Modelling Single and Multi-Phase flow in Non-Uniform Hele-Shaw Channels

  • Jacob Harris

Student thesis: Phd


In this thesis we develop reduced dimensional (depth-averaged) models of the single and multi-phase flows of viscous fluids through Hele-Shaw channels with variable cross sections. We study the significance of including different physical properties in these models, specifically lateral viscous stress effects. Single-phase flow in uniform channels with an extreme width to depth aspect ratio (Hele-Shaw channels) that are modelled with an expression of Darcy's law produces an approximation of the depth-average of the solution to the Stokes equations. This approximation has been sufficiently accurate in much of the published work over the past century. However, this Darcy model is unable to correctly model problems that are dependent on lateral viscous stress effects, such as flow near any lateral walls or channel obstructions. Alternative depth-averaged `Stokes-Brinkman' models introduce lateral stress effects with ad hoc methods, rather than using a systematic approach, and currently the Stokes-Brinkman models in this thesis cannot be derived for non-uniform Hele-Shaw channels. We present the formal derivation of a novel `parabola-weighted' model that includes lateral stress effects in both uniform and non-uniform Hele-Shaw channels. In single-phase problems our parabola-weighted model is shown to approximate the three-dimensional Stokes solution as accurately as the Darcy model away from walls and obstacles, and better than or equal to the Stokes-Brinkman models outlined in this thesis in lateral stress dominated regions. In the multi-phase problem of modelling steadily propagating bubbles in uniform Hele- Shaw channels, the speed and shape of experimentally observed bubbles are correctly pre- dicted with the Darcy model, with exceptions in the small bubble limit and at low surface tensions. We explore the effect of lateral fluid stresses on these exceptions by modelling them with our parabola-weighted model. We show that lateral stress modelling has no significant effect on the bubbles predicted in domains where the Darcy model is known to accurately describe bubbles, as expected. However, these stresses are shown to be essential when modelling the small bubble limit, and we show that they slow bubbles with small surface tensions. Finally, the effect of lateral stress modelling in non-uniform Hele-Shaw channels oc- cluded by a rail is explored. In this domain, the range and interconnectivity of steady bubble solutions are known to be complex and there are limited experimental results. By comparing solutions to our parabola-weighted model to Darcy model solutions, we show that the additional lateral fluid stress effects predict new symmetry-breaking bifurcations and significant quantitative differences in the predicted bubble speed and critical point location for the experimentally unexplored solution branches.
Date of Award1 Aug 2023
Original languageEnglish
Awarding Institution
  • The University of Manchester
SupervisorAndrew Hazel (Supervisor) & Alice Thompson (Supervisor)


  • Parabola-weighted model
  • Darcy's law
  • Computational fluid dynamics
  • Finite element modelling
  • Multi-phase flow modelling
  • Non-uniform Hele-Shaw channels
  • Hele-Shaw channels

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