Past Industrial and Applied Mathematics Seminar

10 November 2016
OCIAM Group Meeting

Ousman Kodio

Lubricated wrinkles: imposed constraints affect the dynamics of wrinkle coarsening

We investigate the problem of an elastic beam above a thin viscous layer. The beam is subjected to
a fixed end-to-end displacement, which will ultimately cause it to adopt the Euler-buckled
state. However, additional liquid must be drawn in to allow this buckling. In the interim, the beam
forms a wrinkled state with wrinkles coarsening over time. This problem has been studied
experimentally by Vandeparre \textit{et al.~Soft Matter} (2010), who provides a scaling argument
suggesting that the wavelength, $\lambda$, of the wrinkles grows according to $\lambda\sim t^{1/6}$.
However, a more detailed theoretical analysis shows that, in fact, $\lambda\sim(t/\log t)^{1/6}$.
We present numerical results to confirm this and show that this result provides a better account of
previous experiments.


Edward Rolls

Multiscale modelling of polymer dynamics: applications to DNA

We are interested in generalising existing polymer dynamics models which are applicable to DNA into multiscale models. We do this by simulating localized regions of a polymer chain with high spatial and temporal resolution, while using a coarser modelling approach to describe the rest of the polymer chain in order to increase computational speeds. The simulation maintains key macroscale properties for the entire polymer. We study the Rouse model, which describes a polymer chain of beads connected by springs by developing a numerical scheme which considers the a filament with varying spring constants as well as different timesteps to advance the positions of different beads, in order to extend the Rouse model to a multiscale model. This is applied directly to a binding model of a protein to a DNA filament. We will also discuss other polymer models and how it might be possible to introduce multiscale modelling to them.

  • Industrial and Applied Mathematics Seminar
3 November 2016
Henry Schenck

One of the fundamental tools in numerical analysis and PDE
is the finite element method (FEM). A main ingredient in
FEM are splines: piecewise polynomial functions on a
mesh. Even for a fixed mesh in the plane, there are many open
questions about splines: for a triangular mesh T and
smoothness order one, the dimension of the vector space
  C^1_3(T) of splines of polynomial degree at most three
is unknown. In 1973, Gil Strang conjectured a formula
for the dimension of the space C^1_2(T) in terms of the
combinatorics and geometry of the mesh T, and in 1987 Lou
Billera used algebraic topology to prove the conjecture
(and win the Fulkerson prize). I'll describe recent progress
on the study of spline spaces, including a quick and self
contained introduction to some basic but quite useful tools
from topology.

  • Industrial and Applied Mathematics Seminar
27 October 2016

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.

  • Industrial and Applied Mathematics Seminar
20 October 2016
Michail Stamatakis

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.


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).

  • Industrial and Applied Mathematics Seminar
13 October 2016
Graham Benham, Nabil Fadai

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.

  • Industrial and Applied Mathematics Seminar
16 June 2016
Suzy Moat

Our everyday usage of the Internet generates huge amounts of data on how humans collect and exchange information worldwide. In this talk, I will outline recent work in which we investigate whether data from sources such as Google, Wikipedia and Flickr can be used to gain new insight into real world human behaviour. I will provide case studies from a range of domains, including disease detection, crowd size estimation, and evaluating whether the beauty of the environment we live in might affect our health.

  • Industrial and Applied Mathematics Seminar
9 June 2016
Javier Buldu, Dave Hewett

Dave Hewett: Canonical solutions in wave scattering

By a "canonical solution" I have in mind a closed-form exact solution of the scalar wave equation in a simple geometry, for example the exterior of a circular cylinder, or the exterior of an infinite wedge. In this talk I hope to convince you that the study of such problems is (a) interesting; (b) important; and (c) a rich source of (difficult) open problems involving eigenfunction expansions, special functions, the asymptotic evaluation of integrals, and matched asymptotic expansions.


  • Industrial and Applied Mathematics Seminar
2 June 2016
Shailesh Naire

Surfactants are chemicals that adsorb onto the air-liquid interface and lower the surface tension there. Non-uniformities in surfactant concentration result in surface tension gradients leading to a surface shear stress, known as a Marangoni stress. This stress, if sufficiently large, can influence the flow at the interface.

Surfactants are ubiquitous in many aspects of technology and industry to control the wetting properties of liquids due to  their ability to modify surface tension. They are used in detergents, crop spraying, coating processes and oil recovery. Surfactants also occur naturally, for example in the mammalian lung. They reduce the surface tension within the liquid lining the airways, which assists in preventing the collapse of the smaller airways. In the lungs of premature infants, the quantity of surfactant produced is insufficient as the lungs are under- developed. This leads to a respiratory distress syndrome which is treated by Surfactant Replacement Therapy.

Motivated by this medical application, we theoretically investigate a model problem involving the spreading of a drop laden with an insoluble surfactant down an inclined and pre-wetted substrate.  Our focus is in understanding the mechanisms behind a “fingering” instability observed experimentally during the spreading process. High-resolution numerics reveal a multi-region asymptotic wave-like structure of the spreading droplet. Approximate solutions for each region is then derived using asymptotic analysis. In particular, a quasi-steady similarity solution is obtained for the leading edge of the droplet. A linear stability analysis of this region shows that the base state is linearly unstable to long-wavelength perturbations. The Marangoni effect is shown to be the dominant driving mechanism behind this instability at small wavenumbers. A small wavenumber stability criterion is derived and it's implication on the onset of the fingering instability will be discussed.

  • Industrial and Applied Mathematics Seminar
26 May 2016
Mason Porter, Robert Van Gorder

A Simple Generative Model of Collective Online Behavior (Mason Porter)

Human activities increasingly take place in online environments, providing novel opportunities for relating individual behaviors to population-level outcomes. In this paper, we introduce a simple generative model for the collective behavior of millions of social networking site users who are deciding between different software applications. Our model incorporates two distinct mechanisms: one is associated with recent decisions of users, and the other reflects the cumulative popularity of each application. Importantly, although various combinations of the two mechanisms yield long-time behav- ior that is consistent with data, the only models that reproduce the observed temporal dynamics are those that strongly emphasize the recent popularity of applications over their cumulative popularity.

This demonstrates --- even when using purely observational data with- out experimental design --- that temporal data-driven modeling can effectively distinguish between competing microscopic mechanisms, allowing us to uncover previously unidentified aspects of collective online behavior.


Bubbles, Turing machines, and possible routes to Navier-Stokes blow-up (Robert van Gorder)

Navier-Stokes existence and regularity in three spatial dimensions for an incompressible fluid... is hard. Indeed, while the original equations date back to the 1840's, existence and regularity remains an open problem and is one of the six remaining Millennium Prize Problems in mathematics that were stated by the Clay Mathematics Institute in 2000. Despite the difficulty, a resolution to this problem may say little about real-world fluids, as many real fluid problems do not seem to blow-up, anyway.
In this talk, we shall briefly outline the mathematical problem, although our focus shall be on the negative direction; in particular, we focus on the possibility of blow-up solutions. We show that many existing blow-up solutions require infinite energy initially, which is unreasonable. Therefore, obtaining a blow-up solution that starts out with nice properties such as bounded energy on three dimensional Euclidean space is rather challenging. However, if we modify the problem, there are some results. We survey recent results on averaged Navier-Stokes equations and compressible Navier-Stokes equations, and this will take us anywhere from bubbles to fluid Turing machines. We discuss how such results might give insight into the loss of regularity in the incompressible case (or, insight into how hard it might be to loose regularity of solutions when starting with finite energy in the incompressible case), before philosophizing about whether mathematical blow-up solutions could ever be physically relevant.

  • Industrial and Applied Mathematics Seminar
19 May 2016

Permanent deformations of crystalline materials are known to be carried out by a large
number of atomistic line defects, i.e. dislocations. For specimens on micron scales or above, it
is more computationally tractable to investigate macroscopic material properties based on the
evolution of underlying dislocation densities. However, classical models of dislocation
continua struggle to resolve short-range elastic interactions of dislocations, which are believed
responsible for the formation of various heterogeneous dislocation substructures in crystals. In
this talk, we start with discussion on formulating the collective behaviour of a row of
dislocation dipoles, which would be considered equivalent to a dislocation-free state in
classical continuum models. It is shown that the underlying discrete dislocation dynamics can
be asymptotically captured by a set of evolution equations for dislocation densities along with
a set of equilibrium equations for variables characterising the self-sustained dislocation
substructures residing on a shorter length scale, and the strength of the dislocation
substructures is associated with the solvability conditions of their governing equilibrium
equations. Under the same strategy, a (continuum) flow stress formula for multi-slip systems
is also derived, and the formula resolves more details from the underlying dynamics than the
ubiquitously adopted Taylor-type formulae.

  • Industrial and Applied Mathematics Seminar