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Thom Gunn’s ‘The Discovery of the Pacific’ (1970) describes a young couple who have driven across America and reached the Californian coast. Their exhilaration as they stare at the ocean recalls another moment of epiphany; when John Keats read George Chapman’s translation of Homer, he felt like a conquistador setting eyes on the Pacific for the first time. In this essay, I argue that Gunn’s poem engages closely with Keats’s ‘On First Looking into Chapman’s Homer’, and that Keats’s sonnet in turn draws on ancient ideas of Homer as being like the ocean. These are ideas Keats seems to have drawn from Alexander Pope’s translation of (and commentary on) the Homeric poems. Pope, Keats, and Gunn all regard the Homer poems as being in some way oceanic—both because of their vastness and capaciousness, and because they have been previously discovered by so many earlier readers.

In his Remarks on Several Parts of Italy (1705), Addison regularly draws on his deep knowledge of Latin poetry in order to ‘compare the natural face of the country with the Landskips that the Poets have given us of it’. Less conventionally, but just as regularly, he elucidates landscape, history, and antiquities through reference to ancient coins. Roughly contemporaneously, Addison wrote a defence of numismatics in the Dialogues Upon the Usefulness of Ancient Medals (published posthumously in 1721), in which one character, Philander, seeks to persuade Cynthio from his view that numismatists are mere ‘critics in Rust’ (Cynthio’s view closely resembling the attacks on Bentley and others by satirists such as William King). Addison, through Philander’s person, sees the poems and medals he juxtaposes as representing ‘the same design executed by different hands’; ‘A reverse often clears up the passages of an old poet, as the poet often serves to unriddle a reverse.’ But coins have, for Addison, a moral as well as an explanatory function, publicizing the characters and deeds of great men and women by keeping them in circulation. This chapter explores the relationship between the moral and the fiscal function of coins, drawing out connections between Addison’s views on ancient numismatics and his approach both to modern British coinage and to the circulation of texts and ideas.

The Boundary Element Method (BEM) is implemented in the simulation of compression, displacement, migration and splitting of intra-tow voids in dual-scale fibrous reinforcements. The last three processes have not been simulated at mesoscopic scale in previous works due to the consideration of a constant pressure in the channels of the Representative Unitary Cell. In this work, both the channels and tows are modeled using the Stokes and Darcy equations, respectively, a pressure gradient is prescribed along the fluid motion, and full air compressibility is deemed, thereby allowing to consider these three processes. The void migration process from the weft towards the channel is analyzed in terms of the ratio between the average air migration velocity and the average liquid velocity, and of the normalized air rate from the weft towards the channel. According to BEM results, the bubble can migrate at both lower and higher velocities with respect to the liquid velocity, and the void removal out of the tows can occur after several stages of compression–displacement–migration–splitting; additionally, the bubble breaks up after several cycles of expansion and compression. BEM results also show that the liquid surface tension, pressure gradient and average channel pressure have important influence in the void migration process.

We investigate immiscible radial displacement in a Hele-Shaw cell with a temperature dependent viscosity using two coupled high resolution numerical methods. Thermal gradients created in the domain through the injection of a low viscosity fluid at a different temperature to the resident high viscosity fluid can lead to the formation of unstable thermo-viscous fingers, which we explore in the context of immiscible flows. The transient, multi-zone heat transfer is evaluated using a newly developed auxiliary radial basis function-finite collocation (RBF-FC) method, which locally captures variation in flux and field variable over the moving interface, without the need for ghost node extrapolation. The viscosity couples the transient heat transfer to the Darcy pressure/velocity field, which is solved using a boundary element - RBF-FC method, providing an accurate and robust interface tracking scheme for the full thermo-viscous problem. We explore the thermo-viscous problem space using systematic numerical experiments, revealing that the early stage finger growth is controlled by the pressure gradient induced by the varying temperature and mobility field. In hot injection regimes, negative temperature gradients normal to the interface act to accelerate the interface, promoting finger bifurcation and enhancing the viscous fingering instability. Correspondingly, cold injection regimes stabilise the flow compared to isothermal cases, hindering finger formation. The interfacial mobility distribution controls the late stage bifurcation mode, with non-uniformities induced by the thermal diffusivity creating alternate bifurcation modes. Further numerical experiments reveal the neutral stability of the thermal effects on the fingering evolution, with classical viscous fingering dynamics eventually dominating the evolution. We conclude the paper with a mechanistic summary of the immiscible thermo-viscous fingering regime, providing the first detailed analysis of the thermal problem in immiscible flows.

We explore the use of Gaussian process emulators (GPE) in the numerical simulation of CO 2 injection into a deep heterogeneous aquifer. The model domain is a two-dimensional, log-normally distributed stochastic permeability field. We first estimate the cumulative distribution functions (CDFs) of the CO 2 breakthrough time and the total CO 2 mass using a computationally expensive Monte Carlo (MC) simulation. We then show that we can accurately reproduce these CDF estimates with a GPE, using only a small fraction of the computational cost required by traditional MC simulation. In order to build a GPE that can predict the simulator output from a permeability field consisting of 1000s of values, we use a truncated Karhunen-Loeve (K-L) expansion of the permeability field, which enables the application of the Bayesian functional regression approach. We perform a cross-validation exercise to give an insight of the optimization of the experiment design for selected scenarios: we find that it is sufficient to use 100s values for the size of training set and that it is adequate to use as few as 15 K-L components. Our work demonstrates that GPE with truncated K-L expansion can be effectively applied to uncertainty analysis associated with modelling of multiphase flow and transport processes in heterogeneous media.

A gas lubricated bearing model is derived which is appropriate for a very small bearing face separation by including velocity slip boundary conditions and centrifugal inertia effects. The bearing dynamics is examined when an external harmonic force is imposed on the bearing due to the bearing being situated within a larger complex dynamical system. A compressible Reynolds equation is formulated for the gas film which is coupled to the bearing structure through an axial force balance where the rotor and stator correspond to spring–mass–damper systems. Surface slip boundary conditions are derived on the bearing faces, characterised by the slip length parameter. The coupled bearing system is analysed using a stroboscopic map solver with the modified Reynolds equation and structural equations solved simultaneously. For a sufficiently large forcing amplitude a flapping motion of the bearing faces is induced when the rotor and stator are in close proximity. The minimum bearing gap over the time period of the external forcing is examined for a range of bearing parameters.

Most implementations of meshless BEMs use a circular integration contours (spherical in 3D) embedded into a local interpolation stencil with the so-called Companion Solution (CS) as a kernel, in order to eliminate the contribution of the single layer potential. However, the Dirichlet Green's Function (DGF) is the unique Fundamental Solution that is identically zero at any given close surface and therefore eliminates the single layer potential. One of the main objectives of this work is to show that the CS is nothing else than the DGF for a circle collocated at its origin. The use of the DGF allows the collocation at more than one point, permitting the implementation of a P-adaptive scheme in order to improve the accuracy of the solution without increasing the number of subregions. In our numerical simulations, the boundary conditions are imposed at the interpolation stencils in contact with the problem boundary instead of at the corresponding integration surfaces, permitting always the use of circular integration contours, even in regions near or in contact with the problem domain where the densities of the integrals are reconstructed from the interpolation formulae that already included the problem boundary conditions. Fil: Power, H.. The University of Nottingham; Reino Unido Fil: Caruso, Nahuel Domingo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina Fil: Portapila, Margarita Isabel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina

A model for a fluid-lubricated bearing is derived for operation under conditions where external forces are subject to random fluctuations that may act to destabilize the bearing. The fluid flow through the bearing is described by a Reynolds equation for incompressible flow and is coupled to the axial displacement of the bearing faces as modelled by spring-mass-damper systems. Representative dynamics of a highly rotating bearing subject to external potentially destabilizing random forcing is developed. An external force characterized by a noise term is imposed on the rotor, where both white noise and coloured noise are considered. For industrial applications, it is important to evaluate potential bearing failure that can arise when the face clearance becomes sufficiently small. Therefore, a quantity of interest is the average time for the face clearance to reach a prescribed tolerance. A computational technique to evaluate the bearing characteristics is implemented based on a simple random walk for a Dirichlet problem for a linear parabolic partial differential equation combined with a Monte Carlo technique. Results of numerical experiments are presented, to give indicative predictions of possible face contact, which has the potential to result in bearing failure.

Developments in industrial applications motivate improvements in fluid lubricated bearing technology, enabling smaller face clearances and increased rotation rates. Associated film lubrication technology aims to improve efficiency and reliability. A bearing model is developed to evaluate the effect of external, potentially destabilising, random forcing applied to a pair of highly rotating axisymmetric bearing faces, separated by a thin fluid film. Two cases of random external force are examined. A first study considers an imposed random force disturbance constrained to a fixed period, where the average minimum face clearance together with the probability it reaches a specified gap tolerance. More general uncertainties are associated with more complex external forcing and takes the form of a white or coloured noise. In this case the average time for the face clearance to reach a prescribed tolerance is examined. Results can inform bearing design, providing an indication of the effect of disturbances on the average lifetime and identify constraints on operating conditions for safe and reliable behaviour.

During the seventeenth century the relationship between monarch and universities was a highly political one. This chapter considers the many collections of verse—in English and Latin—issued by the universities in response to royal successions. The protocols surrounding these volumes allowed for a certain amount of political self-expression. This chapter argues that these volumes became a means by which the universities could establish a relationship with the new monarch. The first half of the chapter charts the emergence and operation of protocols for producing these commemorative volumes. The second half offers a case study of Cambridge’s two commemorative volumes, respectively on the death of Oliver Cromwell in 1658 and the accession of Charles II in 1660. The scholarly exercises contained within these volumes were capable of communicating significant political messages.

The unsteady two-dimensional Navier–Stokes system of equations, for viscous incompressible fluids are solved using a global method of approximated particular solutions MAPS in terms of a Stokes formulation, where the velocity and pressure fields are approximated from a linear superposition of particular solutions of a non-homogeneous Stokes system of equations, with a multiquadric MQ radial basis function RBF as non-homogeneous term. Steady-state solution of the flow problems considered in this work can be unstable at high Reynolds numbers Re, corresponding to bifurcation of solutions that result in the appearance of new stable steady-state or periodic solutions. The main objective of this work is to present a global meshless numerical scheme able to predict these bifurcation points and concurrent new stable or periodic solutions. This is well known to be a very difficult task for any numerical scheme. An implicit first-order time-stepping scheme is used to approximate the transient term and the obtained nonlinear system of algebraic equations is solved by a Newton–Raphson method with variable step. Two steady-state and two transient problems are considered to validate the numerical scheme: the lid-driven cavity and backward-facing step BFS flows steady-state problems and the decaying Taylor–Green vortex and two-sided lid-driven cavity flows transient problems. The first two problems are solved up to Re=10,000 and 2300, respectively. Results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement. Obtained numerical solutions for the decaying vortices at Re=100 shown excellent agreement with the corresponding analytical results. The transient problem of a rectangular two-sided lid-driven cavity flow is solved at Re=700. The influence of the cavity length, l, in determining the different structures of the flow pattern is studied for values of , showing that the scheme is able to reproduce the previously reported change in the flow pattern when l=2. Finally, the global Stokes MAPS are used to carry out nonlinear stability analyses of three steady-state problems: the sudden expansion, lid-driven cavity and BFS flows. Stable and unstable steady-state solutions at Re values greater than critical are predicted with the proposed numerical scheme. Our numerical results are consistent with previously stability analysis reported in the literature.

The Boundary Element Method (BEM) is applied for the solution of the problem of void formation in fibrous reinforcements used in composite materials. Stokes and Darcy formulations for the channel and porous medium are considered, including three main differences with most of the previous models reported in the literature that deal with this type of problem: the consideration of the general form of the Beavers–Joseph slip condition instead of the Saffman simplification, the calculation of the flow direction-dependent capillary pressure in the porous medium without experimental factors and the consideration of the surface traction effects in the channel fluid front. An analytical solution of a simple problem is presented to assess the accuracy and convergence of the BEM solution, obtaining good agreement between the results. In order to evaluate the main differences between the Stokes–Darcy formulation and a dual-scale permeability Darcy approach, a problem of constant pressure filling is considered, showing significant differences in the evolution of the flow field as: filling times, shape of the moving fluid front and size and shape of the formed voids. Finally, several simulations at constant flow rate are carried out to analyze the influence of the capillary number, tow porosity, width of transverse tow, fluid penetrability and RUC porosity on the size, shape and location of the void. In general, the void size and shape are influenced by the considered parameters, but the void location is not.

In this work we present an improvement of the Localized Regular Dual Reciprocity Method (LRDRM). LRDRM is an integral domain decomposition method with two distinguishing features, the boundary conditions are imposed at the local interpolation level and all the calculated integrals are regular. In this work we present an enhancement of this method where the interpolation functions themselves satisfy the partial differential equation to be solved. Results for 1D and 2D convection‐diffusion, 2D Helmholtz and 2D Poisson equations are presented, attaining accuracies two to three orders of magnitude higher than the original version of the LRDRM.

In 1958 Christopher Logue produced his first ‘account’ of the Iliad, ‘Achilles Fights the River’, following encouragement from two BBC producers, Donald Carne-Ross and Xanthe Wakefield. Over the next four decades, Logue produced several more accounts, now known collectively as War Music. Despite being the product of Logue’s idiosyncratic range of interests and affinities, the poem was also shaped through collaboration with two people who had a particular interest in the poem’s performance. Logue continued to correspond into the early 1980s with Carne-Ross, whose constant attention to the sound of the poem was crucial to its development. From 1981, Logue’s collaboration with the actor Alan Howard, who recorded sections of the poem for broadcast on the BBC also helped to shape War Music. This chapter, drawing on archival material, considers the ways in which the poem’s performance history affected its form and its development.

A potential novel working fluid for vapour absorption refrigeration utilising very low grade waste heat, is based on acetone and zinc bromide as the salt solution. A Computational Fluid Dynamics (CFD) model is presented of the fluid with zinc oxide nano-particles in a flat tube flow. A two phase type of model represents the zinc oxide nano-particles as a distinct fluid phase. The cases of laminar and turbulent flow are explored numerically for a wide range of acetone and nanoparticles concentrations. The velocity is varied between 1.5 and 6 ms−1, representing typical heat exchanger conditions. Reynolds number depends significantly on the solution concentration. Heat transfer coefficient increases with Re, by turbulent mixing, and with the concentration of nanoparticles and of acetone by the enhanced thermal diffusivity. The shear wall stress is not affected by changing the concentration of nano-particles. The nano-fluid is demonstrated to work well for heat transfer enhancement over the base fluid; the further issue of suspension of the nano-particles in the solution is explored experimentally. The nano-fluid can be achieved by ultra-sonic excitation, with a settling time in the order of several hours. Subject to the particle suspension time being increased, this fluid combination is a good candidate for the application considered.

We propose a node-based local meshless method for advective transport problems that is capable of operating on centrally defined stencils and is suitable for shock-capturing purposes. High spatial convergence rates can be achieved; in excess of eighth-order in some cases. Strongly-varying smooth profiles may be captured at infinite Peclet number without instability, and for discontinuous profiles the solution exhibits neutrally stable oscillations that can be damped by introducing a small artificial diffusion parameter, allowing a good approximation to the shock-front to be maintained for long travel times without introducing spurious oscillations.The proposed method is based on local collocation with radial basis functions (RBFs) in a "finite collocation" configuration. In this approach the PDE governing and boundary equations are enforced directly within the local RBF collocation systems, rather than being reconstructed from fixed interpolating functions as is typical of finite difference, finite volume or finite element methods. In this way the interpolating basis functions naturally incorporate information from the governing PDE, including the strength and direction of the convective velocity field. By using these PDE-enhanced interpolating functions an "implicit upwinding" effect is achieved, whereby the flow of information naturally respects the specifics of the local convective field. This implicit upwinding effect allows high-convergence solutions to be obtained on centred stencils for advection problems.The method is formulated using a high-convergence implicit timestepping algorithm based on Richardson extrapolation. The spatial and temporal convergence of the proposed approach is demonstrated using smooth functions with large gradients. The capture of discontinuities is then investigated, showing how the addition of a dynamic stabilisation parameter can damp the neutrally stable oscillations with limited smearing of the shock front.

In this work we present an efficient and accurate implementation of the LRDRM. This integral domain decomposition method exploits two advantages: first, it imposes the boundary conditions at the Local RBF interpolation. Second, the integrals to compute are always regular. The approximation of the derivative of the field variable is computed in a posteriori way, directly differentiating the Local RBF interpolation or the local integral equation. The efficient and accurate behaviour of this method are demonstrated by performing numerical examples, with special emphasis on a 1D benchmark convective-diffusion equation. Results for 2D convection-diffusion, 2D Helmholtz and 2D Poisson equations are also presented.

An initial deterministic mathematical model for the dynamic motion of a simple pressurised liquid film bearing is derived and utilised to evaluate the possibility of bearing contact for thin film operation. For a very thin film bearing the flow incorporates a Navier slip boundary condition as parametrised by a slip length that in general is subject to significant variability and is difficult to determine with precision. This work considers the formulation of a modified Reynolds equation for the pressurised liquid flow in a highly rotating coned bearing. Coupling of the axial motion of the stator is induced by prescribed axial oscillations of the rotor through the liquid film. The bearing gap is obtained from solving a nonlinear second-order non-autonomous ordinary differential equation, via a mapping solver. Variability in the value of the slip length parameter is addressed by considering it as a random variable with prescribed mean and standard deviation. The method of derived distributions is used to exactly quantify the impact of variability in the slip length with a parametric study investigating the effect of both the deterministic and distribution parameters on the probability of contact. Additionally, as the axial rotor oscillations also have a random aspect due to possible varying excitations of the system, the probability of contact is investigated for both random amplitude of the periodic rotor oscillations and random slip length, resulting in a two parameter random input problem. The probability of contact is examined to obtain exact solutions and evaluate a range of bearing configurations.

Freezing processes are characterised by sharp changes in specific heat capacity and thermal conductivity for temperatures close to the freezing point. This leads to strong nonlinearities in the governing PDE that may be difficult to resolve using traditional numerical methods. In this work we present a meshless numerical method, based on a local Hermite radial basis function collocation approach in finite differencing mode, to allow the solution of freezing problems. By introducing a Kirchhoff transformation and solving the governing equations in Kirchhoff space, the strength of nonlinearity is reduced while preserving the structure of the heat equation. In combination with the high-resolution meshless numerical method, this allows efficient and stable numerical solutions to be obtained for freezing problems using 3D unstructured datasets. We demonstrate the proposed numerical formulation for the freezing of foodstuffs. By approximating the shape of the thermal conductivity and heat capacity curves in a piecewise linear fashion the temperature-dependent material curves may be described using eight independent parameters. We consider the optimisation of these parameters to match experimental data for the freezing of a hemispherical sample of mashed potato by using a simple manual procedure that requires a minimal number of simulations to be performed. Working in this way, a good approximation is obtained to the temperature profile throughout the sample without introducing instability into the numerical results.

The aim of this paper is to present how to implement a control volume approach improved by Hermite radial basis functions (CV-RBF) for geochemical problems. A multi-step strategy based on Richardson extrapolation is proposed as an alternative to the conventional dual step sequential non-iterative approach (SNIA) for coupling the transport equations with the chemical model. Additionally, this paper illustrates how to use PHREEQC to add geochemical reaction capabilities to CV-RBF transport methods. Several problems with different degrees of complexity were solved including cases of cation exchange, dissolution, dissociation, equilibrium and kinetics at different rates for mineral species. The results show that the solution and strategies presented here are effective and in good agreement with other methods presented in the literature for the same cases. HighlightsA control volume Hermite RBF scheme has been implemented for groundwater reactive problems.The standard SNIA approach was improved by a Richardson extrapolation.The PHREEQ modules are coupled with a transport solution by a local RBF method.The solution by CV-RBF is effective and in agreement with other numerical methods.

The flow of pulverized fuel in a power plant is one example of gas-particle flows in the energy and process industries. Computational Fluid Dynamics (CFD) is a widely used trouble shooting and engineering design tool increasingly deployed within industry to predict the behavior of such flows. Despite wide use, traditional models are often inadequate for solving industrial problems. An example is the concentration of the particulate phase within a coal burner's fuel feed due to particle inertia and inter-particle collisions. This local concentration of the particles can be advantageous, but it is difficult to characterize the flow beyond the dilute limit (volume fraction of 0.001). To address this and other short comings a number of two-phase flow models have been investigated and their applicability and accuracy assessed by comparison with experiments in the literature for a vertical pipe flow with relatively high mass loadings of the discrete phase. The physics investigated includes: particle-wall collisions, particle-particle collisions and structure dependent drag. Models have been implemented in the commercial CFD software ANSYS FLUENT R.14.5 for the Discrete Phase Model. The results show a significant improvement over industry best practice and provide an indication as to the key physics and the effects of scale on confined gas-particle flows. Furthermore, the modeling approach will be applicable in a number of other industrial areas, such as biomass conveying. This paper provides an overview of CFD models for the application to pulverized fuel flows within a coal fired power plant and discusses the gap between academic development and industrial adoption of advanced CFD models.

An incompressible fluid flow model for a thin-film thrust bearing with slip flow is derived, leading to a modified Reynolds equation for a highly rotating rotor that incorporates a slip length shear condition on the bearing faces, extending previous bearing studies for new bearing applications associated with decreasing film thickness. Mathematical and numerical modelling is applied to the coupled process of the pressurized fluid flow through the bearing, with a Navier slip condition replacing a no-slip condition, and the axial motion of the rotor and stator. The derived modified Reynolds equation is coupled with the dynamic motion of the stator through the pressure exerted by the fluid film, with explicit analytical expressions for the pressure and force determined and the equation for the bearing gap reduced to a non-linear second-order non-autonomous ordinary differential equation. A mapping solver is used to investigate the time-dependent bearing gap for prescribed periodic motion of the rotor. A parametric study focuses on bearing operation under close contact motion to examine the minimum film thickness and possibility of bearing face contact.

Based on an approach used to model environmental flows such as rivers and estuaries, we develop a new multi-layered model for thin liquid film flow on a locally-heated inclined plane. The film is segmented into layers of equal thickness with the velocity and temperature of each governed by a momentum and energy equation integrated across each layer individually. Matching conditions applied between the layers ensure the continuity of down-plane velocity, temperature, stress and heat flux. Variation in surface tension of the liquid with temperature is considered so that local heating induces a surface shear stress which leads to variation in the film height profile (the Marangoni effect). Moderate inertia and heat convection effects are also included. In the absence of Marangoni effects, when the film height is uniform, we test the accuracy of the model by comparing it against a solution of the full heat equation using finite differences. The multi-layer model offers significant improvements over that of a single layer. Notably, with a sufficient number of layers, the solution does not exhibit local regions of negative temperature often predicted using a single-layer model. With Marangoni effects included the film height varies however we find heat convection can mitigate this variation by reducing the surface temperature gradient and hence the surface shear stress. Numerical results corresponding to the flow of water on a vertical plane show that very thin films are dominated by the Marangoni shear stress which can be sufficiently strong to overcome gravity leading to a recirculation in the velocity field. This effect reduces with increasing film thickness and the recirculation eventually disappears. In this case heating is confined entirely to the interior of the film leading to a uniform height profile.

The possibility of face contact is examined for a coaxial rotor-stator bearing in dynamic motion constrained by a highly rotating very thin liquid film. A modified Reynolds equation for pressurised flow is coupled to the bearing structure leading to determination of the bearing gap from solving a nonlinear second-order non-autonomous ordinary differential equation. Periodic solutions are found via a mapping solver. Rotor deformation is parametrised by a coning angle and considered a random variable. The method of derived distributions is used to quantify variation in coning angle and examine the probability of rotor-stator contact. Additionally, effects of possible destabilising random aspects on the axial rotor oscillations are investigated. Exact solutions for probability of contact are obtained for various bearing configurations.

A numerical study of voids formation in dual-scale fibrous reinforcements is presented. Flow fields in channels (Stokes) and tows (Brinkman) are solved via direct Boundary Element Method and Dual Reciprocity Boundary Element Method, respectively. The present approach uses only boundary discretization and Dual Reciprocity domain interpolation, which is advantageous in this type of moving boundary problems and leads to an accurate representation of the moving interfaces. A problem admitting analytical solution, previously solved by domain-meshing techniques, is used to assess the accuracy of the present approach, obtaining satisfactory results. Fillings of Representative Unitary Cells at constant pressure are considered to analyze the influence of capillary ratio, jump stress coefficient and two formulations (Stokes---Brinkman and Stokes---Darcy) on the filling process, void formation and void characterization. Filling times, fluid front shapes, void size and shape, time and space evolution of the saturation, are influenced by these parameters, but voids location is not.

The governing equations and mathematical models describing CO2 spreading and trapping in saline aquifers and the related hydro-mechanical and chemical processes were described in Chapt. 3. In this chapter, the focus is on methods for solving the relevant equations. The chapter gives an overview of the different approaches, from high-fidelity full-physics numerical models to more simplified analytical and semi-analytical solutions . Specific issues such as modeling coupled thermo-hydro-mechanical-chemical processes and modeling of small-scale processes , such as convective mixing and viscous fingering , are also addressed. Finally, illustrative examples of modeling real systems, with different types of modeling approaches, are presented.

We consider two-dimensional one-sided convection of a solute in a fluid-saturated porous medium, where the solute decays via a first-order reaction. Fully nonlinear convection is investigated using high-resolution numerical simulations and a low-order model that couples the dynamic boundary layer immediately beneath the distributed solute source to the slender vertical plumes that form beneath. A transient-growth analysis of the boundary layer is used to characterise its excitability. Three asymptotic regimes are investigated in the limit of high Rayleigh number $\mathit{Ra}$, in which the domain is considered deep, shallow or of intermediate depth, and for which the Damköhler number $\mathit{Da}$ is respectively large, small or of order unity. Scaling properties of the flow are identified numerically and rationalised via the analytic model. For fully established high-$\mathit{Ra}$ convection, analysis and simulation suggest that the time-averaged solute transfer rate scales with $\mathit{Ra}$ and the plume horizontal wavenumber with $\mathit{Ra}^{1/2}$, with coefficients modulated by $\mathit{Da}$ in each case. For large $\mathit{Da}$, the rapid reaction rate limits the plume depth and the boundary layer restricts the rate of solute transfer to the bulk, whereas for small $\mathit{Da}$ the average solute transfer rate is ultimately limited by the domain depth and the convection is correspondingly weaker.

The integral equation formulation developed previously by the authors to study isothermal micro flows under shear slip boundary condition is extended in this work to consider the case of non-isothermal micro gas flows with thermal creep effects at finite Peclet numbers. The effect of thermal creep over the flow patterns with and without considering the effect of shear slip is investigated in detail using a boundary element method. In this work the boundary integral approaches for both fluid velocity and temperatures fields are used to solve the problem of shear-driven cavity flow at finite Peclet number. This is obtained by considering the diffusive-convective heat equation using the Dual Reciprocity Method to transform the corresponding volume integral of the convective terms into equivalent surface integrals.

Summary In this work, we describe a meshless numerical method based on local collocation with RBFs for the solution of the poroelasticity equation. The RBF finite collocation approach forms a series of overlapping nodal stencils, over which an RBF collocation is performed. These local collocation systems enforce the governing PDE operator throughout their interior, with the intersystem communication occurring via the collocation of field variables at the stencil periphery. The method does not rely on a generalised finite differencing approach, whereby the governing partial differential operator is reconstructed at the global level to drive the solution of the PDE. Instead, the PDE governing and boundary operators are enforced directly within the local RBF collocation systems, and the sparse global assembly is formed by reconstructing the value of the field variables at the centrepoint of the local stencils. In this way, the solution of the PDE is driven entirely by the local RBF collocation, and the method more closely resembles the approach of the full-domain RBF collocation method. By formulating the problem in this fashion, high rates of convergence may be attained without the computational cost and numerical ill-conditioning issues that are associated with the full-domain RBF collocation approach. An analytical solution is formulated for a 2D poroelastic fluid injection scenario and is used to verify the proposed implementation of the method. Highly accurate solutions are produced, and convergence rates in excess of sixth order are observed for each field variable (i.e. pressure and displacement) and field-variable derivative (i.e. pressure gradients and stresses). The stress and displacement fields resulting from the solution of the poroelasticity equation are then used to describe the formation and propagation of microfractures and microfissures, which may form in the presence of large shear strain, in terms of a continuous damage variable which modifies the mechanical and hydraulic properties of the porous medium. The formation of such hydromechanical damage, and the resulting increase in hydraulic conductivity, is investigated for a pressurised injection into sandstone. Copyright © 2014 John Wiley & Sons, Ltd.

The method of approximate particular solutions (MAPS) is used to solve the two-dimensional Navier–Stokes equations. This method uses particular solutions of a nonhomogeneous Stokes problem, with the multiquadric radial basis function as a nonhomogeneous term, to approximate the velocity and pressure fields. The continuity equation is not explicitly imposed since the used particular solutions are mass conservative. To improve the computational efficiency of the global MAPS, the domain is split into overlapped subdomains where the Schwarz Alternating Algorithm is employed using velocity or traction values from neighboring subdomains as boundary conditions. When imposing only velocity boundary conditions, an extra step is required to find a reference value for the pressure at each subdomain to guarantee continuity of pressure across subdomains. The Stokes lid-driven cavity flow problem is solved to assess the performance of the Schwarz algorithm in comparison to a finite-difference-type localized MAPS. The Kovasznay flow problem is used to validate the proposed numerical scheme. Despite the use of relative coarse nodal distributions, numerical results show excellent agreement with respect to results reported in literature when solving the lid-driven cavity (up to Re = 10,000) and the backward facing step (at Re = 800) problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 777–797, 2015

An efficient accurate Local Method of Approximate Particular Solutions (LMAPS) using Multiquadric Radial Basis Functions (RBFs) for solving convection–diffusion problems is proposed. It consists in adding auxiliary points to the local interpolation stencil at which the governing PDE is enforced, known as PDE points, besides imposing the boundary condition at the stencil in contact with the problem boundary. Two convection–diffusion problems are considered as test problems and solved with two previous local direct RBF collocation schemes (with and without PDE points) and two LMAPS (with and without PDE points), as well as the Global MAPS, in order to compare accuracy, convergence order and their behaviour in terms of the shape parameter. If PDE points are added, the result accuracy is improved as well as the convergence rate when using both local direct and MAPS formulations.

Motivated by evaluating coating oil films within bearing chambers in an aero-engine application, an analysis is presented for the fluid dynamics relevant in their dual capacity as both the coolant and lubricant in highly sheared flows that may approach microscale thickness. An extended model is developed for isothermal rimming flow driven by substantial surface shear within a stationary cylinder. In particular, a partial slip condition replaces the no-slip condition at the wall whilst retaining inertial effects relevant to an intrinsic high speed operation. A depth-averaged formulation is presented that includes appropriate inertial effects at leading-order within a thin film approximation that encompasses a more general model of assessing the impact of surface slip. Non-dimensional mass and momentum equations are integrated across the film depth yielding a one dimensional problem with the a priori assumption of local velocity profiles. The film flow solutions for rimming flow with wall slip are modeled to a higher order than classical lubrication theory. We investigate the impact of wall slip on the transition from pooling to uniform films. Numerical solutions of film profiles are provided for the progressively increased Reynolds number, within a moderate inertia regime, offering evaluation into the effect of film slippage on the dynamics of rimming flow. We find that slip allows non-unique solution regions and existence of multiple possible steady state solutions evaluated in transforming from smooth to pooling film solutions. Additionally, boundary slip is shown to enhance the development of recirculation regions within the film which are detrimental to bearing chamber flows.

Fingering instabilities at the interface between two immiscible fluids can appear during the displacement of a fluid of higher viscosity by another one of lower viscosity. The evolution of the finger structures is determined by the interface kinematic and dynamic matching conditions, which describe mass and momentum conservation across the interface. In the case when the injected fluid is a gas and the resident one is a liquid, dissolution of the injected gas into the displaced liquid can occur at the interface between the two phases. In this case, the transfer velocity of the dissolved gas reduces the interface displacement velocity as described by the kinematic matching condition, delaying the evolution of the fingering. In addition, the momentum flux across the interface, due to the dissolution, modifies the dynamic matching condition with possible changes in the patterns of the fingers structures. This work studies the effects of gas dissolution on the evolution of fingering instabilities during the displacement of a viscous liquid by an immiscible injected gas in a Hele-Shaw cell. A boundary element numerical simulation of the growth of the injected gas bubble is developed and implemented. This numerical model takes into account the dissolution across a sharp interface between the two phases. Our numerical simulations suggest that the inclusion of gas dissolution can lead to the eventual breaking of the fingers. These “shed fingers” become individual bubbles, which move away from the injection source with the velocity of the surrounding fluid and eventually will dissolve into the ambient fluid. New fingers evolve, with their concurrent breaking, resulting in the possibility of a cascade of travelling and dissolving bubbles, instead of a continuous fingering structure.

The two-dimensional Navier-Stokes system of equations for incompressible fluids is solved by the method of approximate particular solutions (MAPS) in its global formulation. The fluid velocity and pressure fields are approximated by a linear superposition of particular solutions of a Stokes non-homogeneous system of equations with multiquadric (MQ) radial basis function as the source term. The nonlinear convective terms of the momentum equations are linearly approximated by using a guess value of the velocity field, and the resulting linear system of equations is solved by a simple direct iterative scheme (Picard iteration), with the velocity guess given by the solution at the previous iteration. Although the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative because the particular solutions exactly satisfy the mass conservation equation. The proposed numerical scheme is validated by comparison of the obtained numerical results with the corresponding analytical solution of the Kovasznay flow problem at different Reynolds numbers, Re. From this analysis, it is observed that the MAPS results are stable and accurate for a wide range of shape parameter values. In addition, lid-driven cavity flow problems in rectangular and triangular domains up to Re=3200 and Re=1000, respectively, and the backward-facing step at Re=800 are solved, and the results obtained are compared with corresponding benchmark numerical solutions, showing excellent agreement.

A global version of the Method of Approximate Particular Solutions (MAPS) is developed to solve two-dimensional Stokes flow problems in bounded domains. The velocity components and the pressure are approximated by a linear superposition of particular solutions of the non-homogeneous Stokes system of equations with a Multiquadric Radial Basis Function as forcing term. Although, the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative since the particular solutions exactly satisfy the mass conservation equation. The present scheme is validated by comparing the obtained numerical result with the analytical solution of two boundary value problems constructed from the Stokeson exterior fundamental solution, i.e. regular everywhere except at infinity. For these two cases, convergence of the method and the influence of the value of the Multiquadric’s shape parameter on the numerical results are studied by computing the relative Root Mean Square (RMS) error for several homogeneous distributions of collocation points and values of the shape parameter. From this analysis is observed that the proposed MAPS results are stable and accurate for a wide range of shape parameter values. In addition, the lid-driven cavity and backward-facing step flow problems are solved and the obtained results compared with the solutions found with more conventional numerical schemes, showing good agreement between them.

This work proposes an alternative decomposition for local scalable meshless RBF collocation. The proposed method operates on a dataset of scattered nodes that are placed within the solution domain and on the solution boundary, forming a small RBF collocation system around each internal node. Unlike other meshless local RBF formulations that are based on a generalised finite difference (RBF-FD) principle, in the proposed ''finite collocation'' method the solution of the PDE is driven entirely by collocation of PDE governing and boundary operators within the local systems. A sparse global collocation system is obtained not by enforcing the PDE governing operator, but by assembling the value of the field variable in terms of the field value at neighbouring nodes. In analogy to full-domain RBF collocation systems, communication between stencils occurs only over the stencil periphery, allowing the PDE governing operator to be collocated in an uninterrupted manner within the stencil interior. The local collocation of the PDE governing operator allows the method to operate on centred stencils in the presence of strong convective fields; the reconstruction weights assigned to nodes in the stencils being automatically adjusted to represent the flow of information as dictated by the problem physics. This ''implicit upwinding'' effect mitigates the need for ad-hoc upwinding stencils in convective dominant problems. Boundary conditions are also enforced within the local collocation systems, allowing arbitrary boundary operators to be imposed naturally within the solution construction. The performance of the method is assessed using a large number of numerical examples with two steady PDEs; the convection-diffusion equation, and the Lame-Navier equations for linear elasticity. The method exhibits high-order convergence in each case tested (greater than sixth order), and the use of centred stencils is demonstrated for convective-dominant problems. In the case of linear elasticity, the stress fields are reproduced to the same degree of accuracy as the displacement field, and exhibit the same order of convergence. The method is also highly stable towards variations in basis function flatness, demonstrating significantly improved stability in comparison to finite-difference type RBF collocation methods.

This work presents a novel meshless numerical approach for the solution of linear elasticity problems, using locally supported RBF collocation. The Kansa (unsymmetric) RBF collocation method is used to form local collocation systems, which enforce the PDE governing and boundary operators. With the displacement values acting as the unknowns in the system, a sparse global system is formed. This global matrix is formed in a manner analogous to a finite difference method, with the displacement values at each internal node defined in terms of the displacements at other nodes within the local stencil. In contrast to traditional finite difference methods, here the RBF collocation assumes the role traditionally played by polynomial interpolants. The RBF collocation does itself satisfy the governing PDE operator at some collocation points, and therefore allows for a significantly more accurate reconstruction than is found from simple polynomial interpolants. In addition, the boundary operators (for applied displacement and applied surface traction) are enforced directly within the local RBF collocation systems, rather than being enforced at the global matrix. In contrast to traditional finite difference methods based on polynomial interpolation, the RBF collocation does not require a regular arrangement of nodes. Therefore, the proposed numerical method is directly applicable to unstructured datasets.

This study provides an extended approach to the mathematical simulation of thin-film flow on a flat inclined plane relevant to flows subject to high surface shear. Motivated by modelling thin-film structures within an industrial context, wave structures are investigated for flows with moderate inertial effects and small film depth aspect ratio, epsilon. Approximations are made assuming a Reynolds number, Re ~ O(1/epsilon), and depth-averaging used to simplify the governing Navier-Stokes equations. A parallel Stokes flow is expected in the absence of any wave disturbance and a generalisation for the flow is based on a local quadratic profile. This approach provides a more general system which includes inertial effects and is solved numerically. Flow structures are compared with studies for Stokes flow in the limit of negligible inertial effects. Both two-tier and three-tier wave disturbances are used to study film profile evolution. A parametric study is provided for wave disturbances with increasing film Reynolds number. An evaluation of standing wave and transient film profiles is undertaken and identifies new profiles not previously predicted when inertial effects are neglected.

The two-dimensional linear elasticity equations are solved by the global method of approximate particular solution as a new meshless option to the conventional finite element discretization. The displacement components are approximated by a linear combination of the elasticity particular solutions and the stress tensor is obtained by differentiating the displacement expressions in terms of the particular solutions. The multiquadric radial basis function RBF is employed as the non-homogeneous term in the governing equation to compute the particular solutions. The cantilever beam and the infinite plate with a hole problem are solved to verify the implemented meshless method. For each situation, the trend of the root mean square error is assessed in terms of the shape parameter and the number of nodes. Unlike most of the RBF collocation strategies, it is found that numerical results are in good agreement with the analytical solutions for a wide range of shape parameter values.

A number of recent studies have shown that the electric field and current density induced in the human body by movement in and around magnetic resonance imaging installations can exceed regulatory levels. Although it is possible to measure the induced electric fields at the surface of the body, it is usually more convenient to use numerical models to predict likely exposure under well-defined movement conditions. Whilst the accuracy of these models is not in doubt, this paper shows that modelling of particular rotational movements should be treated with care. In particular, we show that v × B rather than −(v · ∇)A should be used as the driving term in potential-based modelling of induced fields. Although for translational motion the two driving terms are equivalent, specific examples of rotational rigid-body motion are given where incorrect results are obtained when −(v · ∇)A is employed. In addition, we show that it is important to take into account the space charge which can be generated by rotations and we also consider particular cases where neglecting the space charge generates erroneous results. Along with analytic calculations based on simple models, boundary-element-based numerical calculations are used to illustrate these findings.