L3

The study of moving contact lines is challenging for various reasons: Physically no sliding motion is allowed with a standard no-slip boundary condition over a solid substrate. Mathematically one has to deal with a free-boundary problem which contains certain singularities at the contact line. Instabilities can lead to topological transition in configurations space - their rigorous mathematical understanding is highly non-trivial. In this talk some state-of-the-art modeling and numerical techniques for such challenges will be presented. These will be applied to flows over solid and liquid substrates, where we perform detailed comparisons with experiments.

Modelling catalytic kinetics is indispensable for the design of reactors and chemical processes. However, developing accurate and computationally efficient kinetic models remains challenging. Empirical kinetic models incorporate assumptions about rate-limiting steps and may thus not be applicable to operating regimes far from those where they were derived. Detailed microkinetic modelling approaches overcome this issue by accounting for all elementary steps of a reaction mechanism. However, the majority of such kinetic models employ mean-field approximations and are formulated as ordinary differential equations, which neglect spatial correlations. On the other hand, kinetic Monte Carlo (KMC) approaches provide a discrete-space continuous-time stochastic formulation that enables a detailed treatment of spatial correlations in the adlayer (resulting for instance from adsorbate-adsorbate lateral interactions), but at a significant computation expense.1,2

Motivated by these challenges, we discuss the necessity of KMC descriptions that incorporate detailed models of lateral interactions. Focusing on a titration experiment involving the oxidation of pre-adsorbed O by CO gas on Pd(111), we discuss experimental findings that show first order kinetics at low temperature (190 K) and half order kinetics at high temperature (320 K), the latter previously attributed to island formation.3 We perform KMC simulations whereby coverage effects on reaction barriers are captured by cluster expansion Hamiltonians and Brønsted-Evans-Polanyi (BEP) relations.4 By quantifying the effect of adlayer structure versus coverage effects on the observed kinetics, we rationalise the experimentally observed kinetics. We show that coverage effects lead to the half order kinetics at 320 K, rather than O-island formation as previously thought.5,6

Subsequently, we discuss our ongoing work in the development of approximations that capture such coverage effects but are much more computationally efficient than KMC, making it possible to use such models in reactor design. We focus on a model for NO oxidation incorporating first nearest neighbour lateral interactions and construct a sequence of approximations of progressively higher accuracy, starting from the mean-field treatment and continuing with a sequence of Bethe-Peierls models with increasing cluster sizes. By comparing the turnover frequencies of these models with those obtained from KMC simulation, we show that the mean-field predictions deviate by several orders of magnitude from the KMC results, whereas the Bethe-Peierls models exhibit progressively higher accuracy as the size of the explicitly treated cluster increases. While more computationally intensive than mean-field, these models still enable significant computational savings compared to a KMC simulation, thereby paving the road for employing them in multiscale modelling frameworks.

References

1    M. Stamatakis and D. G. Vlachos, ACS Catal. 2 (12), 2648 (2012).

2    M. Stamatakis, J Phys-Condens Mat 27 (1), 013001 (2015).

3    I. Nakai, H. Kondoh, T. Shimada, A. Resta, J. N. Andersen, and T. Ohta, J. Chem. Phys. 124 (22), 224712 (2006).

4    J. Nielsen, M. d’Avezac, J. Hetherington, and M. Stamatakis, J. Chem. Phys. 139 (22), 224706 (2013).

5    M. Stamatakis and S. Piccinin, ACS Catal. 6 (3), 2105 (2016).

6    S. Piccinin and M. Stamatakis, ACS Catal. 4, 2143 (2014).

Graham Benham

The Fluid Mechanics of Low-Head Hydropower Illuminated by Particle Image Velocimetry

We study a new type of hydropower which is cost-effective in rivers and tides where there are small pressure drops. The concept goes as follows: The cost of water turbines scales with the flow rate they deal with.  Therefore, in order to render this hydropower desirable, we make use of the Venturi principle, a natural fluid mechanical gear system which involves splitting the flow into two streams. The turbine deals with a small fraction of the flow at slow speed and high pressure, whilst the majority avoids the turbine, going at high speed and low pressure. Now the turbine feels an amplified pressure drop, thus maintaining its power output, whilst becoming much cheaper. But it turns out that the efficiency of the whole system depends strongly on the way in which these streams mix back together again.

Here we discuss some new experimental results and compare them to a simplified mathematical model for the mixing of these streams. The experimental results were achieved using particle image velocimetry (PIV), which is a type of flow visualisation. Using a laser sheet and a high speed camera, we are able to capture flow velocity fields at high resolution. Pressure measurements were also taken. The mathematical model is derived from the Navier Stokes equations using boundary layer theory alongside a flow-averaging method and reduces the problem to solving a set of ODE’s for the bulk components of the flow.

Nabil Fadai

Asymptotic Analysis of a Multiphase Drying Model Motivated by Coffee Bean Roasting

Recent modelling of coffee bean roasting suggests that in the early stages of roasting, within each coffee bean, there are two emergent regions: a dried outer region and a saturated interior region. The two regions are separated by a transition layer (or drying front). In this talk, we consider the asymptotic analysis of a multiphase model of this roasting process which was recently put forth and studied numerically, in order to gain a better understanding of its salient features. The model consists of a PDE system governing the thermal, moisture, and gas pressure profiles throughout the interior of the bean. Obtaining asymptotic expansions for these quantities in relevant limits of the physical parameters, we are able to determine the qualitative behaviour of the outer and interior regions, as well as the dynamics of the drying front. Although a number of simplifications and scaling are used, we take care not to discard aspects of the model which are fundamental to the roasting process. Indeed, we find that for all of the asymptotic limits considered, our approximate solutions faithfully reproduce the qualitative features evident from numerical simulations of the full model. From these asymptotic results we have a better qualitative understanding of the drying front (which is hard to resolve precisely in numerical simulations), and hence of the various mechanisms at play as heating, evaporation, and pressure changes result in a roasted bean. This qualitative understanding of solutions to the multiphase model is essential if one is to create more involved models that incorporate chemical reactions and solid mechanics effects.

We propose several optimal preconditioners for systems defined by some functions $g$ of Toeplitz matrices $T_n$. In this paper we are interested in solving $g(T_n)x=b$ by the preconditioned conjugate method or the preconditioned minimal residual method, namely in the cases when $g(T_n)$ are the analytic functions $e^{T_n}$, $\sin{T_n}$ and $\cos{T_n}$. Numerical results are given to show the effectiveness of the proposed preconditioners.

Kerdock matrices are an attractive choice as deterministic measurement matrices for compressive sensing. I'll explain how Kerdock matrices are constructed, and then show how they can be adapted to one particular  strategy for quantizing measurements, in which measurements exceeding the desired dynamic range are rejected.

The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of N scalars with a random potential of fourth order. We show that the scaling of the typical minimal bounce action with N is readily understood. We discuss the extension to more realistic landscape models as well as the effects of gravity. 

Given some class of "geometric spaces", we can make a ring as follows. Additive structure: when U is an open subset a space X,  [X] = [U] + [X - U]. Multiplicative structure:  [X][Y] = [XxY]. In the algebraic setting, this ring (the "Grothendieck ring of varieties") contains surprising structure, connecting geometry to arithmetic and topology.  I will discuss some remarkable
statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural).  A motivating example will be polynomials in one variable. This is joint work with Melanie Matchett Wood.

The Faraday cage effect is the phenomenon whereby electrostatic and electromagnetic fields are shielded by a wire mesh "cage". Nick Trefethen, Jon Chapman and I recently carried out a mathematical analysis of the two-dimensional electrostatic problem with thin circular wires, demonstrating that the shielding effect is not as strong as one might infer from the physics literature. In this talk I will present new results generalising the previous analysis to the electromagnetic case, and to wires of arbitrary shape. The main analytical tool is the asymptotic method of multiple scales, which is used to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. In the electromagnetic case one observes interesting resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells. This is joint work with Ian Hewitt.

Fluids and solids leave our bodies everyday.  How do animals do it, from mice to elephants?  In this talk, I will show how the shape of urinary and digestive organs enable them to function, regardless of the size of the animal.  Such ideas may teach us how to more efficiently transport materials.  I will show how the pee-pee pipe enables animals to urinate in constant time, how slippery mucus is critical for defecation, and how the motion of the gut is related to the density of its contents, and in turn to the gut’s natural frequency. 

More info is in the BBC news here: http://www.bbc.com/news/science-environment-34278595