Viscous fingering under elastic membranes

Lead Research Organisation: University of Manchester
Department Name: Mathematics


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.

Planned Impact

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.


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Franco-Gómez A (2016) Sensitivity of Saffman-Taylor fingers to channel-depth perturbations in Journal of Fluid Mechanics

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Juel A (2012) Flattened fingers in Nature Physics

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Juel A (2018) Instabilities in Blistering in Annual Review of Fluid Mechanics

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Pihler-Puzovic D (2018) Viscous fingering in a radial elastic-walled Hele-Shaw cell in Journal of Fluid Mechanics

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Pihler-Puzovic D (2013) Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells in Journal of Fluid Mechanics

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Prior C (2016) Ribbon curling via stress relaxation in thin polymer films. in Proceedings of the National Academy of Sciences of the United States of America

Description Blistering occurs when a thin solid layer locally separates from an underlying substrate through cracking of a bulk material, delamination of a composite material, or peeling of a membrane adhered to the substrate by a thin layer of viscous fluid. In this last scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. Such blisters are prone to fluid and solid mechanical instabilities. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. Suppression is mediated by the locally tapered geometry of the blister near the fluid interface, which is imposed by the underlying blistering flow. Buckling/ wrinkling instabilities of the delaminated layer arise for sufficiently thin
membranes and can interact with the fluid mechanical fingering instability. We have characterised these processes using a combination of controlled idealised experiments and numerical simulations of a fully-coupled fluid-structure interaction model.
Exploitation Route Fundamental science. Applications in physiological flows, geophysics, and industrial processes such as roll coating.
Sectors Other

Description Leverhulme Project Grant
Amount £156,000 (GBP)
Organisation The Leverhulme Trust 
Sector Charity/Non Profit
Country United Kingdom
Start 09/2014 
End 09/2017
Description BAMC Meet the Mathematician 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? Yes
Geographic Reach National
Primary Audience Schools
Results and Impact Talk sparked interest in school age children and led to discussions.

Interview published online
Year(s) Of Engagement Activity 2012
Description Big Bang Fair 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Public/other audiences
Results and Impact Public understanding of science
Year(s) Of Engagement Activity 2016
Description LMS Women's Day 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? Yes
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact A talk to encourage female participation in STEM subjects / sparked discussion.

Positive feedback from female participants
Year(s) Of Engagement Activity 2013