Thu, 10 Mar 2016

16:00 - 17:00
L3

Dynamic homogenisation

Richard Craster
(ICL)
Abstract

The aim of this talk is to describe effective media for wave propagation through periodic, or nearly periodic, composites. Homogenisation methods are well-known and developed for quasi-static and low frequency regimes. The aim here is to move to situations of more practical interest where the frequencies are high, in some sense, and to compare the results of the theory with large scale simulations.

Thu, 28 Jan 2016

16:00 - 17:00
L3

Predictive simulations for optimisation of inhaled drug delivery

Laura Nicolaou
(ICL)
Abstract

Respiratory illnesses, such as asthma and chronic obstructive pulmonary disease, account for one in five deaths worldwide and cost the UK over £6 billion a year. The main form of treatment is via inhaled drug delivery. Typically, however, a low fraction of the inhaled dose reaches the target areas in the lung. Predictive numerical capabilities have the potential for significant impact in the optimisation of pulmonary drug delivery. However, accurate and efficient prediction is challenging due to the complexity of the airway geometries and of the flow in the airways. In addition, geometric variation of the airways across subjects has a pronounced effect on the aerosol deposition. Therefore, an accurate model of respiratory deposition remains a challenge.

High-fidelity simulations of the flow field and prediction of the deposition patterns motivate the use of direct numerical simulations (DNS) in order to resolve the flow. Due to the high grid resolution requirements, it is desirable to adopt an efficient computational strategy. We employ a robust immersed boundary method developed for curvilinear coordinates, which allows the use of structured grids to model the complex patient-specific airways, and can accommodate the inter-subject geometric variations on the same grid. The proposed approach reduces the errors at the boundary and retains the stability guarantees of the original flow solver.

A Lagrangian particle tracking scheme is adopted to model the transport of aerosol particles. In order to characterise deposition, we propose the use of an instantaneous Stokes number based on the local properties of the flow field. The effective Stokes number is then defined as the time-average of the instantaneous value. This effective Stokes number thus encapsulates the flow history and geometric variability. Our results demonstrate that the effective Stokes number can deviate significantly from the reference value based solely on a characteristic flow velocity and length scale. In addition, the effective Stokes number shows a clear correlation with deposition efficiency.

Thu, 30 Apr 2015

16:00 - 17:00
L3

Complex Solutions of the Navier-Stokes Equations

Jonathan Mestel
(ICL)
Abstract

It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane. 

We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.

by Jonathan Mestel and Florencia Boshier

 

Thu, 30 Oct 2014

16:00 - 17:00
L3

Mathematical modelling and numerical simulation of LiFePO4 cathodes

Steven Dargaville
(ICL)
Abstract

LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. During discharge, LiFePO4 material can undergo phase separation, between a highly and lowly lithiated form. Discharge of LiFePO4 crystals has traditionally been modelled by one-phase Stefan problems, which assume that phase separation occurs.

Recent work has been using phase-field models based on the Cahn-Hilliard equation, which only phase-separates when thermodynamically favourable. In the past year or two, this work has been having considerable impact in both theoretical and experimental electrochemistry.

Unfortunately, these models are very difficult to solve numerically and involve large, coupled systems of nonlinear PDEs across several different size scales that include a range of different physics and cannot be homogenised effectively.

This talk will give an overview of recent developments in modelling LiFePO4 and the sort of strategies used to solve these systems numerically.

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