Wednesday, 3 January 2018

Why your morning cup of coffee sloshes

Americans drink an average of 3.1 cups of coffee per day (and mathematicans probably even more). When carrying a liquid, common sense says walk slowly and refrain from overfilling the container. But easier said than followed. Cue sloshing.

Sloshing occurs when a vessel of liquid—coffee in a mug, water in a bucket, liquid natural gas in a tanker, etc. - oscillates horizontally around a fixed position near a resonant frequency; this motion occurs when the containers are carried or moved. While nearly all transport containers have rigid handles, a bucket with a pivoted handle allows rotation around a central axis and greatly reduces the chances of spilling. Although this is not necessarily a realistic on-the-go solution for most beverages, the mitigation or elimination of sloshing is certainly desirable. In a recent article published in SIAM Review, Oxford Mathematicians Hilary and John Ockendon use surprisingly simple mathematics to develop a model for sloshing. Their model comprises a mug on a smooth horizontal table that oscillates in a single direction via a spring connection. “We chose the mathematically simplest model with which to understand the basic mechanics of pendulum action on sloshing problems,” John said.

The authors derive their inspiration from a Nobel prize-winning paper describing a basic mechanical model that investigates the results of walking backwards while carrying a cup of coffee. They use both Newton’s laws of physics and the basic properties of hydrodynamics to employ a so-called “paradigm” configuration, which explains how a cradle introduces an extra degree of freedom that in turn modifies the liquid’s response. “The paradigm model contains the same mechanics as the pendulum but is simpler to write down,” Ockendon said. “We found some experimental results on the paradigm model, which meant we could make some direct comparisons.”

The authors evaluate this scenario rather than the more realistic but complicated use of a mug as a cradle that moves like a simple pendulum. To further simplify their model, they assume that the mug in question is rectangular and engaged in two-dimensional motion, i.e., motion perpendicular to the direction of the spring’s action is absent. Because the coffee is initially at rest, the flow is always irrotational. “Our model considers sloshing in a tank suspended from a pivot that oscillates horizontally at a frequency close to the lowest sloshing frequency of the liquid in the tank. Together we have written several papers on classical sloshing over the last 40 years, but only recently were we stimulated by these observations to consider the pendulum effect.”

Variables in the initial model represent (i) a hand moving around a fixed position, (ii) the frequency of walking, typically between 1-2 Hertz, and (iii) a spring connecting the shaking hand to the mug, which slides on the table’s smooth surface. Hilary and John are most interested in the spring’s effect on the motion of the liquid. 

The authors solve the model’s equations via separation of variables and analyse the subsequent result with a response diagram depicting the sloshing amplitude’s dependence on forcing frequency. The mug’s boundary conditions assume that the normal velocity of both the liquid and the mug are the same, and that the oscillation’s amplitude is small. Hilary and John linearize the boundary conditions to avoid solving a nonlinear free boundary problem with no explicit solution. They record the equation of motion for the container to couple the motion of the liquid and the spring. In this case, the spring’s tension and the pressure on the walls of the container are the acting horizontal forces.

The authors discover that including a string or a pendulum between the container and the carrying hand (the forcing mechanism) lessens the rigidity and dramatically decreases the lowest resonant frequency, thus diminishing sloshing for almost all frequencies. “Our model shows that, compared to an unpivoted tank, the amplitude of the lowest resonant response will be significantly reduced, provided the length of the pendulum is greater than the length of the tank.” 

In conclusion, Hilary and John use simplistic modeling and analysis to explain a common phenomenon that nearly everybody experiences. They suggest that future analysts investigate sloshing in a cylindrical rather than rectangular mug, or with vertical rather than horizontal oscillations, as both of these factors complicate the model. One could also examine the spring action’s effect on the system’s nonlinear behavior near resonance. Ultimately, researchers can employ basic ideas from this study to consider the nonlinear response of shallow water sloshing, which has a variety of real-world applications.

A version of this article by Linda Sorg first appeared in SIAM news.

Sunday, 31 December 2017

What do fireflies and viruses have in common?

Oxford Mathematician Soumya Banerjee talks about his current work in progress.

"On warm summer days, fireflies mesmerise us with their glowing lights. They produce this cold light using a light-emitting molecule, the luciferin, and a complementary enzyme, luciferase. This process is known as bioluminescence.

Scientists have now genetically engineered this process in viruses. After infecting cells, these modified viruses (called replicons) produce light using the firefly genes. Recently this was used to study how West Nile virus (similar to Zika virus) infects cells.

I have now developed a mathematical model to analyse how firefly genes produce luminescence in these virus infected cells. The model predicts how the luminescence or brightness would gradually decrease as the cells infected by the virus are slowly killed off. This was then matched to experimental data.

The mathematical models predict that some cells in the lymph node live for about 12 hours after being infected with the virus. These cells also release the most virus into the blood. The work suggests that these particular cell types can be targeted using therapies such as anti-viral medication to fight the infection."

You can read more about the work which is at pre-publication stage here.

Tuesday, 19 December 2017

How understanding Oscillator Networks could help unlock the secrets of brain diseases

Oxford Mathematician Christian Bick talks about his and colleagues' research into oscillator networks and how it could be valuable in understanding diseases such as Parkinson's.

"Many systems that govern crucial aspects of our lives can be seen as networks of interacting oscillators. On a small scale, for example, the human brain consists of individual cells that can send bits of information to each other periodically. On a large scale, the power grid of an entire country can be seen as a network of rotating units (generators and motors). The function of these oscillator networks crucially depends on how the units evolve together. For a power grid to be stable it needs to be synchronized to a common grid frequency—think of the 50Hz coming out of  your power outlets at home. By contrast, too much synchronization in the brain is believed to be detrimental as it has been associated with a range of disorders such as epilepsy and Parkinson’s disease.

Synchronization, where distinct units will behave in the same way as time goes on, can arise spontaneously in networks of interacting oscillators. Already Christian Huygens noticed in 1665 that two of his oscillating pendulum clocks showed an “odd kind of sympathy” when they were allowed to interact: they would swing in unison after some time. Mathematical models can help understand how network interactions—the coupling between oscillators—allow for synchronization to arise in oscillator networks.

In contrast to synchronized dynamics, we are interested in networks where identical oscillators can show distinct dynamics in the following sense. Take a collection of oscillators which would oscillate at the same frequency in isolation. Now when you make these oscillators interact in a particular way, then half of the oscillators will be oscillating at one frequency while the other half is oscillating at another frequency! In other words, there is a separation of frequencies that purely arise due to the network interactions. In work with collaborators at St. Louis University we studied this phenomenon in a network of four oscillators. We first analyzed the effect in a mathematical model. Guided by the theoretical work, we then demonstrated the same effect in an experimental setup of electrochemical oscillators with suitable network interactions.

Our results shed further light on the effect the network structure has on the dynamics of interacting oscillators. In particular, they indicate what network interactions allow for dynamics where only part of the oscillators are synchronized in frequency. Hence, if neural disease is indeed related to abnormal levels of synchrony, these insights could, for example, be useful in devising means to prevent or counteract pathological synchronization (as in Parkinson's disease) by tuning the network interactions."

Thursday, 14 December 2017

Alex Bellos Oxford Mathematics Christmas Public Lecture now online

In our Oxford Mathematics Christmas Public Lecture Alex Bellos challenges you with some festive brainteasers as he tells the story of mathematical puzzles from the Middle Ages to modern day.

Alex is the Guardian’s puzzle blogger as well as the author of several works of popular maths, including Puzzle Ninja, Can You Solve My Problems? and Alex’s Adventures in Numberland.








Monday, 11 December 2017

Philip Maini honoured by Indian National Science Academy

Oxford Mathematician Professor Philip Maini FRS, Professorial Fellow in Mathematical Biology at St John’s College has been elected a Foreign Fellow of the Indian National Science Academy for his mathematical and computational modelling of biological processes relevant to wound healing and vascular tumour growth, scar formation and cancer therapy. Philip's previous work has included influencing HIV/AIDS policy in India through mathematical modelling. The election will be effective from 1 January 2018.

Wednesday, 6 December 2017

Oxford Mathematics Virtual Open Day for Masters' Courses, TODAY, Thursday 7 December, 3pm

Today, Thursday 7th December 2017, Oxford Mathematics will be holding its second Graduate Virtual Open Day, from 15:00-16:00 (UK time). This year, the Virtual Open Day will be focusing on taught masters' courses offered at the Mathematical Institute, which will include the following degrees:

MSc Mathematical and Computational Finance
MSc Mathematics and Foundations of Computer Science
MSc Mathematical Modelling and Scientific Computing
MSc Mathematical Sciences
MSc Mathematical and Theoretical Physics

This will be an interactive livestreamed event, where members of faculty will be providing information on the courses mentioned above and also will be answering your queries. If you are a prospective applicant, please e-mail your questions to and tweet them to @OxUniMaths and we will attempt to answer as many of these questions during the hour as possible.


Monday, 4 December 2017

Andrew Wiles London Public Lecture now online

In the first Oxford Mathematics London Public Lecture, in partnership with the Science Museum, world-renowned mathematician Andrew Wiles lectured on his current work around Elliptic Curves followed by an-depth conversation with mathematician and broadcaster Hannah Fry.

In a fascinating interview Andrew talked about his own motivations, his belief in the importance of struggle and resilience and his recipe for the better teaching of his subject, a subject he clearly loves deeply.









Friday, 1 December 2017

Modelling outbreaks of infectious disease - diagnostic tests key for epidemic forecasting

Precise forecasting in the first few days of an infectious disease outbreak is challenging. However, Oxford Mathematical Biologist Robin Thompson and colleagues at Cambridge University have used mathematical modelling to show that for accurate epidemic prediction, it is necessary to develop and deploy diagnostic tests that can distinguish between hosts that are healthy and those that are infected but not yet showing symptoms. The data derived from these tests must then be integrated into epidemic models.

“We used Ebola virus disease as the main case study in this paper" says co-author Nik Cunniffe, "since at the date of publication it was an important and very timely example of the type of disease we focus on in our research, i.e. one for which reporting is incomplete and where some epidemics die out naturally before infecting a large number of people.”

Robin is currently working on a probabilistic modelling framework for managing outbreaks of diseases such as bovine tuberculosis and foot-and-mouth disease: “I am investigating the optimal time to introduce control of a newly invading pathogen. Early control can be beneficial since the outbreak might be suppressed before the pathogen sweeps through the population. However, later control carries the advantage that it allows transmission parameters to be estimated more accurately and interventions to be optimised. Deciding when to initiate control is therefore an optimal stopping problem, and involves balancing the benefits of waiting against the potential costs of the pathogen becoming widespread.”

The team have won the PLoS Computational Biology Research Prize 2017 for the public impact of their work about diagnostic testing for Ebola. Robin is now a Junior Research Fellow at Christ Church in Oxford. He undertook this project as part of his PhD studies in Cambridge.

Wednesday, 29 November 2017

Assessing the impact of local planning on housing delivery and affordability

The investment decisions made by the construction sector have an obvious impact on the supply of housing. Furthermore, Local Planning Authorities play a fundamental role in shaping this supply via town planning and, in particular, by approving or rejecting planning applications submitted by developers. However, the role of these two factors, as well as their interaction, has so far been largely neglected in models of the housing market. Oxford Mathematicians Adrián Carro and Doyne Farmer, from the Institute for New Economic Thinking at the Oxford Martin School, have been working on a model that tries to capture this interaction. To this end, they have adapted a non-spatial agent-based model of the UK housing market previously developed in collaboration with the Bank of England in order to include all the necessary spatial aspects.

In particular, the new model includes different household types, a banking sector as a mortgage lender, a central government collecting taxes, a central bank setting mortgage regulation, a building sector providing new houses and a set of local governments approving or rejecting planning applications. Furthermore, it models both the sales and the rental market in detail, capturing the interactions between renters and buy-to-let investors. This is the first agent-based model of the housing market to explicitly include a dynamic, endogenous building sector endowed with its own behavioural rules, as well as a set of local governments influencing its activities.

Preliminary results suggest that the relationship between planning application approval rates and housing delivery is highly non-linear. In particular, the effect of a decrease in the approval rate in a certain Local Authority District is, to a certain extent, compensated by an increase in its local prices which encourages the building sector to file more planning applications there. Thus, the loss of housing stock due to a decrease in approval rates, while very significant, is found to be less important than the decrease itself. Finally, our results suggest that the increase in housing and rental prices due to a decrease in approval rates has strong social consequences, pushing a significant fraction of households towards social housing and strongly decreasing home ownership.

Wednesday, 29 November 2017

Developing particle-based software with Aboria

Over the last five decades, software and computation has grown to become integral to the scientific process, for both theory and experimentation. A recent survey of RCUK-funded research being undertaken in 15 Russell Group universities found that 92% of researchers used research software, 67% reported that it was fundamental to their research, and 56% said they developed their own software. As well as the practical use of performing numerical calculations impossible to produce by hand, software is vital for the communication of ideas and methods between scientific disciplines and for knowledge transfer to industry. While traditional scholarly publication can communicate the context, benefits and limitations of a given numerical method, the mathematical and computational details of implementation are often beyond non-specialised users, and software provides a formal language for encoding these ideas in such a way that they can be put to use immediately by potential users.

Biology is one of the many fields that increasingly uses software to inform and test new hypotheses. At smaller length-scales, molecular dynamics is used to model biomolecules to learn more about the structure and functional behaviour. For larger systems, coarse-graining is used to model whole molecules as single particles to study the emergent behaviour of chemical pathways at a sub-cellular level. For whole organs, differential equations are used to model the same pathways taking into account tissue mechanics and structure.

In order to support the wide variety of numerical methods used in biology, Oxford Mathematics researchers Martin Robinson and Maria Bruna have developed Aboria, a high performance software library for particle-based methods. In general, particle-based methods involve the calculation of interactions between particles in dimensional space, where the particles can describe either physical particles (e.g. molecular dynamics), a set of discretization points for solving differential equations (e.g. radial basis functions), or high dimensional data points (e.g. kernel methods in machine learning). Traditional particle-based methods such as molecular dynamics are enabled by complex, highly specialised software packages that are costly to develop and maintain. Within biology in particular, individual particle-based methods often require the development of custom particle interactions that are developed from scratch for each new project, making such high specialised packages unsuitable. Instead, Aboria provides an efficient and easy to use abstraction for the evaluation of both local and long range interactions, while at the same time allowing users to completely specify the nature of the both the particle interactions and how they are integrated over time.

Aboria has previously been used to simulate interacting elliptical particles in a molecular-scale liquid crystal model, diffusion through random porous media, and Brownian particles interacting via soft-sphere potentials. We are currently collaborating with Dyson and Ian Griffiths in Oxford to use Aboria to model how solid particles flow through a filter, and where they are trapped by the filter fibres. For this latter case Aboria is used to not only to evaluate the short-range interactions of particles with the fibres, but also to solve the fluid flow around the fibres, and the long-range electrostatic interactions of the system.

The main image shows a packing of polydisperse spheres using Aboria, where each sphere interacts with the others using a repulsive linear spring force. Each sphere is coloured by its radius.

The image below shows Filter simulation using Aboria. (Left) shows the solid particles in black that move with the flow and diffuse independently. The large coloured circles are the fibres of the filter that capture the solid particles (coloured by number of captured particles). The small red particles show where the solid particles were captured. (Middle) shows computational nodes where fluid flow is calculated, showing the flow inlet at the top, and the outlet at the bottom. (Right) plots the fluid flow at each of the nodes, coloured by velocity magnitude.