# Abstracts - Contributed Talks

Abstracts arranged alphabetically by speaker surname.

Details of the contributed talk session, venue and timing are provided after each abstract.

At the end of each abstract, Session, Room, Day information is provided. Use this to plan (and navigate) your week.

Mohammed Afsar (Strathclyde)

The role of the spanwise correlation length in low-frequency sound radiation by turbulence/surface interaction

The interaction between non-homogeneous turbulence carried by a non-planar mean field and the discontinuity created by an impermeable solid surface embedded within the flow causes an $O(1)$ increase in the low frequency sound above the background jet noise radiated by the turbulence itself.

When the solid surfaces $S(\boldsymbol{y})$ are placed parallel to the level curves of the streamwise mean flow, $U(y_2,y_3)=const.$, generalized Rapid distortion theory (Ref) shows that the far-field pressure fluctuation, $p^\prime(\boldsymbol{x},t)$, at the spacetime field point $(\boldsymbol{x},t) = ({x}_1,{x}_2,{x}_3,t)$ in a three-dimensional Cartesian co-ordinate system with origin at the trailing edge, can be expressed in terms of the Green's function of the adjoint Rayleigh equation and a convected scalar field, $\tilde{\omega}_c$, defined by $D\tilde{\omega}_c /Dt = 0$ where$D /Dt \equiv \partial/\partial t + U(x_2, x_3) \partial/\partial x_1$ is the material derivative.

Since this latter quantity is an arbitrary function of its arguments, its space-time spectrum ($\bar{S}$ below) can be related to the appropriate measured turbulence correlation function in the upstream undisturbed flow and used as the boundary condition to determine the downstream acoustic field.

In reference 1\footnote{M. E. Goldstein, S. J. Leib \& M. Z. Afsar, {Rapid distortion theory on transversely sheared mean flows of arbitrary cross-section}, \textit{J. Fluid Mech.} (2019), vol. 881, pp. 551–584.}; it is shown that the acoustic spectrum ($I_\omega$) takes the form, $I_\omega \sim \int_u \int_{\tilde {u}} D(u,\tilde{u}) \bar{S}(u, \tilde{u};\omega) du d\tilde{u}$, where $D(u,\tilde{u})$ is a directivity factor and $u$ is the real part of a conformal mapping that transforms the nonrectangular cross sectional shape of an axisymmetric jet close to an external surface into a rectangular one for the subsequent Wiener-Hopf technique calculation.

Our main contribution in this paper is to show that the low frequency decay of $I_\omega (\boldsymbol{x})$ is approximately the same as that obtained by taking $l_3 \rightarrow \infty$.

The latter limit allows $\bar{S}$ to take a reduced analytical form. The large-$l_3$ limit corresponds to the allowing the spanwise correlation length to be infinite and therefore that the spectral function $\bar{S}$ is independent of spatial separation in the spanwise (or cross-stream) $3$-direction.

We assess the importance of the spanwise correlation length in controlling the low frequency roll-off of $I_\omega (\boldsymbol{x})$ (i.e. its asymptotic value when the angular frequency, $\omega$, goes to zero starting from the peak sound. Our calculations below show that the maximum value and spatial structure of the integrand, $D(u,\tilde{u}) \bar{S}(u, \tilde{u};\omega)$ when $l_3 \rightarrow\infty$ (left figure) is almost identical to the $l_3 = O(1)$ case (right) at very low frequencies.

The accompanying talk will discuss whether the structure of $\bar{S}$ at $l_3\rightarrow \infty$ can be used to approximate $I_\omega$ at more $O(1)$ values of spanwise turbulence length scale, $l_3$.

CT07 High Reynolds Number, Bute 4, Wednesday

Andrew John Archer (Loughborough)

Coupled dynamics of premelting films, water droplets, and ice crystal growth

In exciting recent studies of the surface of ice at supersaturation in the vicinity of the triple point, subnanometer height crystal terraces spreading across the surface were observed simultaneously to the formation of micron size droplets, and the subsequent emergence of nanometer thick films below the drops [Murata et al. PNAS 113, E6741 (2016)]. We develop a mesoscopic crystal growth model that takes as input the equilibrium premelting film thickness and interfacial free energies obtained in computer simulations and bridges all relevant scales from angstroms to tens of nanometers. We find that the average dynamics of this complex out of equilibrium system can be described by an effective free energy leading to a wetting phenomenology on the growing ice substrate in complete analogy with equilibrium wetting properties on an inert substrate. Our results explain the significance of the kinetic transition lines observed in the experiments, showing that the motion of the underlying solid surface can be conveyed through the premelted liquid-like layer to the outer surface. Moreover, spontaneous fluctuations in the solid surface can nucleate subsequent deterministic growth of liquid droplets.

CT14 Droplets, Clyde 5, Thursday

Andrew Baggaley (Newcastle)

Classical and quantum vortex leapfrogging in two-dimensional channels

The leapfrogging of coaxial vortex rings is a famous effect which has been noticed since the times of Helmholtz. Recent advances in ultra-cold atomic gases show that the effect can now be studied in quantum fluids. The strong confinement which characterises these systems motivates the study of leapfrogging of vortices within narrow channels. Using the two-dimensional point vortex model, we show that in the constrained geometry of a two-dimensional channel the dynamics are richer than in an unbounded domain: alongside the known regimes of standard leapfrogging and the absence of it, we identify new regimes of image-driven leapfrogging and periodic orbits. Moreover, by solving the Gross-Pitaevskii equation for a Bose-Einstein condensate, we show that all four regimes exist for quantum vortices too. Finally, we discuss the differences between classical and quantum vortex leapfrogging which appear when the quantum healing length becomes significant compared to the vortex separation or the channel size, and when, due to high velocity, compressibility effects in the condensate becomes significant.

CT07 High Reynolds Number, Bute 4, Wednesday

Callum Birkett (Dundee)

Magnetic Helicity and the Calabi Invariant

Authors: C.Birkett, G.Hornig

Magnetic helicity is an important tool in the study of both astrophysical and laboratory plasmas. The topological features of the magnetic field confined to the plasma provide estimates on the energy contained within. We introduce the Calabi invariant, an integral quantity closely associated with helicity (Calabi 1970, Gambaudo et al. 2000) and show that this leads to interesting new ways to interpret the helicity and to calculate it in magnetically open domains. The Calabi invariant allows us to assess the topology of a plasma contained in a domain without having to construct the entire field - we only need the associated field line mapping. An application to braiding within a coronal loop is presented.

References

[1] Eugenio Calabi. On the group of automorphisms of a symplectic manifold, from: “problems in analysis (lectures at the sympos. in honor of salomonbochner, princeton univ., princeton, nj, 1969)”, 1970.

[2] Jean-Marc Gambaudo and Maxime Lagrange. Topological lower bounds on the distance between area preserving diffeomorphisms .Boletim da SociedadeBrasileira de Matematica, 31(1):9–27, 2000.

Patrick Bourg (Southampton)

Rotating perfect fluid stars in 2+1 dimensions.

Classical Einstein gravity in 2+1 dimensions may at first appear to be trivial since the Weyl tensor is identically zero. However, this apparent simplicity hides, upon closer examination, interesting features. We will review perhaps one of the simplest axistationary matter solutions: rotating perfect fluid stars. These solutions can be written down explicitly in 2+1 dimensions and we will highlight the similarities and differences to their 3+1 counterparts. If time permits, we will also show how these solutions are relevant to the critical collapse of a perfect fluid.

CT 09 Mathematical physics, Clyde 4, Wednesday

Rodolfo Brandao (Imperial College)

Spontaneous dynamics of Leidenfrost drops

Recent experiments have revealed that Leidenfrost drops (levitated by their vapour above a hot surface) undergo “symmetry breaking”, leading to spontaneous rolling motion in the absence of external gradients/asymmetries (A. Bouillant et al., Nature Physics, 14 1188, 2018). Motivated by these observations, we theoretically investigate the dynamics of Leidenfrost drops on the basis of a simplified two-dimensional model, focusing on near-circular drops small relative to the capillary length. The model couples the equations of motion of the drop, which flows as a rigid wheel, and instantaneous thin-film equations governing the vapour flow, the profile of the deformable vapour-liquid interface, and thus the hydrodynamic forces and torques on the drop. The model predicts an instability of the symmetric Leidenfrost state, manifested by a supercritical pitchfork bifurcation, which leads to spontaneous motion. Our model illuminates several aspects of the experiments, including the origins of the experimentally measured propulsion force and its dependence on the physical parameters of the system, in compelling qualitative agreement with the experiments.

CT14 Droplets, Clyde 5, Thursday

Chris Budd (Bath)

Mathematical models for the ice ages

The ice ages are significant changes to the Earth's climate. For the last half a million years they have demonstrated a strong regularity, with large periodic changes in temperature with a 100 kilo year cycle. Before then the climate showed smaller smaller periodic changes, with a 40 kilo year cycle. Although there are many theories for this behaviour, there is as yet no fully convincing explanation for them. In this talk I will apply recent methods from the theory of non-smooth dynamical systems to gain some insight into this complex phenomenon. In particular I will ask the question of whether the 'min-Pleistocene transition (MPT)' half a million years ago was an example of a grazing bifurcation.

Joint work with Kgomotso Susan Morupisi

CT13 Dynamical Systems, Clyde 4, Thursday

Jeremy Michael Budd (TU Delft)

A semi-discrete scheme for graph Allen--Cahn as a classification/segmentation algorithm

An emerging technique in clustering, segmentation and classification problems is to consider the dynamics of flows defined on finite graphs. In particular Bertozzi and co-authors considered dynamics related to Allen—Cahn flow (Bertozzi, Flenner, 2012) and the MBO algorithm (Merkurjev, Kostic, Bertozzi, 2013) for this purpose. Previous work by the authors (Budd, Van Gennip, in review, arxiv preprint 1907.10774) devised a "semi-discrete" scheme for Allen--Cahn, of which MBO is a special case.

This talk will extend this earlier theory to the case of Allen--Cahn/MBO with fidelity forcing. We will then explore the prospects of this semi-discrete scheme (with fidelity) as an alternative to MBO for segmentation and classification applications.

CT 15 Statistical and Numerical Methods, Hillhead 3, Thursday

Andrew Burbanks (Portsmouth)

Computer-assisted proof of the existence of renormalisation fixed points

We prove the existence of a fixed point to the renormalisation operator for period doubling in maps of even degree at the critical point. Building on previous work, our proof uses rigorous computer-assisted means to bound operations in a space of analytic functions and hence to show that a quasi-Newton operator for the fixed-point problem is a contraction map on a suitable ball.

We bound the spectrum of the derivative of the renormalisation operator at the fixed point, establishing the hyperbolic structure, in which the presence of a single essential expanding eigenvalue explains the universal asymptotically self-similar bifurcation structure observed in the iterations of families of maps with the relevant degree at the critical point.

By recasting the eigenproblem for the Frechet derivative in nonlinear form, we use the contraction mapping principle to gain rigorous bounds on eigenfunctions and their corresponding eigenvalues. In particular, we gain tight bounds on the eigenfunction corresponding to the essential expanding eigenvalue delta.

By employing a recursive scheme based on the fixed-point equation, we exhibit the structure of the domain of analyticity of the renormalisation fixed point.

Our computations use multi-precision arithmetic with rigorous directed rounding modes (conforming to relevant standards IEEE754-2008 and ISO/IEC/IEEE60559:2011) to bound tightly the coefficients of the relevant power series and their high-order terms, and the corresponding universal constants.

CT13 Dynamical Systems, Clyde 4, Thursday

Diogo Caetano (Warwick)

Well-posedness for the Cahn-Hilliard equation on an evolving surface

The classical Cahn-Hilliard equation is a fourth order, semilinear parabolic PDE which was first proposed in 1958 to describe phase separation in binary alloys, and it has since been applied to problems in other areas such as image processing, tumor growth models, among others. In this talk, we describe a functional framework suitable to the formulation of the constant mobility Cahn-Hilliard equation on an evolving surface and establish well-posedness for general regular potentials, the thermodynamically relevant logarithmic potential and a double obstacle potential. It turns out that, for the singular potentials, conditions on the initial data and the evolution of the surfaces are necessary for global-in-time existence of solutions, which arise from the fact that the integral of solutions are preserved over time. Time permitting, related models, examples and open questions will be discussed.

CT17 Solid Mechanics, Hillhead 5, Thursday

Marianna Cerasuolo (Portsmouth)

On the role of diffusion on drug interaction in the treatment of prostate cancer.

A hybrid system of ODEs and PDEs has been implemented to assess the role of cells and chemicals diffusion on the dynamics of prostate cancer in a multistage murine model TRAMP (transgenic adenocarcinoma of the mouse prostate) under different therapeutic strategies. The model describes the interdependence of cancer cells on tumour microenvironment as well as the onset of resistance following treatment with a second generation drug (androgen receptor antagonist) called enzalutamide.

The proposed mathematical model, whose development strongly relied on experimental data and their statistical analysis, represents a theoretical framework to bridge the in vitro and in vivo experiments used to assess the effect of single- or combined-drug therapies on TRAMP mice and TRAMP-derived cells. The model revealed that combination therapies can delay the onset of resistance to enzalutamide, and in the suitable scenario with alternating drug therapies, can eliminate the disease. The model also showed that some of the drug combinations can cause the formation of smaller-size tumour clusters, which could give rise to metastasis.

CT10 Mathematical Biology-2, Clyde 5, Wednesday

Avni Chotai (Imperial College)

The effect of wall compliance upon the stability of annular Poiseuille-Couette flow

Driven by a constant axial pressure gradient, the viscous flow through the concentric annular region between a stationary outer cylinder and a sliding inner cylinder is known as annular Poiseuille-Couette flow (APCF). This flow is of particular relevance in medicine; thread-injection is a minimally invasive technique for the transportation of medical implants into the body. In view of this application, the inner cylinder is modelled as compliant in our study. We consider steady, incompressible APCF between two infinitely long, concentric cylinders. The linear stability of this flow to infinitesimal, axisymmetric disturbances is studied asymptotically at large Reynolds numbers and computationally at finite Reynolds numbers when the inner cylinder possesses a degree of flexibility. Typical no-slip conditions apply on the wall of the outer rigid cylinder, and the inner cylinder is described by a spring-backed plate model.

At finite Reynolds number, our problem forms a generalised eigenvalue problem that can be solved numerically via a Chebyshev collocation method to obtain the wavenumbers of the neutral modes. In addition to showing classical viscous modes, the results show that the compliant nature of the inner cylinder results in the existence of unstable ‘elastic’ modes that are not present in the rigid counterpart of the problem.

Both the viscous and elastic modes can be described using asymptotic analysis, with the viscous modes assuming a multizone structure typical of shear flow instability. A comparison of the asymptotic solutions with the finite-Reynolds number solutions shows increasing agreement at larger Reynolds numbers.

Authors - Avni Chotai, Andrew Walton

CT07 High Reynolds Number, Bute 4, Wednesday

Bobby Clement (Nottingham)

Two fluids, one magnet. An Unstable Tale.

The phenomenon of interfacial instability has been of growing interest since the initial work presented by Lord Rayleigh in his 1882 paper [1] concerning a dense incompressible fluid being supported by a less dense incompressible fluid. The importance of such instabilities are present when understanding multi-scale flows within many areas of research, from the small scale modelling of Inertial Confinement Fusion (ICF) [2] to the very large scale modelling of bubbles rising in galaxy cluster cooling flows [3, 4]. An example of such an instability is when milk is poured into black tea. We can see an interfacial boundary between the hot water and milk. The boundary is a good example of a density-stratified medium exhibiting a growing instability. Here we follow earlier work [5] by experimentally and theoretically investigating a circular interface between two concentric fluid layers of differing density in a circular domain. The objective is to investigate how instabilities on the interface grow when they are driven by a centrifugal, rather than gravitational, force. Hence, the whole system is rotated about its axis of symmetry. The geometry of the flow domain impose significant technical difficulties in the initial experimental setup of the fluid layers. The fluids are separated uniformly using a strong, externally applied, magnetic field. We will look at the experimental technique used to complete the experiments and then look at some of the data acquired using verified image processing techniques. This will then be compared with a high viscosity Stokes flow approximation using Darcy’s law for a depth averaged velocity in a circular Hele-Shaw cell. Our aim is to provide a suitable first order theoretical prediction of how the interface behaves in an axisymmetric domain.

CT03 Viscous Fluid Dyamis 2, Clyde 4, Wednesday

Levitation of a cylinder by a thin viscous fluid

We demonstrate that it is possible to levitate a circular cylinder placed horizontally on a vertical belt covered in a thin layer of oil by moving the belt upwards at a specific speed. The cylinder rotates and is balanced at a fixed location on the belt. Levitation occurs solely through viscous lubrication effects. We present the results of an experimental, asymptotic, and numerical study of this fluid-structure interaction. In particular, we show that understanding the asymptotic structure is, somewhat surprisingly, integral to understanding levitation.

CT08 Industrial fluids, Bute 5, Wednesday

Hannah-May D'Ambrosio (Strathclyde)

Deposition from an evaporating sessile droplet

Hannah-May D’Ambrosio, Stephen K. Wilson, Brian R. Duffy and Alexander W. Wray

The evaporation of sessile droplets occurs in a wide variety of physical contexts, with numerous applications in nature, industry and biology. The key interest in most scientific and industrial processes is the deposit that is left behind after evaporation. For applications such as inkjet printing, the ability to control the distribution of solute during the drying process is extremely important with a particular desire for uniform deposits. Over the past 20 years there has been an explosion of research into the deposition of an evaporating droplet, with particular interest in the well-known coffee-ring effect which results in the solute within the droplet forming a ring deposit at the contact line. In this talk I will investigate the effect of spatial variation of the evaporative flux on the final deposition pattern of an evaporating droplet. I propose a one parameter family of quasi-static evaporative fluxes, which include as special cases diffusion-limited evaporation and uniform evaporation, as well as fluxes with maxima at the centre of the droplet. I solve analytically the general problem for the flow velocity, the concentration of solute inside the droplet and the evolution of the contact line deposit ring. I will show that the deposit is directly linked to the evaporative flux profile with the possibility of either coffee-ring deposits, paraboloidal deposits or mountain/bullseye deposits at the centre of the droplet. I will also examine several interesting cases in detail.

CT14 Droplets, Clyde 5, Thursday

Jonathan Dawes (Bath)

Are the Sustainable Development Goals self-consistent and mutually achievable?

The UN’s Sustainable Development Goals (SDGs), launched in 2015, present a global 'to do list' of Challenges. Everyone agrees that there are linkages between them. No two reports agree on what they are. This talk discusses how one might analyse those interlinkages mathematically, and what conclusions we might be able to draw.

Reference:

J.H.P. Dawes, Are the Sustainable Development Goals self-consistent and mutually achievable? Sustainable Development. 2019;1–17. https://doi.org/10.1002/sd.1975

CT 15 Statistical and Numerical Methods, Hillhead 3, Thursday

Sean Edwards (Manchester)

Localised thermally forced streak solutions for a Falkner-Skan boundary layer

The downstream streak response over a semi-infinte flat plate to upstream localised (weak) thermal forcing is presented. We work in the high Reynolds number Boussinesq regime, which allows for efficient streamwise parabolic marching within the localised boundary region. Steady thermally-forced 3D streaks are embedded in an otherwise 2D Falkner-Skan boundary layer, and we consider the effect of a favourable pressure gradient. It is shown that for sufficiently small upstream temperature forcing, the downstream response is characterised by a bi-global eigenvalue problem. The growth/decay of the streak is shown to be dependent on the Prandtl number and the pressure gradient, and a neutral curve is presented. Finally, the stability of the streak to linear time-harmonic variations in upstream forcing is considered.

CT07 High Reynolds Number, Bute 4, Wednesday

Laila Elatrash (Salford)

Translational small-amplitude vibration of a sphere moving in a uniform flow field at low Reynolds number.

Consider translational small-amplitude vibration of a sphere moving in a uniform flow field at low Reynolds number. This problem is decomposed into a steady solution and a translational vibration solution. The steady solution is given by the near-field Stoke flow past a sphere with a matching far-field Oseen flow. The translational vibration solution is a near-field Stokes flow given by Pozrikidis in terms of an unsteady stokeslet and quadrupole. In this talk, we give the matching far-field Oseen flow by an unsteady oseenlet and quadrupole. The solution is visualized by using matlab, and shown to agree with existing solutions when the frequency of vibration tends to zero (steady solution), and when the far-field length tends to zero (Pozrikidis near-field solution).

CT03 Viscous Fluid Dyamis 2, Clyde 4, Wednesday

Sarah Ferguson Briggs (Imperial College)

Linear Stability of a Column of Ferrofluid Centred Along a Rigid Wire

The linear stability of a Newtonian ferrofluid centred on a rigid wire, is investigated in a fully three-dimensional setting. An electric current runs through the wire, generating an azimuthal magnetic field. For both a highly viscous and inviscid system, we consider an inner column of ferrofluid surrounded by another ferrofluid with different magnetic susceptibilities, producing a magnetic stress at the interface of the fluids. When the inner fluid has a larger magnetic susceptibility, the system is linearly unstable to axisymmetric perturbations only, and is stabilised by a sufficiently large magnetic field. When the outer fluid has a larger magnetic susceptibility, we find the system is unstable to both axisymmetric and non-axisymmetric modes. Moreover, in the viscous regime, we find the non-axisymmetric mode to be the most unstable mode. Then, considering a ferrofluid, whose magnetic susceptibility varies radially, we produce stability conditions for axisymmetric and non-axisymmetric modes in both the viscous and inviscid regimes.

Dmitri Finkelshtein (Swansea)

Structural properties of the mean-field expansion

The classical mean-field scheme is widely used both in statistical physics and in the study of stochastic dynamics of complex systems (individual-based models of population ecology, epidemiology, social sciences etc.). It states that when an appropriately chosen small parameter (e.g. the inverse to the number of interacting elements or a space scale parameter) tends to 0, the second and higher-order spatial correlations factorise within the dynamics, provided that the factorisation took place initially (a.k.a. the propagation of chaos property). Therefore, the spatial correlations, being expanded in the power series w.r.t. the small parameter, have an explicit leading term of the expansion, that is the product of solutions to a certain (nonlinear) kinetic equation. In the talk, I will present a new approach which describes all terms of the expansion in the small parameter through solutions to recurrent systems of linear evolution equations. The approach can be applied to a rather general class of dynamics.

CT 09 Mathematical physics, Clyde 4, Wednesday

Tobias Grafke (Warwick M)

Metastability in Fluid Turbulence

Fluid in its turbulent state is one of the most complex interacting systems in physics, with an incredibly large number of strongly and nonlinearly interacting degrees of freedom. This becomes different when scale separation introduces coherent, long-lived structures into the fluid flow, which induces order and stability. I will present two very different scenarios of that type: (1) atmospheric jets on gas giants sustained by the turbulent background fluctuations, and (2) turbulent puffs at the onset of turbulence in pipe flows. In both cases, the coherent structure might exist in a multitude of metastable configurations, and turbulent fluctuations drives transitions between them. I will analyze these metastable states, their basins of attraction, separating hypersurfaces, transition states and most likely transition trajectories from the perspective of instanton calculus and Freidlin-Wentzell large deviation theory.

CT07 High Reynolds Number, Bute 4, Wednesday

Raffaele Grande (Cardiff)

Stochastic control problems and evolution by horizontal mean curvature flow

I will briefly introduce the notation of evolution by horizontal mean curvature flow in sub-Riemannian manifolds and in particular in the Heisenberg group, the associated level set equation and suitably associated stochastic control problems. Then I will introduce a regularizing p-problem and show some results on the structure of the optimal controls both for the p-problem and for the $\infty$-problem. At the end I will show some applications of these results to the Riemannian approximation for the horizontal mean curvature flow.

James Gregory (Manchester)

A collagen recruitment model of failure in tendons and ligaments

Collagen fibrils are microstructural components of tendons and ligaments that we model as elasto-plastic solids, assumed to be continuously distributed within the tissue. By choosing a simple form for the plastic stress for an individual fibril, we can derive an analytic expression for the stress in the entire tendon and, hence, a macroscopic strain energy function. The model agrees well with experimental data and provides a microscopic description of the macroscopic post-yield behaviour of tendons and ligaments in that the onset of yielding on the macroscale occurs when the first individual fibril yields.

CT 16 Mathematical Biology-3, Hillhead 4, Thursday

Information Length Analysis of Linear Autonomous Stochastic Processes

When studying the behaviour of complex dynamical systems, a statistical formulation can provide useful insights. In particular, information geometry is a promising tool for this purpose. In this talk, we introduce the information length for n-dimensional linear autonomous stochastic processes, providing a basic theoretical framework that can be applied to a large set of problems in engineering and physics. Besides, we discuss the application of the metric to a harmonically bound particle system with the natural oscillation frequency Ω, subject to a damping γ and a Gaussian white-noise, exploring how the information length depends on Ω and γ. This elucidates the role of the critical damping γ = 2 Ω in information geometry. Finally, in the long time limit, we show that the information length reflects the linear geometry associated with the Gaussian statistics in a linear stochastic process.

CT 09 Mathematical physics, Clyde 4, Wednesday

Carsten Gundlach (Southampton)

Naked singularity formation at the threshold of gravitational collapse

Can naked singularities form in general relativity during the time evolution of regular initial data? It seems the answer is "yes, for generic initial data right on the boundary between collapse (formation of a black hole) and non-collapse". This is well-understood for matter coupled to gravity in spherical symmetry, but much work remains to be done beyond spherical symmetry, and in particular for pure gravity.

CT 09 Mathematical physics, Clyde 4, Wednesday

Optimal turning gaits for undulators

An organism’s ability to efficiently traverse and search their surroundings can be important to its survival. This has inspired the study of optimal gaits and locomotion strategies, in particular for the case of undulatory movement of slender bodies. The primary focus has been on finding optimal waveforms for moving forwards along straight paths. However, the ability to turn and manoeuvre is also relevant to survival. We revisit this problem in the context of low Reynolds number hydrodynamics and obtain the optimal waveforms for undulators along curved trajectories. For shallow turning angles, we obtain small perturbations of a travelling wave as optimal. For larger turning angles, however, the optimal gait can be radically different, with the undulator abruptly curling and uncurling itself. We believe that these results can lend insight into the search behaviours of simple organisms, such as C. elegans, as well as be a tool for phenotyping their behaviour across mutant strains and under different environmental conditions.

CT01 Viscous Fluid Dyamis 1, Bute 4, Wednesday

Jacob Harris (Manchester)

Modelling flow in non-uniform Hele-Shaw cells

It is often useful to model fluid flows using simpler expressions than the Navier-Stokes equations to reduce the time and computational resources required to find solutions. In channels with large width to height aspect ratios, considering the height averaged fluid velocity gives a simple yet accurate description of the flow. The most well-known approach yields a single second-order partial differential equation, known as Darcy's law, that describes the fluid behaviour in the small space between two infinite parallel plates (a Hele-Shaw cell). Unfortunately, in this model, no slip and no penetration conditions cannot both be imposed on all boundaries. For problems that are strongly dependent on such conditions, those involving obstructions to the channel or side wall interactions, an ad hoc class of problem-specific, fourth-order equations have been employed instead. These equations often vary in their derivation and are rarely usable outside of their niche. We will propose a generalised fourth order alternative equation, derived from first principles that can give highly accurate descriptions of fluid behaviour in Hele-Shaw cells for a far greater range of geometries than the Darcy's law description.

CT03 Viscous Fluid Dyamis 2, Clyde 4, Wednesday

Stewart Haslinger (Liverpool/Imperial College)

Application of geometrical theory of diffraction to scattering by the tips of rough cracks

Stewart Haslinger (Liverpool), Michael Lowe (Imperial), Peter Huthwaite(Imperial), Richard Craster (Imperial), Fan Shi (Hong Kong University of Science and Technology)

The scattering of elastic waves by the tips and/or edges of smooth cracks is well understood. The geometrical theory of diffraction (GTD) was introduced by Keller [1] and is a useful method to compute asymptotic approximations to diffracted fields, for high frequency and/or large distances from a diffracting inclusion, in both two-dimensional and three-dimensional applications [2,3].

In non-destructive evaluation (NDE), modelling techniques are implemented in the technical justification reports for inspection qualification. The application of GTD for straight and flat defects is well established but in many cases, for example in environments with extremes in temperature and pressure, cracks are likely to be rough. A pragmatic approach is to add a safety factor to compensate for the uncertainty but we use stochastic and numerical methods to derive statistical predictions for the expected diffraction amplitudes as roughness is varied. The models presented here are for the two-dimensional case and include validations using high-fidelity finite element software Pogo [4].

References

[1] J.B.Keller, Geometrical theory of diffraction, JOSA (1962), 52,116–130.

[2] J. Achenbach, Wave Propagation in Elastic Solids, North-Holland Publishing Company/American Elsevier, 1973.

[3] J. Achenbach and A.K. Gautesen, Geometrical theory of diffraction for threeD elastodynamics, Journal of the Acoustical Society of America, (1977), 61(2), 413-421.

[4] P. Huthwaite, Accelerated finite element elastodynamic simulations using the GPU, Journal of Computational Physics, (2014), 257, 687–707, www.pogo.software.

Andrew Hazel (Manchester)

Fluid-Structure Interaction, Interfaces and Instabilities In A Simplified Model of Pulmonary Airway Reopening

We study an idealised model of pulmonary airway reopening, previously studied experimentally by Ducloué

et al (2017). An elasto-rigid Hele-Shaw cell, a uniform channel of rectangular cross-section with an elastic sheet as the upper wall, is filled with viscous oil and collapsed to a prescribed level by adjusting the pressure difference over the elastic sheet. Air is driven into the channel at constant volume flux which results in the development of an air finger that reopens the channel as it propagates.

We derive a depth-averaged fluid-structure interaction model in the frame of reference of the moving finger, which we explore via steady and unsteady numerical simulations using the software library oomph-lib. As we vary the initial collapse of the channel, the resulting depth variations alter the morphology of the propagating finger and promote a variety of instabilities from tip-splitting to small-scale fingering. We find remarkable direct agreement between our model and Ducloué et al's experiments, and the model has a complex solution structure with multiple stable and unstable steady and time periodic fingers. We conjecture that the interaction between the stable and unstable solutions underpins the complex fingering behaviour.

CT03 Viscous Fluid Dyamis 2, Clyde 4, Wednesday

Nathaniel Henman (UCL)

Pre-impact dynamics of a droplet impinging on a deformable surface

The non-linear interaction between air and a water droplet just prior to impingement on a surface is a phenomenon that has been researched extensively and occurs in a number of industrial settings. The role that surface deformation plays in an air cushioned impact of a liquid droplet is considered here. Assuming small density and viscosity ratios between the liquid and air, a reduced system of integro-differential equations is derived governing the liquid droplet free-surface shape and the pressure in the thin air film. To close the system, a membrane-type model is used for the shape of the deformable surface. The magnitude and shape of the surface deformation is determined by the surface properties, including surface stiffness. We solve the set of governing equations numerically and present a parametric study by varying the stiffness of the surface. It is found that lowering the surface stiffness results in reduced contact pressure, but bubble entrapment still occurs and in fact increases in magnitude (for a water droplet of radius 1 mm and approach speed of 1 m/s, a reduction in surface stiffness corresponding to a maximum surface deflection of 6.05 µm results in a 29% increase in the initial horizontal bubble extent and a 40% increase in the height, when compared to an impact with a flat rigid surface). Finally, a case is considered in the limit of large surface deformations, where an implied simpler pressure-shape relationship leads to a large time analysis, with good agreement to the numerical results. A quantitative connection with recent experiments is also found.

CT08 Industrial fluids, Bute 5, Wednesday

Alice Hodson (Warwick)

Construction of projectors for the conforming and non-conforming virtual element method

The virtual element method (VEM) began as an extension of the finite element method (FEM), and has gained a lot of attention since its first appearances in the literature. VEM is desirable to use in place of FEM due to the method allowing the considered domain to be decomposed into more general polygonal meshes. The ease in which VEM spaces can be constructed to handle higher order continuity conditions is another highly desirable property.

In contrast to the finite element setting, it is unclear whether it is possible to create a uniform framework to generalise the virtual element setting. Since each VEM construction depends on the prospective problem, the projections and discrete forms need to be set up each time. We explore the possibilities of defining the projection operators in a way that; (a) does not depend on the bilinear form, and (b) are computable using only the degrees of freedom. This would then ease implementation and create an elegant theory for the VEM framework.

The aim of this talk is to describe the generalisation of the construction of the projectors needed in the virtual element discretisation of polyharmonic problems. The generalisation is considered in two spatial dimensions, in both the conforming and non-conforming cases.

CT 15 Statistical and Numerical Methods, Hillhead 3, Thursday

Curtis Hooper (Loughborough)

Undular bores generated by fracture

We demonstrate for the first time, using high-speed pointwise photoelasticity, the generation of undular bores in solid (polymethylmethacrylate) pre-strained bars by fracture [1]. The development of the oscillations is strongly dependent on the strain rate of the release wave which propagates outwards from the fracture site. Both the nonlinear Gardner equation and its linearised (near the level of the pre-strain) version are used to model the wave propagation, and are shown to provide good agreement with the key observed experimental features for suitable choice of elastic parameters [1]. We also consider viscoelastic corrections which bring the modelling results in an even better agreement with experiments. The experimental and theoretical approaches presented open new avenues and analytical tools for the study and application of dispersive shock waves in solids.

Reference:

[1] C.G. Hooper, P.D. Ruiz, J.M. Huntley, K.R. Khusnutdinova, Undular bores generated by fracture, arXiv: 2003.06697v2 [nlin.PS] 4 Aug 2020.

Authors: C.G. Hooper, P.D. Ruiz, J.M. Huntley, K.R. Khusnutdinova

CT17 Solid Mechanics, Hillhead 5, Thursday

Matthew Hunt (Warwick)

Free surface flows in electrohydrodynamics with prescribed vorticity.

Free surface flows have classically used the assumption of irrotationality in the derivation of various different models which includes the celebrated Korteweg-de Vries(KdV) equation. In recent years there has been interest in the inclusion of constant vorticity to the model. This has lead to the use of various potentials to try and get a model described the free surface.

The approach taken here is to isolate the vertical component of the velocity and use that as the master equation. This will allow the introduction of not only a constant vorticity but a variable one. The results presented today will be a derivation of the method, linear and weakly nonlinear theory for a variety of different vorticity distributions for both linear and weakly nonlinear cases.

Pallav Kant (Twente, Netherlands)

Fast-Freezing Kinetics of Drops impacting on Cold Surfaces

Impact of a droplet on an undercooled solid surface instigates a number of physical processes simultaneously, including drop scale fluid motion, heat transfer between the liquid and the substrate, and the related phase transition. Whereas a large number of studies have investigated the corresponding interface deformations and the spreading of a droplet after it impinges onto an undercooled surface, the kinetics of phase transition within the impacting droplet has been addressed only in a few. Moreover, among the studies concerning solidification kinetics, only the regimes where phase transition effects are slower than the fast dynamics of droplet impact have been investigated. Here, we explore freezing behaviours that arise due to the rapid solidification of an impacting droplet at a sufficiently high substrate undercooling. Such scenarios are encountered in several industrial processes ranging from additive manufacturing to thermal plasma spraying of ceramics and metallic materials etc. In the present work, we adapt the total-internal-reflection (TIR) technique to visualise the phase transition in the vicinity of the liquid-substrate interface after a droplet impacts onto an undercooled transparent surface. This technique reveals the existence of distinct freezing morphologies at different undercooling. In particular, we describe/discuss a peculiar freezing behaviour, involving sequential advection of the solidified phase in the form of frozen-fronts, that emerges from the complex interplay between solidification and droplet-scale fluid motion. Besides, we present a simplified model that combines elements of nucleation theory and hydrodynamics to rationalize the observed freezing behaviour.

CT14 Droplets, Clyde 5, Thursday

Matthew Keith (Strathclyde)

Analysis of Thin Leaky Dielectric Layers Subject to an Electric Field

The application of an electric field can have a significant effect on the behaviour of fluids. For example, electrohydrodynamic (EHD) instabilities can lead to droplet formation or nonlinear pattern formation which have an abundance of industrial applications such as inkjet printing and the production of micro-electronic devices. The recent review by Papageorgiou [1] gives an overview of the recent work on EHD instabilities. In the present work, we investigate a bilayer of liquid and gas contained between two solid walls subject to a normal electric field, adopting the full Taylor-Melcher leaky dielectric model [2]. This work builds on that of Wray et al. [3] who considered the enhancement and suppression of EHD instabilities, investigating the axisymmetric problem of a fluid layer coating the outside of a solid cylinder. Using the long-wave approximation, we explore the linear stability of the system. An investigation of the nonlinear regime highlights four characteristic behaviours of the system, namely, asymptotic thinning, contact with the upper wall, the return of the interface to its flat state, and singular touchdown behaviour. Numerically calculated plots of appropriate parameter planes are obtained. Of particular interest are the critical transitions between these characteristic states, which we investigate both analytically and numerically. Furthermore, we investigate the long-time behaviour in the asymptotic thinning regime and the onset of sliding, and explore the self-similar dynamics of the liquid-gas interface during touchdown and upper-wall contact. Finally, we also explore the physically relevant perfectly conducting limiting case which simplifies the system and allows for additional analytical and numerical progress.

[1] Papageorgiou, D.T., 2019. Film flows in the presence of electric fields. Annual Review of Fluid Mechanics, 51, pp.155-187.

[2] Saville, D.A., 1997. Electrohydrodynamics: the Taylor-Melcher leaky dielectric model. Annual Review of Fluid Mechanics, 29, pp.27-64.

[3] Wray, A.W., Papageorgiou, D.T. and Matar, O.K., 2013. Electrified coating fibres: enhancement or suppression of interfacial dynamics. Journal of Fluid Mechanics, 735, pp.427-456.

CT02 Geophysical Fluid Dynamics, Bute 5, Wednesday

Oliver Kerr (City, London)

Double-diffusive instabilities at a suddenly heated sidewall

When a large body of salt-stratified fluid is heated from the side at a vertical boundary it causes the fluid near to the wall to rise. This results in the fluid near the wall being both warmer and saltier than the fluid at the same level further from the wall, giving rise to horizontal temperature and salinity gradients as well as vertical shear. If the heating is strong enough then instabilities can form. These have been observed experimentally.

A previous criterion for instabilities when the salinity gradient is strong and there is a relatively gradual increase in the wall temperature was determined by a quasi-static stability analysis. However, for weaker salinity gradients and/or faster heating the background state is intrinsically unsteady at the time of instability, and the quasi-static assumption is no longer valid and a stability analysis has been absent. We will look at these instabilities using the approach of Kerr and Gumm (2017) in their investigation of heating of fluid at isolated boundaries. They found the optimal evolution of a quadratic energy-like measure of the amplitude of the instabilities. The choice of the measure is not predetermined, but selected to minimize this optimal growth. This approach has been used previously for investigating double-diffusive instabilities at a heated horizontal boundary. Here we will apply it to the double-diffusive sidewall problem.

We will show that there are transitions between three basic modes of instability: the small and large Prandtl number modes found previously for the purely thermal problem, and a double-diffusive mode. We also find that varying the time from which the start of the growth of the instabilities is measured can be significant, and by delaying this a significant enhancement of the growth may be observed in some cases.

We will compare our findings with the results of earlier experiments.

CT08 Industrial fluids, Bute 5, Wednesday

Eun-jin Kim (Coventry)

Information Geometry in non-equilibrium processes

A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs), it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability. This metric structure then provides a key link between stochastic systems and geometry. We define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time, and is called the information length. We apply this method to atmospheric data obtained from the global circulation model as well as the self-organising process in laboratory plasmas. In particular, we show that time-dependent PDFs are non-Gaussian in general, and the information length calculated from these PDFs shed us a new perspective of understanding variabilities, correlation among different variables and regions.

CT 09 Mathematical physics, Clyde 4, Wednesday

Anna Kirpichnikova (Stirling)

Construction of artificial point sources for linear wave equation and application for inverse problems

We study the wave equation on a bounded domain $R^m$ or on a compact Riemannian manifold with boundary.

Let us assume that we do not know the coefficients of the wave equation but are only given the hyperbolic Neumann-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. We show that it is possible to construct a sequence of Neumann boundary values so that at a time $t_0$ the corresponding waves converge to zero while the time derivative of the waves converge to a delta distribution. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point where the energy of wave focuses is known in travel time coordinates, and satisfies a certain geometrical condition.

Nickolay Korabel (Manchester)

Non-homologues repair process driven by anomalous dynamics of DNA double strand breaks

The end joining process during the nonhomologous repair of the DNA double strand breaks after radiation damage is considered. It has been experimentally established that double strand break ends do not follow simple diffusion but move subdiffusively. However, it is still an open question which subdiffusion model is more appropriate. To decipher the mechanism of sub diffusion, we extend the DNA Mechanistic Repair Simulator (DaMaRiS) developed recently by implementing two additional sub diffusion models, fractional Brownian motion and a fractional Langevin equation, to compliment theContinuous Time Random Walk model already present. We compare predictions of biologically measurable end points such as repair protein recruitment kinetics

and DSB repair kinetics for each sub diffusion model with published experimental data. We further calculate the probabilities of formation of a dicentric chromosome and an acentric chromosome fragment as a result of misrepair of DNA strand breaks.

CT10 Mathematical Biology-2, Clyde 5, Wednesday

Revealing phase space structures in the presence of high instabilities

Investigations of transport in Hamiltonian systems require the identification of invariant phase space structures that govern the dynamics. These phase space structures are the stable and unstable invariant manifolds of NHIMs (unstable periodic orbits for systems with 2 degrees of freedom). We introduce a method [5] for approximating individual branches of stable and unstable invariant manifolds of (but not limited to) highly unstable periodic orbits. The method enabled the investigation of roaming reaction dynamics [1,3,7] in Chesnavich's CH4+ model subject to the Hamiltonian isokinetic thermostat [2,6]. Certain of the periodic orbits that govern roaming in the system [4] have Lyapunov exponents ~10^21.

[1] J. M. Bowman and B. C. Shepler. Roaming radicals. Annu. Rev. Phys. Chem., 62:531–553, 2011.

[2] C. P. Dettmann and G. P. Morriss. Hamiltonian formulation of the Gaussian isokinetic thermostat. Phys. Rev. E, 54, 1996.

[3] V. Krajňák and H. Waalkens. The phase space geometry underlying roaming reaction dynamics. J. Math. Chem., 56, 2018.

[4] V. Krajňák, G. S. Ezra, and S. Wiggins. Roaming at constant kinetic energy: Chesnavich’s model and the Hamiltonian isokinetic thermostat. Regul. Chaotic Dyn., 24 2019.

[5] V. Krajňák, G. S. Ezra, and S. Wiggins. Using Lagrangian descriptors to uncover invariant structures in Chesnavich's Isokinetic Model with application to roaming. accepted by Int. J. Bifurcation Chaos, 2020.

[6] G. P. Morriss and C. P. Dettmann. Thermostats: Analysis and application. Chaos, 8, 1998.

[7] R. D. van Zee, M. F. Foltz, and C. B. Moore. Evidence for a second molecular channel in the fragmentation of formaldehyde. J. Chem. Phys., 99, 1993.

CT13 Dynamical Systems, Clyde 4, Thursday

Andrew Krause (Oxford)

Pattern Formation in Heterogeneous Environments: Turing's Theory of Successive Patterning

Motivated by experimental work on multiscale patterns in biology, I will discuss recent progress extending Turing's theory of diffusion-driven morphogenesis to the case of a spatially heterogeneous domain. We employed a WKBJ ansatz to study the emergence of Turing-type instabilities from a spatially heterogeneous steady state in regions which locally satisfy conditions for Turing instability, permitting pattern formation in an intuitive way across a spatially-varying domain. Such an extended theory allows us to account for successive patterning events during development, explaining microstructures within periodic patterns such as on the skin of jaguars. Our theory separates mechanisms of patterning due to environemntal forces from those related to population interactions, and hence may also be valuable for elucidating the causes of colony formation and niche partitioning in complex ecosystems.

CT10 Mathematical Biology-2, Clyde 5, Wednesday

Jochen Kursawe (St Andrews)

Quantitative models of gene expression dynamics during embryonic cell fate decisions

Understanding and regulating cell fate decisions will be crucial in many bio-medical applications, for example the growth of artificial organs or stem cell therapies. Traditionally, cell states are identified and described by investigating static gene expression profiles. However, in recent years it has become increasingly clear that dynamic patterns of gene expression, such as oscillations, can play important roles in cellular decision making. For example, gene expression oscillations have been proposed to control the timing of cell differentiation during embryonic neurogenesis, i. e. the generation of nerve cells. The mathematical analysis of gene expression dynamics may be hindered by sparse data and parameter uncertainty. Here, we combine Bayesian inference and quantitative experimental data on mouse and zebrafish neurogenesis to explore mechanisms controlling aperiodic and oscillatory gene expression dynamics during cell differentiation. We find that quantitatively accurate model predictions are possible despite high parameter uncertainty. We identify examples of stochastic amplification, where oscillations are enhanced by intrinsic noise and we show how such oscillations can be initiated by changes in biophysical parameters. We further consider mechanisms that may enable the down-stream interpretation of dynamic gene expression. Our analysis illustrates how quantitative modelling can help unravel fundamental mechanisms of dynamic gene regulation.

CT 16 Mathematical Biology-3, Hillhead 4, Thursday

Jake Langham (Bristol)

Stability of steady erosive shallow flows

All geophysical flows deposit and erode sediment to and from the environment. Strong and heavily sediment-laden flows such as volcanic lahars represent dangerous natural hazards; predictive physical models play a key role in understanding these flows and mitigating their effects. We present stability analyses of steady sediment-laden flows over constant grade, in a general shallow-water formulation with bed transfer terms. A key finding is that a diffusive term is always required to regularise these models (which otherwise become ill-posed at unit Froude number, leading to an inability to compute solutions under general initial and boundary conditions). Given sensible generic model closures, we observe two coexistent steady solutions (one stable, one unstable) with different sediment concentrations and discuss the geophysical implications of such states.

CT02 Geophysical Fluid Dynamics, Bute 5, Wednesday

Christopher Lanyon (Nottingham)

A model to investigate the impact of farm practice on antimicrobial resistance in UK Dairy Farms.

Dairy farm slurry tanks are repositories of solid and liquid bovine waste as well as farm waste and antibiotic contaminated milk. The antimicrobial resistance genes, antimicrobial resistant microbes and antibiotics discharged by cows, combined with the natural presence of bacteria, mean that the slurry tank is potentially a site for the development and proliferation of AMR. Slurry is spread onto crops as a fertiliser, making dairy farms a potential source of AMR to the environment.

Mathematical modelling can be used to quickly and efficiently identify important process factors using computer simulation. Particularly in the case of slurry storage, where practical experiments may conflict with a farmer's needs, models can be used to simulate changes in farm practice over a number of years without having to enact those changes.

To assess the impact of farm practice on the development of AMR we have designed a new ordinary differential equation (ODE) model of the prevalence and spread of AMR within the dairy slurry tank environment. We model the chemical fate of bacteriolytic and bacteriostatic antibiotics within the slurry and their effect on a population of Escherichia coli (E.coli) bacteria, which are capable of resistance to both types of antibiotic. Through our analysis we find that changing the rate at which a slurry tank is emptied may delay the proliferation of multidrug-resistant bacteria by up to five years depending on conditions. This finding has implications for farming practice and the policies that influence waste management practices. We also find that, within our model, the development of multidrug-resistance is particularly sensitive to the use of bacteriolytic antibiotics, rather than bacteriostatic antibiotics, and this may be cause for controlling the usage of bacteriolytic antibiotics in agriculture.

Our model demonstrates the influence of farm practices on the development and spread of AMR within the slurry tank. The model itself has implications outside of the UK, especially in countries such as America and China, where intensive dairy farming operations are particularly prevalent. This work will cover the mathematical development of the model, including our approaches to each of the dynamics which affect the prevalence and spread of antimicrobial resistance within the E.coli community. This will be followed by a discussion of our model analysis and recommendations for future research and data collection.

CT04 Mathematical Biology-1, Clyde 5, Wednesday

Joseph Leedale (Liverpool)

Multiscale modelling of drug transport and metabolism in liver spheroids

In early preclinical drug development, potential candidates are tested in the laboratory using isolated cells. These in vitro experiments traditionally involve cells cultured in a two-dimensional monolayer environment. However, cells cultured in three-dimensional spheroid systems have been shown to more closely resemble the functionality and morphology of cells in vivo. While the increasing usage of hepatic spheroid cultures allows for more relevant experimentation in a more realistic biological environment, the underlying physical processes of drug transport, uptake and metabolism contributing to the spatial distribution of drugs in these spheroids remain poorly understood. The development of a multiscale mathematical modelling framework describing the spatiotemporal dynamics of drugs in multicellular environments enables mechanistic insight into the behaviour of these systems. Here, our analysis of cell membrane permeation and porosity throughout the spheroid reveals the impact of these properties on drug penetration, with maximal disparity between zonal metabolism rates occurring for drugs of intermediate lipophilicity. Our research shows how mathematical models can be used to simulate the activity and transport of drugs in hepatic spheroids, and in principle any organoid, with the ultimate aim of better informing experimentalists on how to regulate dosing and culture conditions to more effectively optimise drug delivery.

CT10 Mathematical Biology-2, Clyde 5, Wednesday

Jane Lyle (Surrey)

Symmetric Projection Attractor Reconstruction: Multi-dimensional Embedding of Physiological Time Series

The proactive recording of many physiological signals, such as the electrocardiogram (ECG), is increasingly routine, generating large quantities of data, from which diagnostic and predictive information must be derived. Traditional analysis of such time series data tends to identify local extrema and intervals between them, and therefore discards the morphological detail of the whole waveform. Our aim is to capture the waveform shape information of non-stationary, approximately periodic signals as a bounded two-dimensional attractor to allow simple interpretation and quantification.

Takens' method of delay coordinates takes N points from a single continuous variable to reconstruct an attractor in an N-dimensional phase space. We have previously described our Symmetric Projection Attractor Reconstruction (SPAR) approach that adapts this to extract information from any approximately periodic waveform through an embedding in a three-dimensional phase space and subsequent projection to a two-dimensional image which can easily be quantified.

We present an extension of our SPAR method for embeddings in any dimension N2, whilst retaining a two-dimensional output, and obtain a simple generalised result for a periodic signal. The derivation of this reveals that the two-dimensional attractor is a visualisation of the Fourier coefficients of the trigonometric interpolating polynomial of the embedding vector, which aids our understanding of the attractor properties, and its relationship to the underlying signal.

We then extend our generalised result into the space of approximately periodic signals. The ECG is a complicated signal reflecting various processes. Generating attractors in multiple higher dimensions provides information on subtle changes in different parts of the ECG morphology. We have previously applied machine learning to discriminate gender in the ECG, and we build on this to show that an individual's response to treatment with cardioactive drugs can be quantified from the trajectory of changes observed in the attractor.

The extension of the SPAR method into higher dimensions whilst retaining a two-dimensional projection has produced an interesting theoretical result which we have been able to apply and interpret in approximately periodic signals. In the clinical context, SPAR analysis of physiological signals has simple visual appeal, and provides support for a stratified approach to the diagnosis and management of patients.

CT 15 Statistical and Numerical Methods, Hillhead 3, Thursday

Fiona Macfarlane (St Andrews)

From a discrete model of chemotaxis with volume-filling to a generalised Patlak-Keller-Segel model

We present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased random walk. In order to take into account possible alterations in cellular motility observed at high cell densities (i.e. volume-filling), we let the probabilities of cell movement be modulated by a decaying function of the cell density. We formally show that a general form of the celebrated Patlak-Keller-Segel (PKS) model of chemotaxis can be formally derived as the appropriate continuum limit of this discrete model. We carry out a systematic quantitative comparison between numerical simulations of the discrete model and numerical solutions of the corresponding PKS model. The results obtained indicate that there is excellent quantitative agreement between the spatial patterns produced by the two models.

CT10 Mathematical Biology-2, Clyde 5, Wednesday

Green's Functions in Discrete Flexural and Elastic Systems

The Green's function is the canonical object of study for many problems associated with wave propagation in structured solids; it contains the fundamental information of the dynamic response of the system. There has previously been a substantial amount of literature on Green's functions for scalar systems, such as photonics, acoustics, and platonics, particularly in statics. There has been comparatively little work on constructing Green's functions for dynamic vector systems corresponding to multi-scale mechanical materials. This area of research has applications in elastic metamaterials, non-destructive evaluation, one-way edge waves, cloaking, and seismic protection.

This talk discusses the unstudied problem of propagating flexural waves through discrete lattice systems. We construct the dynamic lattice Green's function for one- and two-dimensional flexural lattices. The dispersion equation is found explicitly, and then used to locate the band gaps of the system. The band gap Green's function for the vertical displacement motion of a localised defect mode is found explicitly for the system. The rotational effect on the masses at the connecting nodes as flexural waves travel down the system is also studied. The novel results presented lead to interesting phenomena, including localised defect states, dynamic anisotropy, and tensorial inertia matrices -- providing links to micropolar and Willis-type media.

Andrew Mair (Heriot-Watt)

Modelling the relationship between soil moisture and root density dynamics

Be it through carbon fixation or the provision of food and medicines, plants play a crucial role in supporting animal life on earth. Biological knowledge of the interactions between plants and their environment is rich and mathematical models of such interactions can provide valuable information regarding the effects of environmental stresses on plant life. In this project we derive and numerically simulate a model for the relationship between soil moisture and root density dynamics. The model consists of Richards' equation for water flow in soil, coupled with a system of transport equations describing root density dynamics within the soil.

There are many sources of empirical evidence which suggest that root density levels have a significant effect upon the hydraulic properties of soil. In our work we have used multi-scale analysis (homogenization) methods to suggest how the dependence on root density could be encoded into Richards' equation. Mathematical models for root density dynamics are usually transport equations involving both spatial and angular dimensions. We propose a root density model with only spatial dimensions so that, when coupled with Richards' equation, the system is numerically tractable.

The last section of the talk will describe the development of a numerical scheme to simulate our system of nonlinear partial differential equations. There are inherent challenges in numerically approximating a coupled system of Richards' equation and transport equations, the numerical methods employed to address these challenges will be discussed.

CT06 Porous media, Hillhead 5, Wednesday

Rebecca McKinlay (Strathclyde)

Coating Flow on a Rotating Horizontal Circular Cylinder Subject to a Radial Electric Field

The two-dimensional dynamics of a thick film of an electrified, perfectly conducting Newtonian fluid flowing on the surface of a rotating horizontal cylinder are studied. The rotating cylinder is an electrode held at a constant potential and a concentric outer electrode whose potential is allowed to vary spatially encloses the system, inducing electrostatic forces at the interface. The long-wave approximation is used along with the method of weighted residuals to derive a model that incorporates the effects of the electric stress, rotation, gravity, viscosity, inertia, and capillarity. This model is investigated both numerically and analytically in appropriate limits to validate against known results. Novel second-order models governing the electric potential are derived by projecting onto asymptotically accurate polynomials and are validated against direct numerical simulations.

CT01 Viscous Fluid Dyamis 1, Bute 4, Wednesday

Richard Mcnair (Manchester)

Amazing Marangoni: Maze solving with surfactant dynamics

Authors: Richard Mcnair, Oliver Jensen, Julien Landel

Experiments (Temprano-Coleto et al., Phys. Rev. Fluids 3:100507, 2018) have shown how exogenous surfactant introduced to the entrance of a maze filled with a shallow liquid layer will induce a flow in the liquid which spontaneously finds the longest path through the maze, effectively solving the maze with minimal flow into dead end sections of the maze. Here we test the hypothesis that this is due to the dynamics of endogenous contaminant surfactants already present in the liquid. We propose a model based on lubrication theory to derive equations capturing the Marangoni flow induced by the exogenous and endogenous surfactants at the film surface. The equations reduce to a nonlinear diffusion equation which describes the late-time spreading of the exogenous surfactant in the presence of endogenous surfactant. A further equation is needed to track the leading edge of the introduced exogenous surfactant. Numerical simulations of the equations describing the flow and exogenous surfactant front location qualitatively capture the experimental data. A numerical method for solving the nonlinear diffusion equation on a network using tools from graph theory and discrete calculus is also presented.

Matthias Mimault (James Hutton)

Smoothed Particle Hydrodynamics Method to model root development

Plant organs develop from a group of undifferentiated cells called the meristem. Meristematic cells operate sequences of elongation, division and differentiation from which emerge complex morphologies and tissue architectures. The kinematics of development in plants is well characterised but understanding the coordination of cell activity is challenging. Indeed, a mathematical framework encompassing the discrete nature of cellular processes while solving biophysical dynamics at the macroscopic scale remains out of reach.

Here, we propose a cell-based model capable to simulate entire root meristems. The model is based on the Smoothed Particle Hydrodynamics (SPH) method and allows to solve equations for the growth at tissue scale from a flexible distribution of cells (particles). We developed a new approach to simultaneously compute field values such as turgor pressure, stress and strain rates while predicting the establishment of the root anatomy during growth. We show the influence of soil mechanical constraints on the development of the root and the reorganisation of the cell architecture in the meristem. The model could also predict growth patterns observed experimentally and use 3D microscopy data as input for numerical simulations.

In the future, our model will foster a better understanding of complex soil biological processes that are especially difficult to observe. This technique has the potential to incorporate the key biophysical processes of the root in a robust root-soil interaction model.

CT06 Porous media, Hillhead 5, Wednesday

Giorgos Minas (St Andrews)

Stochastic modelling, simulation and analysis of oscillatory biological systems: the NF-κB case study

Cells constantly receive a multitude of different signals from their external environment. They use networks of interacting molecules to respond to these signals and trigger the appropriate actions. An important target of molecular biology is to identify and study the key components of these networks that are often found to be therapeutic targets. An important example is the NF-κB signalling system that responds to a variety of signals related to stress and inflammation in order to activate a large number (500) of different genes. The NF-κB network is noisy and complex with oscillatory dynamics, involving multiple feedback loops, and therefore, mathematically, very interesting. In this talk, I am going to introduce the NF-kB signalling pathway, discuss a stochastic model for oscillatory systems and describe an analytical framework for assessing the ability of a stochastic signaling system to distinguish between simultaneously received signals.

CT 16 Mathematical Biology-3, Hillhead 4, Thursday

Andrew James Mitchell (Strathclyde)

The effect of the Lower Boundary on Porous Squeeze-Film Flow

Squeeze-film flow of a Newtonian fluid (that is, flow of a layer of fluid in the gap between two rigid impermeable plates that approach each other) is a classical problem in fluid mechanics, with applications in, for example, the squeezing of synovial fluid in the knee joint or between hydrolic clutch plates. As is well known, an infinite time is required for all of the fluid to be squeezed out of the gap, i.e. for the plates to make contact. Knox et al. (“Porous Squeeze-Film Flow”, IMA J. Appl. Math. 80 2015, 376-409) generalised classical squeeze-film flow to the case of fluid being squeezed between an impermeable plate and a porous layer with an impermeable base. In particular, using the Darcy equation in the porous layer and the Navier–Stokes equation in the fluid layer, and imposing the Beavers–Joseph condition at the interface between the fluid and porous layer, Knox et al. (2015) showed that the contact time is finite and depends on the permeability of the porous layer. The current work extends that of Knox et al. (2015) for both an axisymmetric and a two-dimensional geometry to consider

what happens when there is no impermeable base below the porous layer. Like Knox et al. (2015), we find that there is a finite contact time, which depends on the permeability of the porous layer. The behaviour in the limits of small and large permeability is also analysed. We find that as a result of the lower impermeable base being removed the contact of the impermeable plate and the porous layer happens much sooner than in the problem considered by Knox et al. (2015).

CT06 Porous media, Hillhead 5, Wednesday

Matthew Moore (Oxford)

How solute diffusion counters advection in the early stages of coffee ring formation

We study the initial evolution of the coffee ring that is formed by the evaporation of a thin surface tension-dominated droplet containing a dilute solute. When the solutal Péclet number is large, we show that diffusion close to the droplet contact line controls the coffee-ring structure in the initial stages of evaporation. We perform a systematic matched asymptotic analysis for two evaporation models - a simple, non-equilibrium, one-sided model (in which the evaporative flux is taken to be constant across the droplet surface) and a vapour-diffusion limited model (in which the evaporative flux is singular at the contact line) - valid during the early stages in which the solute remains dilute. We call this the 'nascent coffee ring' and describe the evolution of its features, including the size and location of the peak concentration and a measure of the width of the ring. We consider a variety of droplet shapes to asymptotically investigate the role played by contact line curvature in the formation of the ring. Moreover, we use the asymptotic results to investigate when the assumption of a dilute solute breaks down and the effects of finite particle size and jamming are expected to become important.

CT14 Droplets, Clyde 5, Thursday

Shibabrat Naik (Bristol)

Tilting and Squeezing: Implications of Hamiltonian saddle-node bifurcation for reaction dynamics

Authors: Shibabrat Naik, Víctor J. García-Garrido, Wenyang Lyu, Stephen Wiggins

Hamiltonian models provide insights into phase space mechanisms of chemical reactions which have a characteristic potential energy surface. We present the normal form of a two-degree-of-freedom Hamiltonian that undergoes saddle-node bifurcation. We discuss how the changes in the geometry, that is depth and flatness, of the potential energy surface leads to the saddle-node bifurcation. Then, we show how the bifurcation affects the geometry of the phase space structures that mediate reactions in this Hamiltonian system. We discuss how Lagrangian descriptors can be used for detecting the qualitative changes in the phase space structures with changes in the depth and flatness of the potential energy surface. Then, we address the qualitative and quantitative changes due to depth and flatness from a reaction dynamics perspective.

CT13 Dynamical Systems, Clyde 4, Thursday

Michael Negus (Oxford)

High-speed droplet impact onto deformable substrates: analysis and simulations

The impact of a high-speed droplet onto a substrate is a highly non-linear, multiscale phenomenon and poses a formidable challenge to model. In addition, when the substrate is deformable, such as a spring-suspended plate or an elastic sheet, the fluid-structure interaction introduces an additional layer of complexity. We present two modeling approaches for droplet impact onto deformable substrates: matched asymptotics and direct numerical simulations. In the former, we use Wagner's theory of impact to derive analytical expressions which approximate the behaviour during the early stages of the impact. In the latter, we use the open source volume-of-fluid code Basilisk to conduct direct numerical simulations designed to both validate the analytical framework and provide insight into the later times of impact. Through both methods, we are able to observe how the properties of the substrate, such as elasticity, affect the behaviour of the flow. We conclude by showing how these methods are complementary, as a combination of both can lead to a thorough understanding of the droplet impact across timescales.

CT14 Droplets, Clyde 5, Thursday

Matthew Nethercote (Manchester)

Edge Diffraction of Acoustic Waves by Periodic Composite Metamaterials: The Hollow Wedge

Normally in metamaterial research, some form of infinite periodicity is assumed which allows us to restrict the study to a small portion known as the unit cell. This has led to many studies which increase the complexity of the unit cell and reconstruct the global scattering using the periodicity of the metamaterial. An alternative approach looks into the case where infinite periodicity is no longer assumed. This means that the metamaterial will have well-defined boundaries that can symbolise many different interfaces such as edges and corners.

In this presentation, the scattering of an acoustic pressure wave by a hollowed out wedge is studied where, for simplicity, the unit cells will be sound-soft cylinders with infinite height and a small radius. This configuration can also be viewed as two separate semi-infinite gratings with two sets of scattering coefficients to determine. We will construct an iterative scheme from the resulting infinite system of equations and find a solution using the discrete Wiener-Hopf technique. We shall also discuss some tools that are useful for computations such as tail-end asymptotics and rational approximations.

Koji Ohkitani (Sheffield)

Revisiting self-similarity for the 3D Navier-Stokes equations

We investigate the scale-invariance property the Navier-Stokes equations theoretically and numerically, as a basis for studying their statistical solutions. (See a related reference below.) We distinguish critical scale-invariance of 'the first kind' (deterministic) and that of 'the second kind' (statistical), the latter of which is related to the so-called source-type solution (that is, a nonlinear counterpart of fundamental solutions to linear PDEs.)

Mathematically a self-similar solution is known to exist in a function class which contains singular solutions like 1/|x|, but its explicit functional form is yet to be determined. We formulate a forward self-similar problem for the 3D Navier-Stokes equations in velocity, vorticity and vorticity gradient and introduce successive approximations. We work out the self-similar decaying profile to the leading-order and determine it approximately to compare it with those of the Burgers and 2D Navier-Stokes equations.

(This is a joint work with Dr Riccardo Vanon.)

Reference:

K. Ohkitani, "Study of the Hopf functional equation for turbulence: Duhamel principle and dynamical scaling," Phys. Rev. E 101, 013104(2020).

CT07 High Reynolds Number, Bute 4, Wednesday

Philip Pearce (Harvard Medical School)

Emergent robustness of bacterial quorum sensing in fluid flow

Bacteria use intercellular signalling, or quorum sensing (QS), to share information and respond collectively to aspects of their surroundings. The autoinducers that carry this information are exposed to the external environment; consequently, they are affected by factors such as removal through fluid flow, a ubiquitous feature of bacterial habitats ranging from the gut and lungs to lakes and oceans. Here, we develop and apply a general theory that identifies and quantifies the conditions required for QS activation in fluid flow by systematically linking cell- and population-level genetic and physical processes. We predict that, when a subset of the population meets these conditions, cell-level positive feedback promotes a robust collective response by overcoming flow-induced autoinducer concentration gradients. By accounting for a dynamic flow in our theory, we predict that positive feedback in cells acts as a low-pass filter at the population level in oscillatory flow, allowing a population to respond only to changes in flow that occur over slow enough timescales. Our theory is readily extendable, and provides a framework for assessing the functional roles of diverse QS network architectures in realistic flow conditions.

CT10 Mathematical Biology-2, Clyde 5, Wednesday

Nicholas Pearce (Coventry)

Modelling cardiac mechano-electric feedback response to perturbations using a modified multi-scale model of heart dynamics.

Cardiac dynamics are a function of many scales and physical processes housed entirely within the heart itself. Feedback mechanisms across these scales helps maintain the regulation of blood flow. The two most important mechanisms are the mechanoelectric feedback (MEF) and the electro-contraction coupling (ECC). In the former, microscopic changes in the mechanical environment of the muscle cell starts electrochemical processes leading to alterations in the macroscale mechanical regulation of blood flow. The ECC can be described as the opposite to the MEF. Whilst MEF helps maintain heart stability, it has been shown to both promote and discourage arrhythmia and asystole. In some instances, a mechanical stimulation of the heart may even lead to death.

Here, a multiscale lumped-parameter model of the heart that incorporates the MEF mechanism is modified to investigate the response to a sudden mechanical stimulation of muscle cells. The model has previously shown chaotic behaviour resulting from a dysfunction of MEF. We find that that the timing of an impulse is a critical indicator for the resulting behaviour. Though the predominant response to an impulse is found to be short term arrhythmia, it is also found that a short mechanical perturbation delivered during diastole can regularise a dysfunctional heart. This contrasts with recent studies finding mechanical stimulation leads only to fibrillation.

CT 16 Mathematical Biology-3, Hillhead 4, Thursday

Buckling of chiral rods due to coupled axial and rotational growth

Most existing works modelling non­-planar configurations in growing filaments stick to isotropic rods. Such models usually rely on differential growth, or the presence of an external elastomeric matrix, multi-­rod composites, or phototropism to model the generation of curvature and torsion in non-­planar deformations. Growth in chiral rods can be another way to obtain such non­planar configurations; this has not been explored in the literature. In this work, we focus on axial growth coupled with rotation of cross­sections. This can lead to non­planar configurations if the material of the rod exhibits some sort of twist­-extension coupling, simply with a boundary condition that arrests relative axial rotation at the ends. We present a homogeneous growth model for special Cosserat rods with two controls, one for lengthwise growth and the other for rotations. This is explored in greater detail for straight rods with transverse hemitropy and helicalmaterial symmetry by introducing the assumption of symmetry preserving growth to account for the microstructure. The example of a guided-­guided rod possessing chiral material symmetry is considered to illustrate the occurrence of out­-of­-plane buckling at certain stages of growth (or atrophy). These solutions obtained are flip symmetric and chiral. A complete mirroring of the rod, including both growth and constitutive properties gives a solution with opposite chirality, under the same deformation. End-­to-­end distance for different combinations of growth and material chiralities are examined to understand the effect of twisting growth on the constitutive twist­-extension coupling.

Preprint: Pradhan, S.P. and Saxena, P., 2020. Buckling of chiral rods due to coupled axial and rotational growth.arXiv preprint arXiv:2009.02037

CT17 Solid Mechanics, Hillhead 5, Thursday

Christopher Prior (Durham)

Wavelet decompositions of helicity

Fourier decompositions of magnetic helicity have been used to provide information on the cascade to small scales of magnetic topology in resistive MHD systems, such as Dynamo and driven turbulence models. The helicity and energy of the magnetic field at each Fourier scale are shown to be strongly linked and, in homogeneously driven turbulence, one can even find a scale-based linear decomposition of the energy into its helicity and a correlation tensor of the field. We have recently derived a set of similar tools for relating the helicity and energy content of a magnetic field using wavelet decompositions. This has the relative advantages over Fourier analysis of being applicable in highly inhomogeneous systems as well as providing information about the toology-energy relationship in localised regions of space; highly advantageous in, say, Dynamo models, where helicity must be spatial transported away from the region of dynamo formation. I will introduce these tools and give a short practical guide to methods for applying them.

Clare Rees-Zimmerman (Cambridge)

Modelling diffusiophoresis: a promoter of stratification in drying films

Stratification in drying films – how a mixture of differently-sized particles arranges itself upon drying – is examined. It is seen experimentally that smaller particles preferentially accumulate at the top surface, but it is not understood why. Understanding this could allow the design of formulations that self-assemble during drying to give a desired structure. Potential applications are across a wide range of industries, from a self-layering paint for cars, to a biocidal coating in which the biocide stratifies to the top surface, where it is required.

On the basis of diffusional arguments alone, it would be expected that larger particles stratify to the top surface. However, other physical processes, including diffusiophoresis, may also be important. By deriving transport equations, the magnitude of different contributions can be compared, and numerical solutions for the film profile are produced. Asymptotic solutions are derived for the film profile in the high Péclet number (fast evaporation compared to diffusion) regime.

Diffusiophoresis is the migration of particles along a concentration gradient of a different solute species. A particular diffusiophoresis mechanism that has been hypothesised to cause small-on-top stratification is an excluded volume effect. This work probes the significance of this type of diffusiophoresis: to the diffusional model, a diffusiophoresis term is added that can be varied in strength. For hard spheres, it is predicted that diffusiophoresis counteracts the effect of diffusion, resulting in approximately uniform films. When the diffusiophoresis strength is increased, the small particles are predicted to stratify to the top surface. This suggests that diffusiophoresis does contribute to experimental observations of small-on-top stratification, but it might not be the only promoting factor.

CT01 Viscous Fluid Dyamis 1, Bute 4, Wednesday

Jean Reinaud (St Andrews)

Breaking of baroclinic tori of potential vorticity

Large scale oceanic and atmospheric flows are strongly influenced by the background planetary rotation as well as as the stable density stratification of the fluid. When these two effects are dominant, the flow dynamics can be fully described by the single scalar quantity: the potential vorticity (PV). The latter is materially conserved in absence of diabatatic effects and often negligle frictional effects. Eddies in the oceans can be seen as compact volumes of PV. The contribution of meso-scale eddies to mass transport in the oceans is comparable to the wind-driven or the thermohaline circulations, hence such structure play key roles in the oceans. Hetons are an example of baroclinic eddies which allow to transport quantities over large distances. Hetons are structures consisting of two eddies rotating in opposite directions (a cyclone and an anti-cyclone) lying at different depths. One possible mechanism of formation of hetons is the breaking of larger distributions of PV such as PV rods and PV tori. We discuss the stability and the breaking of baroclinic tori of PV and the subsequent formation of diverging hetons, in the so-called quasi-geostrophic regime.

CT02 Geophysical Fluid Dynamics, Bute 5, Wednesday

Martin Richter (Nottingham)

Convergence Properties of Transfer Operators for Billiards with a Mixed Phase-Space

We analyse the convergence properties of a ray-tracing approach to transfer operators. The investigation focuses on a two-dimensional Hamiltonian system with a mixed-phase space, i.e. coexisting integrable and chaotic dynamics. More precisely, we focus on a two-dimensional billiard domain in which the corresponding wave problem has Dirichlet boundary conditions. As we focus on mid- to high-frequency regimes, we construct the transfer operator by means of a ray-tracing approach. We then solve the propagation problem dynamically and investigate the rate of convergence. We accompany this analysis with an investigation of the dynamics in phase space in terms of the associated boundary map in Birkhoff-coordinates and investigate spectral properties of the transfer operator. We compare our findings with recent proofs carried out for a circular domain and conclude with an outlook about its applicability for real-world problems.

CT13 Dynamical Systems, Clyde 4, Thursday

Jonna Roden (Edinburgh)

PDE-Constrained Optimization for Multiscale Particle Dynamics

There are many industrial and biological processes, such as beer brewing, nano-separation and bird flocking, which can be described by integro-PDEs. These PDEs define the dynamics of a particle density within a fluid bath, under the influence of diffusion, external forces and particle interactions, and often include complex, nonlocal boundary conditions.

A key challenge is to optimize these types of processes. For example, in nano-separation, it is of interest to determine the optimal inflow rate of particles (the control), which leads to high separation of the particles (the target), at a minimal financial cost. Mathematically, this requires tools from PDE-constrained optimization. A standard technique is to derive a system of optimality conditions and solve it numerically. Due to the nonlinear, nonlocal nature of the governing PDE and boundary conditions, the optimization of multiscale particle dynamics problems requires the development of new theoretical and numerical methods.

I will present the system of nonlinear, nonlocal integro-PDEs that describe the optimality conditions for such an optimization problem. Furthermore, I will introduce a numerical method, which combines pseudospectral methods with a multiple shooting approach. This provides a tool for the fast and accurate solution of these optimality systems. Finally, some examples of future industrial applications will be given. This is joint work with Ben Goddard and John Pearson.

Mario Sandoval (Autonoma Metropolitana)

’Maxwell-Boltzmann’ velocity distribution for noninteracting active matter

We theoretically and computationally find a Maxwell-Boltzmann-like velocity distribution for noninteracting active matter (NAM). To achieve this, mass and moment of inertia are incorporated into the corresponding noninteracting active Fokker-Planck equation (NAFP), thus solving for the first time, the underdamped scenario of NAM following a Fokker-Planck formalism. This time, the distribution results in a bimodal symmetric expression that contains the effect of inertia on transport properties of NAM. The analytical distribution is further compared to experiments dealing with vibrobots. A generalization of the Brinkman hierarchy for NAFP is also provided and used for systematically solving the NAFP in position space. This work is an important step toward characterising active matter using an equivalent non-equilibrium statistical mechanics.

CT 09 Mathematical physics, Clyde 4, Wednesday

Osian Shelley (Warwick)

Transaction tax in a general equilibrium model

In this talk, we consider the effects of a quadratic tax rate levied against two agents with heterogeneous risk aversions in a continuous-time, risk-sharing equilibrium model. The goal of each agent is to choose a trading strategy which maximises the expected changes in her wealth, for which an optimal strategy exists in closed form, as the solution to an FBSDE. This tractable set-up allows us to analyse the utility loss incurred from taxation. In particular, we show why in some cases an agent can benefit from the taxation before redistribution. Moreover, when agents have heterogeneous beliefs about the traded asset, we discuss if taxation and redistribution can dampen speculative trading and benefit the agents, respectively.

CT 15 Statistical and Numerical Methods, Hillhead 3, Thursday

Jemma Shipton (Exeter)

Modelling freely decaying shallow water turbulence with compatible finite element methods.

We present results from recent compatible finite element simulations of freely decaying shallow water turbulence. Compatible finite element methods are a form of mixed finite element methods (meaning that different finite element spaces are used for different fields) that allow the exact representation of the standard vector calculus identities div-curl=0 and curl-grad=0. This necessitates the use of div-conforming finite element spaces for velocity, such as Raviart-Thomas or Brezzi-Douglas-Marini, and discontinuous finite element spaces for pressure. The development of these methods for numerical weather prediction has been motivated by the parallel scalability bottleneck of the standard latitude-longitude grid. Cotter and Shipton [2012] demonstrated that compatible finite element discretisations for the linear shallow water equations satisfy the basic conservation, balance and wave propagation properties listed in Staniforth and Thuburn [2012], without the restriction that the grid is orthogonal. The linear equations dictate our choice of finite element spaces; we then need to construct stable and accurate advection schemes to solve the nonlinear equations. Here we focus on the challenging problem of modelling freely decaying turbulence. In this situation, vortices form, interact, and like-signed vortices merge. In the spherical case the flow evolves into zonal bands. We investigate the effect of different spatial and temporal discretisations on the properties of the flow.

CT02 Geophysical Fluid Dynamics, Bute 5, Wednesday

Matthew Shirley (Oxford)

Heat transfer by gas flow in silicon furnaces

In silicon furnaces, channels are observed to form through the solid raw materials and hot gases created by chemical reactions flow through these channels. The gases play a vital role in heating the incoming raw materials, both through diffusive heat transfer and through exothermic chemical reactions between the gas and solid. Understanding the gas flow is therefore crucial to building a realistic model of heat transfer in the furnace, and in turn optimising the furnace's efficiency.

In this talk we present a mathematical model for the laminar flow of compressible gas in such a channel. Using the slenderness of the channel and the small pressure drop driving the flow, we asymptotically obtain a simplified system of equations which we solve to predict how the temperature of both the gas and the surrounding solid varies with depth. We discuss how changing the geometry of the channel, and the operating parameters of the furnace affects the gas flow and thermal distribution.

CT01 Viscous Fluid Dyamis 1, Bute 4, Wednesday

Josephine Solowiej-Wedderburn (Surrey)

A mathematical model for a contractile mechanosensory mechanism within cells

Cells interact with their environments through a variety of chemical and physical signalling mechanisms. It is becoming increasingly clear that physical force and the mechanical properties of their microenvironment play a crucial role in determining cellular behaviour and coordination. Understanding these differences is crucial for tissue engineering applications and to determine how the mechanical microenvironment may affect, for example, cancer growth and invasion. We use a continuum elasticity-based model with an active contractile component to describe the mechanosensory mechanism of a cell or cell layer adhered to a substrate. The model context focuses on the most common biophysical experimental set-ups investigating cellular contractility, using experimentally determined knowledge of the mechanical response of the designed substrates to infer the cell-generated force from observed substrate deformations. The mathematical model is analysed and solved using both analytical approaches (exploiting approximations and symmetry arguments) and Finite Element Methods. We use the model to explain observed cellular adaptations to changes in the mechanical properties of the underlying gel. In particular, we consider the distribution of adhesion and contractility throughout a cell. For experimentally realistic distributions of adhesion points, the model is capable of recreating cell shapes and deformations that are consistent with those experimentally observed. Furthermore, energy considerations are shown to have significant implications for the optimisation of cell adhesion and the organisation of cellular contractility.

CT 16 Mathematical Biology-3, Hillhead 4, Thursday

Emma Southall (Warwick)

Evaluation of lead time prediction methods for detecting critical transitions using timeseries data

Early-warning signals are widely used in many fields to anticipate a critical threshold prior to reaching it. A systems undergoes the phenomenon known as critical slowing down as it crosses through a bifurcation. Theory predicts that fluctuations away from the mean will recover more slowly as the system approaches a critical transition. This is key in infectious disease modelling to assess when the basic reproduction number is reduced below the threshold of one.

Recent theoretical advances have shown indicators of critical transitions in epidemiology such as measuring the lag-1 autocorrelation in synthetic disease data. An effective early-warning signal would be able to predict an impending critical transition of this type with a suitable lead time in order to act on the current path of the disease.

We validate several empirical studies which offer lead time predictions for ecological and infectious diseases systems when using this theory practice. Our work highlights several challenges when applying lead time methodologies to simulated models. We find poor specificity, falsely reporting defections of a critical transitions in simulations at steady state.

In this talk we present an extension to these methods and our results show promising potential for calculating early-warning signals of elimination on real-world noisy data.

CT04 Mathematical Biology-1, Clyde 5, Wednesday

Helena Stage (Manchester)

Multi-Scale Superinfection Models in Evolutionary Epidemiology

The study of evolutionary epidemiology is vital to understand and control the spread of anti-microbial resistance, but is inherently challenging because pathogen evolution is driven by forces acting at multiple scales: for example, HIV needs to escape the immune system within a host, but also needs to maintain the ability to be transmitted efficiently between hosts. Time-since-infection models are much more flexible than ODEs if we want to allow for realistic enough aspects of both within- and between-host scales, but capturing the feedback loops between such scales is a formidable challenge.

We will discuss the main technical challenges in developing a general theory for time-since-infection models that allow for superinfection (e.g. multi-strain systems with partial cross-immunity), starting from the problem of characterising the system’s steady states. We will distinguish between the cases when superinfection of the host facilitates the coexistence of two (or more) infections that interact synergistically by fuelling each other’s spread (syndemic), and when these infections hinder each other. We show how in the former case multiple stable steady states are possible, while in the latter case the stable steady state is unique but possibly harder to compute. We discuss the consequent implications for public health control measures.

CT04 Mathematical Biology-1, Clyde 5, Wednesday

Peter Stewart (Glasgow)

Nonlinear Rayleigh--Taylor instability in aqueous foam fracture

A monolayer of gas-liquid foam can exhibit a brittle fracture mode when subjected to a large driving pressure, analogous to brittle fracture of crystalline atomic solids. The brittle crack advances through successive rupture of liquid films, each driven by the Rayleigh--Taylor instability. We analyse the linear stability of a uniform foam lamella accelerated in the direction perpendicular to its interfaces, constructing a reduced approximation for the growth rate of Rayleigh--Taylor instability in the limit of large surface tension. We show that in this limit the asymptotic description can be extended to form a system of one-dimensional partial differential equations which govern the nonlinear growth of the instability and subsequent film rupture. The nonlinear model predicts that, following the onset of instability, the liquid film thins exponentially at its edges and bulges at its centre. As a result the time taken for the film to rupture can be shown to be almost twice that predicted by linear theory alone.

CT08 Industrial fluids, Bute 5, Wednesday

Matteo Taffetani (Bristol)

Stokes - Biot coupling in a perfusion bioreactor with inhomogeneous porosity

Authors: Matteo Taffetani (Oxford), Ricardo Ruiz Baier (Monash, Australia) , Sarah Waters (Oxford)

A perfusion bioreactor for tissue growth can be simplified as a fluid-poroelastic system constituted by of two domains, one fluid and one porous, that interface with each others through a single boundary and with a prescribed flux imposed at the inlet. Cells are mechanosensitive entities so the local cellular mechanical environment may be tuned by controlling the geometry of the two phase poroelastic scaffold in terms of spatial distribution of the porosity.

Our modelling approach is motivated by this latter aspect and we describe the porous domain using the equations of linear poroelasticity with the permeability and the shear modulus of the poroelastic material dependent upon the initial porosity field and we determine the impact of the heterogeneity on the kinematic variables (fluid velocity and solid displacement) and on the associated load distribution (pressure and stresses). We first show how the pressure and the strain fields are related to the permeability and the elastic properties solving the proper one dimensional problem and evaluating the the lubrication limit in the small Reynolds number regime. Then we compute the full 2D fields by means of finite element simulations.

CT06 Porous media, Hillhead 5, Wednesday

Jack Thomas (Warwick)

Tight Binding Models for Insulators: locality of interatomic forces & geometry optimisation

The tight binding model is a minimalistic electronic structure model for predicting properties of materials and molecules. We show that the potential energy surface of this model can be decomposed into exponentially localised site energy contributions, thus providing qualitatively sharp estimates on the interatomic interaction range which justifies a range of multi-scale models. For insulators, we demonstrate that “pollution” of the spectral gap by a point spectrum caused, for example, by local defects in the crystal, only weakly affects the locality estimates. Numerical tests confirm our analytical results. Time permitting, we shall extend our results to a self-consistent (non-linear) tight binding model.

We then describe atomistic geometry relaxation for point defects in the tight binding model. A limiting model as Fermi-temperature is sent to zero is formulated, and an exponential rate of convergence for the nuclei configuration is established. Finally, we discuss the non-trivial relationship between taking zero temperature and thermodynamic limits in the finite Fermi-temperature models.

This work extends [Chen, Ortner. Multiscale Model. Simul., 2016] and [Chen, Lu, Ortner. Arch. Rational Mech. Anal., 2018] to the case of zero Fermi-temperature as well as strengthening the locality results proved therein.

Joint work with Christoph Ortner and Huajie Chen.

CT17 Solid Mechanics, Hillhead 5, Thursday

Miloslav Torda (Liverpool)

Geometry of the n-torus stochastic trust region method for materials discovery.

Crystal structure prediction (CSP) is a widely used approach in materials science to guide new materials discovery. Current methods of CSP involve global searches in the space crystals. Given pressure-temperature conditions and a crystallographic space group the goal is to find a structure that minimizes lattice energy often represented as force fields that give rise to complicated energy landscapes with many basins of attraction. We propose a novel global search method based on stochastic trust region approach, as a variant of the natural gradient descent that exploits the structure of crystallographic space groups. We propose a parametric family of probability distributions defined on a n-dimensional torus, called extended multivariate von Mises distribution, that has a structure of a statistical manifold, and perform the search on this functional space. To further improve the performance of the stochastic trust region method we define an adaptive selection quantile and show a connection to the simulated annealing approach where the stochastic trust region method could be regarded as running multiple simulated annealing schedules in parallel. We examine the geometry of the stochastic trust region method with the adaptive selection quantile and show that it is a form of entropic proximal algorithm where the kernel in the infimal convolution is given by Kullback-Leibler divergence where in fact a minimization of Kullback-Leibler divergence between two statistical manifolds is performed in parallel. As a proof of concept we apply our method to the problem of finding densest packings of convex polygons in 2-dimensional space groups that is closely related to the crystal structure prediction problem.

Matt Tranter (Nottingham Trent)

Initial-value problem for the Boussinesq-Klein-Gordon and coupled Boussinesq equations

In this talk I will discuss the so-called “zero-mass contradiction” that can emerge when constructing weakly-nonlinear solutions to Boussinesq-type equations for periodic functions on a finite interval. Firstly I will overview the recent results for the Boussinesq-Klein-Gordon equation, where a solution was constructed that takes account of this issue and numerical results will justify this approach. I will then consider the coupled Boussinesq equations. Solutions will be constructed for the cases when the characteristic speeds in the equations are close and when they differ significantly. Numerical results for solitary and cnoidal wave initial conditions will be presented and the energy in the system will be examined. These equations can model the propagation of long nonlinear longitudinal bulk strain waves in a two-layered elastic waveguide with a soft bonding between the layers, where the material in the layers determines the type of equation. This is joint work with Karima Khusnutdinova.

Linear system identification by feedback control

A problem of restoration of time-dependent coefficients of a linear ordinary differential equation, based on the noised output measurements, is considered. This problem is ill-posed and needs a regularized solution. In the proposed algorithm, the coefficient vector is approximated by an optimal feedback control time realization in the auxiliary tracking problem. Two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields a linear-quadratic optimal control problem having a known explicit solution. Note that these derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter needs a numerical solution of a corresponding Bellman equation. Simulation results show that a bilinear model provides more accurate coefficients estimates.

Using mathematics to understand stem cell pluripotency regulation

Human pluripotent stem cells, hPSCs, hold great promise for developments in regenerative medicine and are at the forefront of modern biological research due to their ability to differentiate into any type of human adult cell (the pluripotency property) and their potential to self-renew indefinitely through repeated divisions. As part of an interdisciplinary team at Newcastle University, I am working to optimise experiments by modelling the behaviour of hPSC colonies using a combination of agent-based and stochastic techniques. Having already modelled colony proliferation to optimise colony clonality experimentally, we now focus on modelling the defining property of stem cells, pluripotency. We use a stochastic logistic equation containing both multiplicative and additive noise to capture the temporal shifts in pluripotency over time as seen experimentally. Analysis of the experimental data also shows pluripotency is anti-persistent, revealing self-regulation in the system. Representative models of individual cell pluripotency further our understanding of the inherent biological behaviours of stem cells, provide the basis for modelling pluripotency spatially across colonies and assist in experimental optimisation.

CT 16 Mathematical Biology-3, Hillhead 4, Thursday

Chen Wang (Exeter)

Instability in shallow-water magnetohydrodynamics with magnetic shear

In this study, we consider the linear instability of shallow water shear flow with a magnetic field. Shallow water MHD was first proposed by Gilman (2000, \textit{J. Astrophys.} 544, L79) as a reduced system for studying certain astrophysical flows, particularly those in the solar tachocline. The basic magnetic field is parallel to the basic flow velocity and has a shear in the cross-stream direction. The flow admits critical levels where the neutral modes become singular at the locations where the phase velocity relative to the basic flow equals the Alfv{\'e}n wave velocity. Previous studies have shown that a constant magnetic field generally has a stabilizing effect (Mak \textit{et al.} 2016, \textit{J.Fluid Mech.} 788, 767). Here we show that in general, this is also the case when the magnetic field varies with space. However, in situations where two critical levels are close to each other, the magnetic field could generate a new instability. The second-order spatial derivative of the magnetic field plays the crucial role, in a mechanism similar to the critical-layer vorticity gradient of hydrodynamic instability.

Danyang Wang (Glasgow)

The energetics of flow in a flexible channel with nonlinear fluid-beam model

We consider flow along a finite-length collapsible channel, where one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to a uniform external pressure. A modified constitutive law is used to ensure that the elastic beam model is both geometrically and materially nonlinear, while remaining energetically conservative. A parabolic inlet flow with constant flux is driven through the channel, with a prescribed fluid pressure at the downstream end. We apply the finite element method with adaptive meshing to numerically solve the fully nonlinear steady and unsteady systems [1]. In line with previous studies, we show that across the parameters investigated the system always has at least one static solution; in addition there is a narrow region of the parameter space where the system exhibits two stable static configurations simultaneously, consisting of an upper branch (where the steady beam is almost entirely inflated) and a lower branch (where the steady beam is collapsed) [2]. These two branches of static solutions are connected in the parameter space by an unstable intermediate branch, joined at a pair of limit point bifurcations. We show that both the upper and lower static configurations can each become unstable to self-excited oscillations, whereas only lower branch instabilities had previously been noted in this fluid-beam system. We consider a detailed study along two slices through the parameter space, showing that for fixed beam elasticity and increasing Reynolds number the upper branch is first to become unstable to oscillations via a supercritcal Hopf bifurcation; these oscillations onset close to, but just outside, the region of parameter space with multiple static states. As the Reynolds number increases further the unstable branch enters the region with multiple steady states and eventually stabilises again very close to the limit point of the upper branch static solutions. As the Reynolds number increases further the lower branch of static solutions becomes unstable to oscillations via a supercritical Hopf bifurcation, again in a region of parameter space close to, but just outside, the region with multiple static states. Furthermore, our new formulation allows us to self-consistently calculate a detailed energy budget over a period of fully developed oscillation. We show that, for both the upper and lower branch instabilities, the oscillation requires an increase in the work done by the upstream pressure to overcome the corresponding increase in dissipation in the oscillatory flow.

[1] Luo, X. Y., Cai, Z. X., Li, W. G., & Pedley, T. J. (2008). The cascade structure of linear instability in collapsible channel flows. Journal of Fluid Mechanics, 600, 45-76.

[2] Stewart, P. S. (2017). Instabilities in flexible channel flow with large external pressure. Journal of Fluid Mechanics, 825, 922-960.

CT17 Solid Mechanics, Hillhead 5, Thursday

Cat Wedderburn (Edinburgh)

Burn, baby, burn: Mathematical firefighting to reduce potential disease spread

The Firefighter game offers a simple, discrete time model for the spread of a perfectly infectious disease and the effect of vaccination. A fire breaks out on a graph at time 0 on a set F of f vertices. At most d non-burning vertices are then defended and can not burn in future. Vertices, once either burning or defended remain so for the rest of the game. At each subsequent time step, the fire spreads deterministically to all neighbouring

undefended vertices and then at most d more vertices can be defended. The game ends when the fire can spread no further. Determining whether k vertices can be saved is NP-complete. I focus on finding maximal minimal damage (mmd) graphs - graphs which have the least burning if the fire starts in the worst place and the defenders defend optimally.

I shall present some new and old results linking mmd graphs to optimal graphs for the Resistance Network Problem of finding graphs where all F-sets of vertices have limited neighbourhoods; a new framework for proving graphs are mmd and a new algorithm for optimal defense of a graph under certain conditions. I shall then present some results using the discharge method on the edge defence version of the Firefighter game and compare them to results in the original game.

CT04 Mathematical Biology-1, Clyde 5, Wednesday

A size-structured filtration model

Filters are used in industry to separate impurities and harmful particulates from solution, with applications ranging from high-volume industrial emissions abatement to the processing of blood samples. W.L. Gore and Associates supply businesses with particle filtration products. These are constructed from fibres that form a complex network of pores, within which particles can become trapped as the filtration process progresses. However, as the pores capture contaminants, they become blocked. The filters gradually clog, preventing further particle removal. Clogging greatly increases downtime and running costs of filtration processes, leading to decreased overall productivity. It also often leads to filter disposal, to the detriment of the environment.

We will present a size-structured mathematical model that predicts contaminant removal and clogging. In our model, we track the concentration distribution of particles and pores. The model comprises two coupled integro-partial-differential equations for the particles and pores, along with Darcy’s equation for the flow. We solve the model numerically in a number of scenarios, and compare and contrast the results.

CT08 Industrial fluids, Bute 5, Wednesday

Christophe Wilk (Manchester)

Wetting and dewetting dynamics of a thin liquid film spreading on an immiscible liquid surface

Macroscopic thin liquid films are present in many processes ranging from coating flow technology to biological systems such as fluid lining in the pulmonary airways [1] or in the formation of tear film. Thin liquid films also provide a model mesoscopic system to study directly the effects of fundamental scientific issues such as intermolecular forces. In this study, experiments on the spreading of surfactant-laden immiscible droplets over liquid substrates demonstrated a striking and unusual dewetting instability [2]. As the film spreads over the liquid substrate due to Marangoni forces, it reaches, at least locally, a critical thickness under which intermolecular forces becomes important [3]. Then, a van der Waals induced thinning process leads to the appearance and growth of holes (regions of minimal film thickness) at the surface of the spreading liquid film while its leading edge is still expanding. After the end of the spreading phase, these apparent holes continue to form and grow. The interplay between Marangoni and intermolecular forces enables the system to wet and apparently dewet the substrate at the same time, which seems counter-intuitive. We have studied the system both experimentally and theoretically. Analysis of the experimental data shows that the spreading is driven by a Marangoni flow and that the dewetting velocity (the rate at which the holes grow) is constant across the film interface. The pattern of holes formed in the process was analysed using Minkowski functionals which showed that holes nucleate following random processes rather than spinodal dewetting, in agreement with past literature on solid substrate configurations [4]. Our model, based on the lubrication approximation [5], attempts to explain the key physical mechanisms responsible for the dewetting phenomena observed experimentally. Linear stability analysis of a system consisting of two superposed layers, a thin film on a thick substrate, of immiscible liquids resting on a solid substrate is performed [6]. The model includes intermolecular forces (with a long-range attractive component and a short-range repulsive one), capillary forces, and insoluble surfactant in the film. We investigate the most unstable modes to identify the key parameters that can lead to the dewetting phenomena observed experimentally. We discuss our model predictions in comparison with our experimental findings.

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Bibliography:

[1] Jensen and Grotberg, J. Fluid Mech. 240, 259 (1992).

[2] Shahidzadeh et al., Phys. Rev. E. 64, 021911 (2001).

[3] Sharma and Bandyopadhyay, J.Chem. Phys. 125, 054711 (2006).

[4] Becker et al., Nat. Mater. 2, 59 (2003).

[5] Craster and Matar, Langmuir 23, 5, 2588-2601 (2007).

[6] Karapetsas et al., Phys. Fluids 23, 122106 (2011).

CT08 Industrial fluids, Bute 5, Wednesday

Exponential synchronization of a class of fractional order complex chaotic systems and application through digital cryptography

In this article, exponential synchronization between a class of fractional-order chaotic systems has been studied. The exponential synchronization is analyzed by using exponential stability theorem for the fractional-order system, and the stability analysis has been done with the help of a new lemma, which is given for the Lyapunov function for the fractional-order system. The fractional-order Lorenz and Lu complex chaotic systems are considered to illustrate the exponential synchronization. The numerical simulations and graphical results are also presented to verify the effectiveness and reliability of exponential synchronization. The application in communication through digital cryptography is also discussed between the sender (transmitter) and receiver using the exponential synchronization. A well-secured key system of a message is obtained in a systematic and very simple way.

CT13 Dynamical Systems, Clyde 4, Thursday

Anthony Yeates (Durham)

Revisiting Taylor relaxation

Turbulent magnetic relaxation is an important candidate mechanism for coronal heating and some types of solar flare. By developing turbulence that reconnects the magnetic field throughout a large volume, magnetic fields can spontaneously self-organize into simpler lower-energy configurations. We are using resistive MHD simulations to probe this relaxation process, in particular to test whether a linear force-free equilibrium is reached. Such an end state would be predicted if one assumes the classic Taylor hypothesis: that the only constraints on the relaxation come from conservation of total magnetic flux and helicity. In fact, a linear force-free state is not reached in our simulations, despite the conservation of these total quantities. Instead, the end state is better characterised as a state of (locally) uniform field-line helicity.