Showing papers in "Ima Journal of Applied Mathematics in 2020"

A second-order asymptotic model for Rayleigh waves on a linearly elastic half plane

Abstract: We derive a second-order correction to an existing leading-order model for surface waves in linear elasticity. The same hyperbolic–elliptic equation form is obtained with a correction term added to the surface boundary condition. The validity of the correction term is shown by re-examining problems which the leading-order model has been applied to previously, namely a harmonic forcing, a moving point load and a periodic array of compressional resonators.

Stable Asymmetric Spike Equilibria for the Gierer-Meinhardt Model with a Precursor Field

Abstract: Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction-kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1-D Gierer-Meinhardt (GM) model we show that a simple precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric, two-spike patterns, corresponding to localized peaks in the activator of different heights. This is a qualitatively new phenomena for the GM model, in that asymmetric spike patterns are all unstable in the absence of the precursor field. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach we analyze the spectrum of the linearization of the GM model around the two-spike steady-state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis a new class of vector-valued nonlocal eigenvalue problem (NLEP) is derived and analyzed.

Kuwabara-Kono numerical dissipation: a new method to simulate granular matter

Abstract: A new method is introduced for the simulation of multiple impacts in granular media using the Kuwabara-Kono (KK) contact model, a nonsmooth (not Lipschitz continuous) extension of Hertz contact that accounts for viscoelastic damping. We use the technique of modified equations to construct time-discretizations of the nondissipative Hertz law matching numerical dissipation with KK dissipation at different consistency orders. This allows us to simulate dissipative impacts with good accuracy without including the nonsmooth KK viscoelastic component in the contact force. This tailored numerical dissipation is developed in a general framework, for Newtonian dynamical systems subject to dissipative forces proportional to the time-derivative of conservative forces. Numerical tests are performed for the simulation of impacts in Newton’s cradle and on alignments of alternating large and small balls. Resulting wave phenomena (oscillator synchronization, propagation of dissipative solitary waves, oscillatory tails) are accurately captured by implicit schemes with tailored numerical dissipation, even for relatively large time steps.

Fractional phase-field crystal modelling: analysis, approximation and pattern formation

Abstract: We consider a fractional phase-field crystal (FPFC) model in which the classical Swift–Hohenberg equation (SHE) is replaced by a fractional order Swift–Hohenberg equation (FSHE) that reduces to the classical case when the fractional order $\beta =1$. It is found that choosing the value of $\beta $ appropriately leads to FSHE giving a markedly superior fit to experimental measurements of the structure factor than obtained using the SHE ($\beta =1$) for a number of crystalline materials. The improved fit to the data provided by the fractional partial differential equation prompts our investigation of a FPFC model based on the fractional free energy functional. It is shown that the FSHE is well-posed and exhibits the same type of pattern formation behaviour as the SHE, which is crucial for the success of the PFC model, independently of the fractional exponent $\beta $. This means that the FPFC model inherits the early successes of the FPC model such as physically realistic predictions of the phase diagram etc. and, therefore, provides a viable alternative to the classical PFC model. While the salient features of PFC and FPFC are identical, we expect more subtle features to differ. The prediction of grain boundary energy arising from a mismatch in orientation across a material interface is another notable success of the PFC model. The grain boundary energy can be evaluated numerically from the PFC model and compared with experimental measurements. The grain boundary energy is a derived quantity and is more sensitive to the nuances of the model. We compare the predictions obtained using the PFC and FPFC models with experimental observations of the grain boundary energy for several materials. It is observed that the FPFC model gives superior agreement with the experimental observation than those obtained using the classical PFC model, especially when the mismatch in orientation becomes larger.

Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions)

Abstract: This paper considers the use of compliant or Robin boundary conditions to provide a homogenised model of a finite array of collinear plates, modelling a perforated screen or grating. This geometry forms a canonical model in scattering theory, with applications from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenised boundary conditions in this case, especially as the homogenised model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial), and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for applying each method, of broader interest to those considering applying the methods. We verify the appropriateness of the constant compliance given in previous theoretical research to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime in which the homogenisation is valid, which we describe. We further investigate tapering the compliance near the extreme endpoints of the screen, and find that tapering with tanh functions reduces the error in the approximation of the far-field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber have significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.

A mathematical multi-organ model for bidirectional epithelial–mesenchymal transitions in the metastatic spread of cancer

Abstract: Funding: Engineering and Physical Sciences Research Council (EPSRC) [to L.C.F.]; EPSRC Grant No. EP/N014642/1 (EPSRC Centre for Multiscale Soft Tissue Mechanics WithApplication to Heart & Cancer) [to M.A.J.C.].

Non-Newtonian channel flow—exact solutions

Abstract: In this short communication, exact solutions are obtained for a range of non-Newtonian flows between stationary parallel plates. The pressure-driven flow of fluids with a variational viscosity that adheres to the Carreau governing relationship are considered. Solutions are obtained for both shear-thinning (viscosity decreasing with increasing shear-rate) and shear-thickening (viscosity increasing with increasing shear-rate) flows. A discussion is presented regarding the requirements for such analytical solutions to exist. The dependence of the flow rate on the channel half width and the governing non-Newtonian parameters is also considered.

Shape optimization of stirring rods for mixing binary fluids

Abstract: Mixing is an omnipresent process in a wide-range of industrial applications, which supports scientific efforts to devise techniques for optimising mixing processes under time and energy constraints. In this endeavor, we present a computational framework based on nonlinear direct-adjoint looping for the enhancement of mixing efficiency in a binary fluid system. The governing equations consist of the non-linear Navier-Stokes equations, complemented by an evolution equation for a passive scalar. Immersed and moving stirrers are treated by a Brinkman-penalisation technique, and the full system of equations is solved using a Fourier-based pseudospectral approach. The adjoint equations provide gradient and sensitivity information which is in turn used to improve an initial mixing strategy, based on shape, rotational and path modifications. We utilise a Fourier-based approach for parameterising and optimising the embedded stirrers and consider a variety of geometries to achieve enhanced mixing efficiency. We consider a restricted optimisation space by limiting the time for mixing and the rotational velocities of all stirrers. In all cases, non-intuitive shapes are found which produce significantly enhanced mixing efficiency.

Steady states of thin film droplets on chemically heterogeneous substrates

Abstract: We study steady-state thin films on a chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the one-dimensional steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady-states on each branch. We show using perturbation expansions, that leading order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. The analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast on linear stability and dynamics. Results are also applied to describe two-dimensional droplets on hydrophilic square patches and striped regions used in microfluidic applications.

Uniqueness in inverse electromagnetic scattering problem with phaseless far-field data at a fixed frequency

Abstract: This paper is concerned with uniqueness in inverse electromagnetic scattering with phaseless far-field pattern at a fixed frequency. In our previous work [{\em SIAM J. Appl. Math.} {\bf 78} (2018), 3024-3039], by adding a known reference ball into the acoustic scattering system, it was proved that the impenetrable obstacle and the index of refraction of an inhomogeneous medium can be uniquely determined by the acoustic phaseless far-field patterns generated by infinitely many sets of superpositions of two plane waves with different directions at a fixed frequency. In this paper, we extend these uniqueness results to the inverse electromagnetic scattering case. The phaseless far-field data are the modulus of the tangential component in the orientations $\mathbf{e}_\phi$ and $\mathbf{e}_\theta$, respectively, of the electric far-field pattern measured on the unit sphere and generated by infinitely many sets of superpositions of two electromagnetic plane waves with different directions and polarizations. Our proof is mainly based on Rellich's lemma and the Stratton--Chu formula for radiating solutions to the Maxwell equations.

Gradient-induced droplet motion over soft solids

Abstract: Fluid droplets can be induced to move over rigid or flexible surfaces under external or body forces. We describe the effect of variations in material properties of a flexible substrate as a mechanism for motion. In this paper, we consider a droplet placed on a substrate with either a stiffness or surface energy gradient, and consider its potential for motion via coupling to elastic deformations of the substrate. In order to clarify the role of contact angles and to obtain a tractable model, we consider a two-dimensional droplet. The gradients in substrate material properties give rise to asymmetric solid deformation and to unequal contact angles, thereby producing a force on the droplet. We then use a dynamic viscoelastic model to predict the resulting dynamics of droplets. Numerical results quantifying the effect of the gradients establish that it is more feasible to induce droplet motion with a gradient in surface energy. The results show that the magnitude of elastic modulus gradient needed to induce droplet motion exceeds experimentally feasible limits in the production of soft solids and is therefore unlikely as a passive mechanism for cell motion. In both cases, of surface energy or elastic modulus, the threshold to initiate motion is achieved at lower mean values of the material properties.

Homogenization and hypocoercivity for Fokker–Planck equations driven by weakly compressible shear flows

Abstract: We study the long-time dynamics of two-dimensional linear Fokker-Planck equations driven by a drift that can be decomposed in the sum of a large shear component and the gradient of a regular potential depending on one spatial variable. The problem can be interpreted as that of a passive scalar advected by a slightly compressible shear flow, and undergoing small diffusion. For the corresponding stochastic differential equation, we give explicit homogenization rates in terms of a family of time-scales depending on the parameter measuring the strength of the incompressible perturbation. This is achieved by exploiting an auxiliary Poisson problem, and by computing the related effective diffusion coefficients. Regarding the long-time behaviour of the solution of the Fokker-Planck equation, we provide explicit decay rates to the unique invariant measure by employing a quantitative version of the classical hypocoercivity scheme. From a fluid mechanics perspective, this turns out to be equivalent to quantifying the phenomenon of enhanced diffusion for slightly compressible shear flows.

Multiple solutions and their asymptotics for laminar flows through a porous channel with different permeabilities

Abstract: The existence and multiplicity of similarity solutions for the steady, incompressible and fully developed laminar flows in a uniformly porous channel with two permeable walls are investigated. We shall focus on the so-called asymmetric case where the upper wall is with an amount of flow injection and the lower wall with a different amount of suction. We show that there exist three solutions designated as type $I$, type $II$ and type $III$ for the asymmetric case. The numerical results suggest that a unique solution exists for the Reynolds number $0\leq R 14.10$. The corresponding asymptotic solution for each of the multiple solutions is constructed by the method of boundary layer correction or matched asymptotic expansion for the most difficult high Reynolds number case. Asymptotic solutions are all verified by their corresponding numerical solutions.

High-contrast approximation for penetrable wedge diffraction

Abstract: The important open canonical problem of wave diffraction by a penetrable wedge is considered in the high-contrast limit. Mathematically, this means that the contrast parameter, the ratio of a specific material property of the host and the wedge scatterer, is assumed small. The relevant material property depends on the physical context and is different for acoustic and electromagnetic waves for example. Based on this assumption, a new asymptotic iterative scheme is constructed. The solution to the penetrable wedge is written in terms of infinitely many solutions to (possibly inhomogeneous) impenetrable wedge problems. Each impenetrable problem is solved using a combination of the Sommerfeld–Malyuzhinets and Wiener–Hopf techniques. The resulting approximated solution to the penetrable wedge involves a large number of nested complex integrals and is hence difficult to evaluate numerically. In order to address this issue, a subtle method (combining asymptotics, interpolation and complex analysis) is developed and implemented, leading to a fast and efficient numerical evaluation. This asymptotic scheme is shown to have excellent convergent properties and leads to a clear improvement on extant approaches.

The effect of compressibility on the behaviour of filter media

Abstract: A filter comprises porous material that traps contaminants when fluid passes through under an applied pressure difference. One side effect of this applied pressure, however, is that it compresses the filter. This changes the permeability, which may affect its performance. As the applied pressure increases, the flux of fluid processed by the filter will also increase but the permeability will decrease. Eventually, the permeability reaches zero at a point in the filter and the fluid flux falls to zero. In this paper, we derive a model for the fluid transport through a filter due to an applied pressure difference and the resulting compression. We use this to determine the maximum operating flux that can be achieved without the permeability reaching zero and the filter shutting down. We determine the material properties that balance the desire to maximize flux while minimizing power use. We also show how choosing an initial spatially dependent permeability can lead to a uniformly permeable filter under operation and we find the permeability distribution that maximizes the flux for a given applied pressure, both of which have desirable industrial implications. The ideas laid out in this paper set a framework for modelling more complex scenarios such as filter blocking.

Free-stream coherent structures in the unsteady Rayleigh boundary layer

Abstract: Results are presented for nonlinear equilibrium solutions of the Navier–Stokes equations in the boundary layer set up by a flat plate started impulsively from rest. The solutions take the form of a wave–roll–streak interaction, which takes place in a layer located at the edge of the boundary layer. This extends previous results for similar nonlinear equilibrium solutions in steady 2D boundary layers. The results are derived asymptotically and then compared to numerical results obtained by marching the reduced boundary-region disturbance equations forward in time. It is concluded that the previously found canonical free-stream coherent structures in steady boundary layers can be embedded in unbounded, unsteady shear flows.

Degenerate elliptic equations for resonant wave problems

Abstract: The modeling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coeffcient alpha and compact terms. The coeffcient alpha changes sign. The region where alpha is positive is propagative, and the region where alpha is negative is non propagative and elliptic. The two models are coupled through the line Sigma, corresponding to alpha equal to zero. Generically, it is an ill-posed problem, and additional information must be introduced to get a satisfactory treatment at Sigma. In this work we define the solution by relying on the limit absorption principle (alpha is replaced by alpha + i0^+) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular part and a singular part, which originates at Sigma, and on quasi-solutions. It leads to a new well-posed mixed variational formulation with coupling. As we design explicit quasi-solutions, numerical experiments can be carried out, which illustrate the good properties of this new tool for numerical computation.

A lumped-parameter model for kidney pressure during stone removal

Abstract: In this paper, we consider a lumped-parameter model to predict renal pressures and flow rate during a minimally invasive surgery for kidney stone removal, ureterorenoscopy. A ureteroscope is an endoscope designed to work within the ureter and the kidney and consists of a long shaft containing a narrow, cylindrical pipe, called the working channel. Fluid flows through the working channel into the kidney. A second pipe, the ‘access sheath’, surrounds the shaft of the scope, allowing fluid to flow back out of the urinary system. We modify and extend a previously developed model ( Oratis et al., 2018) through the use of an exponential, instead of linear, constitutive law for kidney compliance and by exploring the effects of variable flow resistance, dependent on the presence of auxiliary ‘working tools’ in the working channel and the cross-sectional shapes of the tools, working channel, scope shaft and access sheath. We motivate the chosen function for kidney compliance and validate the model predictions, with ex vivo experimental data. Although the predicted and measured flow rates agree, we find some disagreement between theory and experiment for kidney pressure. We hypothesize that this is caused by spatial pressure variation in the renal pelvis, i.e. unaccounted for in the lumped-parameter model. We support this hypothesis through numerical simulations of the steady Navier–Stokes equations in a simplified geometry. We also determine the optimal cross-sectional shapes for the scope and access sheath (for fixed areas) to minimize kidney pressure and maximize flow rate.

Analysis of a model for the formation of fold-type oscillation marks in the continuous casting of steel

Abstract: This paper investigates the different possible behaviours of a recent asymptotic model for oscillation-mark formation in the continuous casting of steel, with particular focus on how the results obtained vary when the heat transfer coefficient ( $m$ ), the thermal resistance ( $R_{mf}$ ) and the dependence of the viscosity of the flux powder as a function of temperature, $\mu _{f}\left ( T\right ),$ are changed. It turns out that three different outcomes are possible: (I) the flux remains in molten state and no solid flux ever forms; (II) both molten and solid flux are present, and the profile of the oscillation mark is continuous with respect to the space variable in the casting direction; (III) both molten and solid flux are present, and the profile of the oscillation mark is discontinuous with respect to the space variable in the casting direction. Although (I) gave good agreement with experimental data, it suffered the drawback that solid flux is typically observed during actual continuous casting; this has been rectified in this work via alternative (II). On the other hand, alternative (III) can occur as a result of hysteresis-type phenomenon that is encountered in other flows that involve temperature-dependent viscosity; in the present case, this manifests itself via the possibility of multiple states for the oscillation-mark profile at the instants in time when solid flux begins to form and when it ceases to form.

Rankine-type cylinders having zero wave resistance in infinitely deep flows

Abstract: We prove the existence of a family of immersed obstacles that have zero wave resistance in the context of the 2D Neumann–Kelvin problem. We first build a waveless potential by superposing a source and a sink in a uniform flow for an appropriate choice of parameters. The obstacle is obtained by a combination of streamlines of the waveless potential. Numerical simulations show that the construction is valid for a large set of parameters.

Homogenization approximations for unidirectional transport past randomly distributed sinks

Abstract: Transport in biological systems often occurs in complex spatial environments involving random structures. Motivated by such applications, we investigate an idealised model for solute transport past an array of point sinks, randomly distributed along a line, which remove solute via first-order kinetics. Random sink locations give rise to long-range spatial correlations in the solute field and influence the mean concentration. We present a non-standard approach to evaluating these features based on rationally approximating integrals of a suitable Green's function, which accommodates contributions varying on short and long lengthscales and has deterministic and stochastic components. We refine the results of classical two-scale methods for a periodic sink array (giving more accurate higher-order corrections with non-local contributions) and find explicit predictions for the fluctuations in concentration and disorder-induced corrections to the mean for both weakly and strongly disordered sink locations. Our predictions are validated across a large region of parameter space.

Markov chain models of refugee migration data

Abstract: The application of Markov chains to modelling refugee crises is explored, focusing on local migration of individuals at the level of cities and days. As an explicit example we apply the Markov chains migration model developed here to UNHCR data on the Burundi refugee crisis. We compare our method to a state-of-the-art `agent-based' model of Burundi refugee movements, and highlight that Markov chain approaches presented here can improve the match to data while simultaneously being more algorithmically efficient.

Analysis of the susceptible-infected-susceptible epidemic dynamics in networks via the non-backtracking matrix

Abstract: We study the stochastic susceptible-infected-susceptible model of epidemic processes on finite directed and weighted networks with arbitrary structure. We present a new lower bound on the exponential rate at which the probabilities of nodes being infected decay over time. This bound is directly related to the leading eigenvalue of a matrix that depends on the non-backtracking and incidence matrices of the network. The dimension of this matrix is $N+M$ , where $N$ and $M$ are the number of nodes and edges, respectively. We show that this new lower bound improves on an existing bound corresponding to the so-called quenched mean-field theory. Although the bound obtained from a recently developed second-order moment-closure technique requires the computation of the leading eigenvalue of an $N^2\times N^2$ matrix, we illustrate in our numerical simulations that the new bound is tighter, while being computationally less expensive for sparse networks. We also present the expression for the corresponding epidemic threshold in terms of the adjacency matrix of the line graph and the non-backtracking matrix of the given network.

Asymptotic analysis of internal relaxation-oscillations in a conceptual climate model

Abstract: We construct a dynamical system based on the KCG (Kallen, Crafoord, Ghil) conceptual climate model which includes the ice-albedo and precipitation-temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover, it follows from the general theory that the periodic orbit exhibits relaxation-oscillations which are a characteristic feature of the Pleistocene ice-ages. We provide an asymptotic analysis of their behaviour and derive a formula for the period along with several estimates. They, in turn, are in a decent agreement with paleoclimatic data and are independent of any parametrization used. Whence, our simple but robust model shows that a climate may exhibit internal relaxation-oscillations without any external forcing and for a wide range of parameters.

A theory of porous media and harmonic wave propagation in poroelastic body

Abstract: The present work is based on a mixture theory of poroelastic media which is consistent with the classical Darcy’s law and uplift force in soil mechanics. In addition, it also results in having an inertial effect on the motion of solid constituent as commonly expected, in contrast to Biot’s theory, where relative acceleration is introduced as an interactive force between solid and fluid constituents to account for the apparent inertial effect. The propagation of plane harmonic waves in homogeneously deformed region is considered. For different poroelastic models with either incompressible solid or incompressible fluid constituent, phase speeds and attenuation coefficients are analysed and numerically determined with convenient data from a nonlinear material model for comparison with some available results in the literature.

Time-harmonic elastic singularities and oscillating indentation of layered solids

Abstract: A new approach is proposed for obtaining the dynamic elastic response of a multilayered elastic solid caused by axisymmetric, time-harmonic elastic singularities. The method for obtaining the elastodynamic Green’s functions of the point force, double forces and center of dilatation is presented. For this purpose, the boundary conditions in an infinite solid at the plane passing through the singularity are derived first by using Helmholtz potentials. Then the Green’s function solution for layered solids is obtained by solving a set of simultaneous linear algebraic equations using the boundary conditions for both the singularities and for the layer interfaces. The application of the point force solution for the oscillating normal indentation problem is also given. The solution of the forced normal oscillation is formulated by integrating the point force Green’s function over the contact area with unknown surface traction. The dual integral equations of the unknown surface traction are established by considering the boundary conditions on the contact surface of the multilayered solid, which can be converted into a Fredholm integral equation of the second kind and solved numerically.

Reconstruction of inclusions in electrical conductors

Abstract: We investigate the reciprocity gap functional method, which has been developed in the inverse scattering theory, in the context of electrical impedance tomography. In particular, we aim to reconstruct an inclusion contained in a body, whose conductivity is different from the conductivity of the surrounding material. Numerical examples are given, showing the performance of our algorithm.

A mathematical model of asynchronous data flow in parallel computers

Abstract: We present a simplified model of data flow on processors in a high performance computing framework involving computations necessitating inter-processor communications. From this ordinary differential model, we take its asymptotic limit, resulting in a model which treats the computer as a continuum of processors and data flow as an Eulerian fluid governed by a conservation law. We derive a Hamilton-Jacobi equation associated with this conservation law for which the existence and uniqueness of solutions can be proven. We then present the results of numerical experiments for both discrete and continuum models; these show a qualitative agreement between the two and the effect of variations in the computing environment's processing capabilities on the progress of the modeled computation.

Blake, bubbles and boundary element methods

Abstract: Professor John Blake spent a considerable part of his scientific career on studying bubble dynamics and acoustic cavitation. As Blake was a mathematician, we will be focusing on the theoretical and numerical studies (and much less on experimental results). Rather than repeating what is essentially already known, we will try to present the results from a different perspective as much as possible. This review will also be of interest for readers who wish to know more about the boundary element method in general, which is a method often used by Blake and his colleagues to simulate bubbles. We will, however, not limit the discussion to bubble dynamics but try to give a broad discussion on recent advances and improvements to this method, especially for potential problems (Laplace) and wave equations (Helmholtz). Based on examples from Blake’s work, we will guide the reader and show some of the mysteries of bubble dynamics, such as why jets form in collapsing bubbles near rigid surfaces. Where appropriate, we will illustrate the concepts with examples drawn from numerical simulations and experiments.