Viscous fingering under elastic membranes

The research proposed here is motivated by recent striking and unexpected observations of the suppression of a viscous fingering instability in elastic-walled Hele-Shaw cells. The dendritic patterns that readily develop when a thin fluid layer contained in a narrow gap between two rigid plates is displaced by a less viscous fluid, is an archetype for front-propagating, pattern-forming phenomena. We have observed that elastic deformations of the plates that bound the fluid can have a dramatic effect on the onset and nonlinear development of this instability. Specifically, in an elastic-walled Hele-Shaw cell where one of the bounding plates is replaced by a latex membrane, the interface remains axisymmetric for values of the injection rate at which the rigid system already exhibits strongly nonlinear interfacial growth. The critical injection rate beyond which the axisymmetrically expanding interface becomes unstable in the elastic-walled system is approximately 1000 times larger than the corresponding value in the rigid system. Wall elasticity not only affects the onset of the instability but also has a strong effect on the structure of the fingers that develop subsequently. Moreover, if the variations in fluid pressure along the bounding plate are sufficiently large (relative to the plate's stiffness), the fluid loading can cause the plate to buckle (or wrinkle), leading to a strong interaction between two fluid- and solid-based instabilities. We propose to employ a combination of experimental, theoretical and computational approaches to provide a comprehensive understanding of the mechanisms responsible for these novel elasto-hydrodynamic phenomena.

1) The UK fluid dynamics community will benefit from the training of a PDRA with a unique inter-disciplinary outlook. The multi-disciplinary programme of research outlined in this proposal will provide the PDRA with scientific training of outstanding quality both in breadth and depth, which meets the recommendations of the IRMS 2010 for fluid dynamics, in combining computational fluid dynamics with experiments. This will help to develop the UK's expertise and the output will be a highly-skilled individual suitable for employment in industry or academia.

2) The project will also be of a more general benefit to the applied scientific community, both in academia and industry, because oomph-lib, the scientific computing library within which the numerical methods will be implemented, is available as "open source" software.The library is already widely used (the oomph-lib mailing list currently has more than 350 subscribers) and is employed for research projects in academia and industry (e.g. by Thales Underwater Ltd. who are involved in an EPSRC-funded KTA project that explores its use in sonar applications). Following the approach adopted in many previous oomph-lib-based research projects, the newly-developed functionality (particularly the automatic mesh re-generation and solution transfer that is required for the simulation of large-displacement free-boundary problems) will be augmented by detailed tutorials and then incoporated into the library, allowing its re-use in many other applications.

3) The subject lends itself to the type of public engagement activities that the PI is already involved in on a regular basis and for which funds are requested as part of this proposal. The complex dendritic patterns generated by repeated tip-splitting are fascinating to a general audience, and the change in character of the instability induced by the presence of elastic boundaries is striking; comparisons between experimental and theoretical/numerical results demonstrate the power of mathematical modelling in a real-world problem.