L3

The challenge is to produce a reduced order model which predicts the maximum temperature rise of a thermally conducting object subjected to a power deposition profile supplied by an external code. The target conducting object is basically cuboidal but with one or more shaped faces and may have complex internal cooling structures, the deposition profile may be time dependent and exhibit hot spots and sharp edged shadows among other features. An additional feature is the importance of radiation which makes the problem nonlinear, and investigation of control strategies is also of interest. Overall there appears to be a sequence of problems of degree of difficulty sufficient to tax the most gifted student, starting with a line profile on a cuboid (quasi-2D) with linearised radiation term, and moving towards increased difficulty.

We present analysis and computations of a non-local thin film model developed by Kalogirou et al (2016) for a perturbed two-layer Couette flow when the thickness of the more viscous fluid layer next to the stationary wall is small compared to the thickness of the less viscous fluid. Travelling wave solutions and their stability are determined numerically, and secondary bifurcation points identified in the process. We also determine regions in parameter space where bistability is observed with two branches being linearly stable at the same time. The travelling wave solutions are mathematically justified through a quasi-solution analysis in a neighbourhood of an empirically constructed approximate solution. This relies in part on precise asymptotics of integrals of Airy functions for large wave numbers. The primary bifurcation about the trivial state is shown rigorously to be supercritical, and the dependence of bifurcation points, as a function of Reynolds number R and the primary wavelength 2πν−1/2 of the disturbance, is determined analytically. We also present recent results on time periodic solutions arising from Hoof-Bifurcation of the primary solution branch.

(This work is in collaboration with D. Papageorgiou & E. Oliveira ) 
 

Optical super-resolution microscopy enables the observations of individual bio-molecules. The arrangement and dynamic behaviour of such molecules is studied to get insights into cellular processes which in turn lead to various application such as treatments for cancer diseases. STFC's Central Laser Facility provides (among other) public access to super-resolution microscope techniques via research grants. The access includes sample preparation, imaging facilities and data analysis support. Data analysis includes single molecule tracking algorithms that produce molecule traces or tracks from time series of molecule observations. While current algorithms are gradually getting away from "connecting the dots" and using probabilistic methods, they often fail to quantify the uncertainties in the results. We have developed a method that samples a probability distribution of tracking solutions using the Metropolis-Hastings algorithm. Such a method can produce likely alternative solutions together with uncertainties in the results. While the method works well for smaller data sets, it is still inefficient for the amount of data that is commonly collected with microscopes. Given the observations of the molecules, tracking solutions are discrete, which gives the proposal distribution of the sampler a peculiar form. In order for the sampler to work efficiently, the proposal density needs to be well designed. We will discuss the properties of tracking solutions and the problems of the proposal function design from the point of view of discrete mathematics, specifically in terms of graphs. Can mathematical theory help to design a efficient proposal function?

Here is the scientific program.

Keynote speakers:

Roger Kamm, Cecil and Ida Green Distinguished Professor of Biological and Mechanical Engineering, MIT

Alicia El Haj,  Interdisciplinary Chair of Cell Engineering, Healthcare Technology Institute, University of Birmingham

Invited speakers (confirmed to date):

Davide Ambrosi, Politecnico di Torino, Italy

Anthony Callanan, University of Edinburgh, UK

Ruth Cameron, University of Cambridge, UK

Sonia Contera, University of Oxford, UK

Linda Cummings, New Jersey Institute of Technology, USA

Mohit Dalwadi, University of Oxford, UK

John Dunlop, University of Salzburg, Austria

John King, Nottingham, UK

Nati Korin, Technion, Israel

Catriona Lally, Trinity College Dublin, Ireland

Sandra Loerakker, TU Eindhoven, Netherlands

Ivan Martin, University of Basel, Switzerland

Scott McCue, Queensland University of Technology, Australia

Pierre-Alexis Mouthuy, University of Oxford, UK

Tom Mullin,  University of Oxford, UK

Ramin Nasehi, Politecnico di Milano, Italy

Reuben O'Dea, University of Nottingham, UK

James Oliver, University of Oxford, UK

Ioannis Papantoniou, KU Leuven, Belgium

Ansgar Petersen, Julius Wolf Institute Berlin, Germany

Luigi Preziosi, Politecnico di Torino, Italy

Rebecca Shipley, University College London, UK

Barbara Wagner, Weierstrass Institute for Applied Analysis and Stochastics, Berlin

Cathy Ye, Oxford University, UK

Feihu Zhao, TU Eindhoven, Netherlands

Thin film flows of nematic liquid crystal will be considered, using the Leslie-Ericksen formulation for nematics. Our model can account for variations in substrate anchoring, which may exert a strong influence on patterns that arise in the flow. A number of simulations will be presented using an "in house" code, developed to run on a GPU. Current modeling directions involving flow over interlaced electrodes, so-called "dielectrowetting", will be discussed.

The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry. 

Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. 

By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such meta-stable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.

The problem of a bubble moving steadily in a Hele-Shaw cell goes back to Taylor and Saffman in 1959.  It is analogous to the well-known selection problem for Saffman-Taylor fingers in a Hele-Shaw channel.   We apply techniques in exponential asymptotics to study the bubble problem in the limit of vanishing surface tension, confirming previous numerical results, including a previously predicted surface tension scaling law.  Our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane.