Viscous fingering on soft substrates

The research proposed here is motivated by recent striking observations of the viscous fingering instability on soft substrates. This instability readily develops if a viscous fluid is displaced by a less viscous fluid in a narrow gap between rigid plates, and typically results in complex highly branched patterns of fingers on the interface between the two fluids. Viscous fingering has been of long-standing research interest because it serves as an archetype for front-propagating pattern-forming phenomena, as diverse as the growth of bacterial colonies and the propagation of flame fronts. We find that this instability can be manipulated by replacing the bottom bounding plate with a soft substrate. Firstly, we observe that the instability can be delayed, so that the interface remains axisymmetric for injection rates at which the pattern in the corresponding rigid system already exhibits nonlinear growth. This is similar to other systems in which the fluid flow interacts with a compliant boundary, for example, with thin elastic membranes, although soft substrates deform differently compared to those. Secondly, the thickness and the stiffness of the soft substrate can be tuned to change the structure of the fingers that do develop once the circular interface becomes unstable to non-axisymmetric perturbations. The resulting patterns range from the highly branched fingering observed in rigid systems to the stubby fingers that develop on the meniscus between two cylinders co-rotating with the same speed. Thus, the instability is strongly influenced by the deformation of the soft substrate, which in turn deforms in response to the two-phase flow, and is likely to have viscoelastic properties. We propose to employ a combination of experimental, theoretical and computational approaches to fully-characterise the complex fluid-structure interactions that lead to these novel phenomena.

1. The UK fluid mechanics and the soft matter communities will benefit from the PDRA's scientific training, which meets the requirements of the IRMS 2010 for fluid mechanics in combining numerical and analytical research with experiments. This individual will work in an inter- and intradisciplinary research environment of high quality, obtaining skills and scientific expertise well-suited for employment in both academia and industry.

2. The project will also benefit a more general scientific community in academia and industry as it will involve further development of the open-source library oomph-lib, making the codes developed for this project available for re-use in many other applications. The library is regularly used
in industry-driven projects, e.g. by Thales Underwater Ltd., Mondelez International Inc. and Cambridge Display Technology Ltd. who were/are involved in EPSRC-funded KTA projects conducted at the MCND. Following the approach taken in all previous studies that utilised oomph-lib, the
newly developed parts of the packages will be augmented by detailed tutorials and made available at http://www.oomph-lib.org.

3. The subject of the proposal lends itself to the type of public engagements that the PI is involved in on a regular basis, such as outreach activities organised by ScienceGrrl, or the Big Bang Fair. By demonstrating the comparisons between the experimental and theoretical results of this research, the PI will be able to show the power of mathematical modelling in a real-world problem. Further expansion of the outreach activities is already planned, as explained in the "Pathways to Impact" document.