Applied Mathematics Seminar

The Applied Mathematics Seminar usually takes place on Thursdays at 12:00 during term time in either Nuffield G13 or Lecture Theatre B Watson.

Autumn 2019

Low-order divergence-free finite element methods

  • Gabriel Barrenechea, University of Strathclyde
  • Thursday 14 November 2019, 13:00
  • Nuffield G13 

In this talk I will review results on a divergence-free reconstruction of the lowest order pair for the Navier-Stokes equation. More precisely, from a stabilised P1xP0 scheme, a divergence-free velocity field is built as the result of a lift of the pressure jumps, and it is then plugged in the convective term of the momentum equation. This process provides a method that can be proven stable without the need to suppose the mesh refined enough. We first apply this idea to the transient Navier-Stokes equations, where estimates independent of the viscosity are derived. Then, the applicability of this idea is extended to a steady-state generalised Boussinesq system.

Continuum Model for the Collective Behaviour of Gyrotactic Micro-Swimmers

  • Lloyd Fung, Imperial College
  • Thursday 07 November 2019, 13:00
  • Nuffield G13

Bottom-heavy motile micro-organisms orient themselves under the influence of gravitational and viscous torque. In a downflowing pipe, the balance between the two torques will cause the micro-organism to swim towards the centre, forming a focused beam-like structure. The structure, known as a gyrotactic plume, may further break down into blips. In this talk, we will discuss different strategies on modelling the suspension of gyrotactic swimmers under the influence of a uni-directional flow, and why a continuum model is needed in the present case. In particular, we will focus on the use and limitation of the Generalised Taylor Dispersion Theory to create a population-level model. The use of Generalised Taylor Dispersion also leads to some newly discovered bifurcations in the plume solution, which may give some new interpretations to the plumes and blips observed in experiments.

Dynamic modulation of glucose utilisation by glucocorticoid rhythms in health and disease

  • Eder Zavala
  • Thursday 31 October 2019, 13:00
  • Nuffield G13

 

Some of the most pressing challenges in healthcare require a dynamic understanding of the cross-talk interactions between endocrine axes. For example, glucocorticoid hormones mediating the stress response are also important for glycemic control and are known to play a role in the development of metabolic conditions such as type 2 diabetes. While high-resolution continuous sampling techniques have revealed circadian and ultradian rhythmicity of stress hormones (e.g., cortisol), little is known about how these rhythms are decoded by peripheral tissues and endocrine organs, or how their dysregulation leads to disease. In particular, the mechanisms by which hypercortisolism can lead to glucose intolerance and insulin resistance are not well understood.

To address this, we propose a mathematical model of glycemic control that accounts for glucocorticoid and insulin antagonism, with an emphasis on the dynamic effects of glucocorticoid pulsatility. The model predicts differential dynamic responses following oral glucose tolerance tests (OGTTs) that depend on ultradian and circadian timing, are modulated by agonists and antagonists of the glucocorticoid receptor, and shows how hypercortisolism disrupts circadian variability in both glucose and insulin responses to OGT Ts. Lastly, we use the model to explore the mechanisms contributing to the development of secondary type 2 diabetes in patients with Cushing’s syndrome. We envisage this class of mathematical models will guide the interpretation of circadian-dependent clinical diagnosis and therapies.

Wetting-dewetting hysteresis in porous and fractured media

  • Ran Holtzman, Coventry University
  • Thursday 24 October 2019, 13:00
  • Nuffield G13

Hysteretic wetting-dewetting cycles are ubiquitous in various natural and industrial applications, from soil moisture, contamination, carbon sequestration and hydrocarbon recovery to filtering, microfluidics and coating processes. Consequently, the relationship between the key variables in continuum models for multiphase flow (e.g. Richards), capillary pressure and saturation, is hysteretic. Despite its importance, existing retention models remain phenomenological.

We derive a physically-sound model, where all parameters have clear physical meaning. We consider quasi-static displacements in a Hele–Shaw cell of random gap spacing (a simple proxy of a rough fracture), resolving the sequence of interfacial equilibrium configurations from a local pressure balance (energy minima). The model is implemented numerically through simple synchronous updating (RFIM-like) rules of extreme computational efficiency, and validated against laboratory experiments of known, controlled microstructure. Our model provides fundamental understanding of the link between pore-scale capillary processes in disordered media, Haines jumps, and sample-scale hysteresis. Our findings offer constitutive relations required for reservoir simulators, and pave the way for upscaling two-phase displacements in porous and fractured materials.

Multiscale modeling and simulation of spatial reaction-diffusion kinetics

  • Andreas Hellander, Uppsala
  • Thursday 10 October 2019, 13:00
  • Nuffield G13

Lunch with the speaker at staff house 12, tea after the seminar

In this talk I will present two recent developments to speed up stochastic reaction-diffusion simulations. We have developed an algorithm coupling mesoscopic simulations on different levels in a hierarchy of Cartesian meshes. Based on the multiscale nature of the chemical reactions, some molecules in the system will live on a fine-grained mesh, while others live on a coarse-grained mesh. By allowing molecules to transfer from the fine levels to the coarse levels when appropriate, we show that we can save up to three orders of magnitude of computational time compared to microscopic simulations or highly resolved mesoscopic simulations, without losing significant accuracy. We demonstrate this in several numerical examples with systems that cannot be accurately simulated with a coarse-grained mesoscopic model. On a coarser level, we also propose a multiscale model where a compartment-based model approximates a detailed spatial stochastic model. The compartment model is constructed via a first-exit times analysis on the spatial model, thus capturing spatial aspects of the fine-grained simulations. We apply the approach to a model of negative feedback gene regulation and evaluate the approximation accuracy over a wide range of parameters, assessing the situations in which a detailed spatial representation can be replaced by the computationally much cheaper compartment multiscale model. This work is motivated by the need for fast approximations of the stochastic dynamics for use in inference problems, and also to be embedded in center-based multicellular models.

Centrosome positioning and asymmetric division in polarized cells: Torque generation in the early C. elegans embryo

  • Adriana Dawes, Ohio State University
  • Thursday 03 October 2019, 13:00
  • Nuffield G13

Lunch with the speaker at staff house 12, tea after the talk

Asymmetric cell division, where daughter cells inherit unequal amounts of specific factors, is critical for development and cell fate specification. Polarized cells, characterized by factors segregated to spatially distinct regions in the cell, often divide asymmetrically by positioning the centrosomes along the polarity axis. Using an individual-based stochastic model of microtubule dynamics nucleated at the centrosomes, and experiments in the early C. elegans embryo, we explore potential sources of cortical force generation and demonstrate the role of both cortical and centrosomal asymmetries for recapitulating the in vivo dynamics and proper positioning of the centrosomes prior to first division.

Find out more

There is a complete list of talks in the Applied Mathematics Seminar on talks@bham.