The motion of deformable bubbles, drops, capsules and cells surrounded by viscous fluid and confined in a narrow channel has applications in micro-fluidic devices and lab-on-a-chip design. When viewed from above, the motion often appears two dimensional, suggesting that we might be able to use two dimensional, depth-averaged models to predict, design and control system behaviour.
Depth-averaged models are known to be accurate in some situations, such as when the bubble is very flattened and far from any walls, and where thin films are relatively passive. However, the typical assumptions break down if two bubbles are very close to each other (e.g. coalescence or breakup), or to walls, constrictions or obstacles, when it highly likely that three-dimensional effects come into play. Furthermore, the typical depth-averaged model is of lower order and dimension than the Stokes equations and hence neglects certain free boundary effects.
The aim of this project is to establish the limits of 2D modelling, and to determine how we can extend our models to be valid in these extreme scenarios. This project will involve asymptotic analysis, model development, finite element computations of two or three dimensional systems in oomph-lib, and likely also collaborations with experimentalists in the Manchester Centre for Nonlinear Dynamics.