}^{1/3}$, where $\mathrm{Re}$ and $\mathrm{We}$ are the Reynolds and Weber number based on the droplet's pre-impact velocity and radius $R_0$, $\vartheta_{adv}$ is the advancing contact angle (assumed constant in our simplified analysis), and $k_1$ and $k_2$ are constants. We perform several campaigns of simulations using the Volume of Fluid Method to model the droplet impact, and we find that the simulation results are consistent with the theoretical bounds.

physics.flu-dyn↗

arXiv2024

Bounds on the Spreading Radius in Droplet Impact: The Viscous Case

We consider the problem of droplet impact and droplet spreading on a smooth surface in the case of a viscous Newtonian fluid. We revisit the concept of the rim-lamella model, in which the droplet spreading is described by a system of ordinary differential equations (ODEs). We show that these models contain a singularity which needs to be regularized to produce smooth solutions, and we explore the different regularization techniques in detail. We adopt one particular technique for further investigation, and we use differential inequalities to derive upper and lower bounds for the maximum spreading radius $\mathcal{R}_{max}$. Without having to resort to dimensional analysis or scaling arguments, our bounds confirm the leading-order behaviour $\mathcal{R}_{max} \sim \mathrm{Re}^{1/5}$, as well as a correction to the leading-order behaviour involving the combination $\mathrm{We}^{-1/2}\mathrm{Re}^{2/5}$, in agreement with the experimental literature on droplet spread. Our bounds are also consistent with full numerical solutions of the ODEs, as well as with energy-budget calculations, once head loss is accounted for in the latter.

physics.flu-dyn↗

arXiv2024

A Mathematical Framework for Spatio-Temporal Control in Industrial Drying

We introduce two models of industrial drying - a simplified one-equation model, and a detailed three-equation model. The purpose of the simplified model is rigorous validation of numerical methods for PDE-constrained optimal control. The purpose of the detailed model is to be able to predict and control the behaviour of an industrial disk drier. For both models, we introduce a fully validated numerical method to compute the optimal source term to maintain the outlet temperature as close as possible to the set-point temperature. By performing simulations using realistic parameters for industrial driers, we illustrate potential applications of the method.

math.OC↗

arXiv2023

Bounds on the Spreading Radius in Droplet Impact: The Inviscid Case

We consider the classical problem of droplet impact and droplet spread on a smooth surface in the case of an ideal inviscid fluid. We revisit the rim-lamella model of Roisman et al. [\textit{Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences}, 458(2022), pp.1411-1430.]. This model comprises a system of ordinary differential equations (ODEs); we present a rigorous theoretical analysis of these ODEs, and derive upper and lower bounds for the maximum spreading radius. Both bounds possess a $\mathrm{We}^{1/2}$ scaling behaviour, and by a sandwich result, the spreading radius itself also possesses this scaling. We demonstrate rigorously that the rim-lamella model is self-consistent: once a rim forms, its height will invariably exceed that of the lamella. We introduce a rational procedure to obtain initial conditions for the rim-lamella model. Our approach to solving the rim-lamella model gives predictions for the maximum droplet spread that are in close agreement with existing experimental studies and direct numerical simulations.

physics.flu-dyn↗

arXiv2023

Analysis of the Spreading Radius in Droplet Impact: The Two-Dimensional Case

We study droplet-impact problems in a three-dimensional cylindrical or equivalent two-dimensional Cartesian geometry. Such structures do have an approximate experimental realization, and they are often simulated a test-bed for computational methods. We focus on droplet impact on a smooth homogeneous surface as well as head-on collision of two droplets. We perform an energy-budget analysis and introduce a correlation which predicts the maximum spreading radius as a function of Reynolds number and Weber number. We show how the dissipation term in this analysis can be decomposed into boundary-layer dissipation in the droplet lamella (where applicable), and head loss. We use existing results in the literature (simulations and experiments), as well as our own simulation results to validate the correlation. Dissipation by head loss is a key term in the analysis: only by modeling it accurately can one obtain good agreement between the simulations and the theory.

physics.flu-dyn↗

arXiv2023

Symmetry-Breaking in Point-Heated Droplets

We investigate theoretically the stability of thermo-capillary convection within a droplet when heated by a point source from below. To model the droplet, we use a mathematical model based on lubrication theory. We formulate a base-state droplet profile, and we examine its respect to small-amplitude perturbations in the azimuthal direction. Such linear stability analysis reveals that the base state is stable across a wide parameter space. We carry out transient simulations in three spatial dimensions: the simulations reveal that when the heating is slightly off-centered with respect to the droplet center, vortices develop within the droplet. The vortices persist when the contact line is pinned. These findings are consistent with experimental studies of point-heated sessile droplets.

physics.flu-dyn↗

arXiv2023

A new convergence analysis of the particle method for the Camassa-Holm equation

We present a new self-contained convergence analysis of the particle method that can be applied to a range of PDEs, including the Camassa-Holm equation. It is a development of the analysis of Chertock, Liu and Pendleton, which used compactness properties of spaces of functions having bounded variation. In our analysis we establish solutions by applying a metric Arzelà-Ascoli compactness result to a space of measure-valued functions equipped with the bounded Lipschitz metric. All the convergence and regularity results of the previous analysis follow as a consequence and are computationally easier to establish.

math.AP↗

arXiv2023

Convergence Analysis of the Geometric Thin-Film Equation

The Geometric Thin-Film equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon data, admits solutions that are globally defined for all time and are expressible as push-forwards of Borel measurable functions whose behaviour is governed by a set of ordinary differential equations (ODEs). The existence is first demonstrated in the special case of a finite weighted sum of delta functions whose centres evolve over time -- these are known as `particle solutions'. In the general case, we construct a convergent sequence of particle solutions whose limit yields a solution of the above form. Moreover, we demonstrate that all weak solutions constructed in this way are $1/2$-Hölder continuous in time and are uniquely determined by the initial conditions.

math.AP↗

arXiv2021

A mathematical model and mesh-free numerical method for contact-line motion in lubrication theory

We introduce a mathematical model with a mesh-free numerical method to describe contact-line motion in lubrication theory. We show how the model resolves the singularity at the contact line, and generates smooth profiles for an evolving, spreading droplet. The model describes well the physics of droplet spreading -- including Tanner's Law for the evolution of the contact line. The model can be configured to describe complete wetting or partial wetting, and we explore both cases numerically. In the case of partial wetting, the model also admits analytical solutions for the droplet profile, which we present here.

physics.flu-dyn↗

arXiv2019

Linear and Nonlinear Stability Analysis in Microfluidic Systems

In this article we use analytical and numerical modeling to describe parallel viscous two-phase flows in microchannels. The focus is on idealized two-dimensional geometries, with a view to validating the various methodologies for future work in three dimensions. In the first instance, we use analytical Orr--Sommerfeld theory to describe the linear instability which governs the formation of small-amplitude waves in such systems. We then compare the results of this analysis with an in-house Computational Fluid Dynamics (CFD) solver called TPLS. Excellent agreement between the theoretical analysis and TPLS is obtained in the regime of small-amplitude waves. We continue the numerical simulations beyond the point of validity of the Orr--Sommerfeld theory. In this way, we illustrate the generation of nonlinear interfacial waves and reverse entrainment of one fluid phase into the other. We justify our simulations further by comparing the numerical results with corresponding results from a commercial CFD code. This comparison is again extremely favourable -- this rigorous validation paves the way for future work using TPLS or commercial codes to perform extremely detailed three-dimensional simulations of flow in microchannels.

physics.flu-dyn↗

arXiv2019

A large-scale statistical study of the coarsening rate in models of Ostwald-Ripening

In this article we look at the coarsening rate in two standard models of Ostwald Ripening. Specifically, we look at a discrete droplet population model, which in the limit of an infinite droplet population reduces to the classical Lifshitz--Slyozov--Wagner model. We also look at the Cahn--Hilliard equation with constant mobility. We define the coarsening rate as $β=-(t/F)(d F/d t)$, where $F$ is the total free energy of the system and $t$ is time. There is a conjecture that the long-time average value of $β$ should not exceed $1/3$ -- this result is summarized here as $\langle β\rangle\leq 1/3$. We explore this conjecture for the two considered models. Using large-scale computational resources (specifically, GPU computing employing thousands of threads), we are able to construct ensembles of simulations and thereby build up a statistical picture of $β$. Our results show that the droplet population model and the Cahn--Hilliard equation (asymmetric mixtures) are demonstrably in agreement with $\langleβ\rangle\leq 1/3$. The results for the Cahn--Hilliard equation in the case of symmetric mixtures show $\langleβ\rangle$ sometimes exceeds $1/3$ in our simulations. However, the possibility is left open for the very long-time average values of $\langle β\rangle$ to be bounded above by $1/3$. The theoretical methodology laid out in this paper sets a path for future more intensive computational studies whereby this conjecture can be explored in more depth. \end{abstract}

physics.comp-ph↗

arXiv2019

A Geometric Diffuse-Interface Method for Droplet Spreading

This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation. The solution properties of this regularization are investigated via a sequence of numerical simulations whose results lead to new perspectives on thin-film behavior. The new perspectives in large-scale droplet-spreading dynamics are elucidated by comparing numerical-simulation results for the solution properties of the current model with corresponding known properties of three different alternative models. The three specific comparisons in solution behavior are made with the slip model, the precursor-film method and the diffuse-interface model.

physics.flu-dyn↗

arXiv2019

A mathematical framework for determining the stability of steady states of reaction-diffusion equations with periodic source terms

We develop a mathematical framework for determining the stability of steady states of generic nonlinear reaction-diffusion equations with periodic source terms, in one spatial dimension. We formulate an \textit{a priori} condition for the stability of such steady states, which relies only on the properties of the steady state itself. The mathematical framework is based on Bloch's theorem and Poincaré's inequality for mean-zero periodic functions. Our framework can be used for stability analysis to determine the regions in an appropriate parameter space for which steady-state solutions are stable.

math.AP↗

arXiv2019

A differential-equation model shows the proximate cause of the glass ceiling in European academia

We introduce a model based on Ordinary Differential Equations to describe how two mutually exclusive groups progress through a career hierarchy, whether in a single organization, or in an entire economic sector. The intended application is to gender imbalance at the top of the academic hierarchy in European Universities, however, the model is entirely generic and may be applied in other contexts also. Previous research on gender imbalance in European universities has focused on large-scale statistical studies. Our model represents a point of departure, as it is deterministic (i.e. based on Ordinary Differential Equations). The model requires a precise definition of the progression rates for the different groups through the hierarchy; these are key parameters governing the dynamics of career progression. The progression rate for each group can be decomposed into a product: the proportion of group members at a low level in the hierarchy who compete for promotion to the next level a given year, multiplied by the in-competition success rate for the group in question. Either of these two parameters can differ across the groups under consideration; this introduces a group asymmetry into the organization's composition. We introduce a glass-ceiling index to summarize this asymmetry succinctly. Using case studies from the literature, we demonstrate how the mathematical framework can pinpoint the proximate cause of the glass ceiling in European academia.

physics.soc-ph↗

arXiv2018

cuPentBatch -- A batched pentadiagonal solver for NVIDIA GPUs

We introduce cuPentBatch -- our own pentadiagonal solver for NVIDIA GPUs. The development of cuPentBatch has been motivated by applications involving numerical solutions of parabolic partial differential equations, which we describe. Our solver is written with batch processing in mind (as necessitated by parameter studies of various physical models). In particular, our solver is directed at those problems where only the right-hand side of the matrix changes as the batch solutions are generated. As such, we demonstrate that cuPentBatch outperforms the NVIDIA standard pentadiagonal batch solver gpsvInterleavedBatch for the class of physically-relevant computational problems encountered herein.

physics.comp-ph↗

arXiv2010

Interfacial instability in turbulent flow over a liquid film in a channel

We revisit here the stability of a deformable interface that separates a fully-developed turbulent gas flow from a thin layer of laminar liquid. Unlike previous work, the turbulent base state velocity profile proposed here requires only a specification of a flowrate or pressure drop, and no a posteriori choice of parameters. Moreover, the base state contains sufficient detail such that it allows for instability due to a viscosity-contrast mechanism (which turns out to be dominant) as well as instability due to a critical-layer-type mechanism, and it is validated against the experimental and numerical data available in the literature. Furthermore, the effect of perturbations in the turbulent stress distributions is investigated, and demonstrated, for the first time, to be small for cases wherein the liquid layer is thin. The detailed modelling of the liquid layer elicits two unstable modes, and mode competition can occur, although in most cases the instability is due to the viscosity-contrast mechanism. In particular, there is the possibility that surface roughness can reduce the growth rate of the interfacial mode, and promote a liquid-layer instability to the status of most dangerous mode. Our base-state model facilitates a new definition of `slow' and `fast' waves. We use our linear stability analysis to determine the factors that affect the wave speed and demonstrate that the waves are `slow' according to the definition proposed here. Finally, we compare our results with experimental data, and good agreement is obtained.

physics.flu-dyn↗

arXiv2010

Modelling the growth rate of a tracer gradient using stochastic differential equations

We develop a model in two dimensions to characterise the growth rate of a tracer gradient mixed by a statistically homogeneous flow with rapid temporal variations. % % The model is based on the orientation dynamics of the passive-tracer gradient with respect to the straining (compressive) direction of the flow, and involves reducing the dynamics to a set of stochastic differential equations. The statistical properties of the system emerge from solving the associated Fokker--Planck equation. In a certain limiting case, and within the model framework, there is a rigorous proof that the tracer gradient aligns with the compressive direction. This limit involves decorrelated flows whose mean vorticity is zero. % % % Using numerical simulations, we assess the extent to which our model applies to real mixing protocols, and map the stochastic parameters on to flow parameters.

physics.flu-dyn↗