Reduced models in fluid dynamics via complex variable theory

Lead Research Organisation: Imperial College London
Department Name: Dept of Mathematics


Many emerging technologies involving microfluidics, MEMS (micro -electro-mechanical systems) and ``lab-on-chip'' design involve the control and manipulation of fluids at very small scales when the motion and dynamical mechanisms are very different to that of our day-to-day experience of fluids. Viscous, or frictional, effects dominate and lead to a range of new effects that need to be well understood. Such flows are referred to as Stokes flows or low Reynolds number flows. The same class of flows arise in understanding the motion of swimming microorganisms such as spermatozoa or E Coli bacteria. A wide range of experimentally observed behaviour associated with such swimming organisms remains to be properly explained and many of these phenomena have hydrodynamical underpinnings. This research aims to develop simple mathematical models, centred on the use of a powerful set of mathematical techniques from complex analysis, for understanding some of the hydrodynamical mechanisms. In particular, we will focus on how the presence of boundaries - such as no-slip walls where the fluid velocity must vanish or free surfaces on which surface tension is active - can affect the dynamical behaviour of the fluid or the swimming microorganisms.A second component of our project is to study reduced models, again using complex analysis, to understand the dynamics of vorticity when the fluid in which the vorticity is present is compressible. (Sound waves, for example, are a manifestation of a compressible fluid and their interaction with vortical structures associated with aircraft wakes is an important area of study in terms of minimizing noise pollution).

Planned Impact

Future research projects: The proposed collaborative visit is expected to lay the foundations for a broader exchange of ideas and a longer term scientific collaboration. The proposed short academic visit by the PI to UCSD in 2010 will serve to strengthen the scientific interaction between the PI and his collaborators. This initial exchange will lay important groundwork for building proposals of future grant-funded research both here and in the USA. Knowledge transfer: Basic problems of Stokes flows with boundaries have many applications in a wide variety of emerging fields including microfluidics and ``lab-on-a-chip'' technologies, the manufacture of complex microstructured surfaces and even the design of micro-structured optic fibres (MOF's) used in network communication technologies.There is therefore broad scope for knowledge transfer to any of these application areas which have significant societal impact and are consistent with EPSRC's mission objectives. Broadened network of contacts: The PI's interaction to date with researchers he has initiated collaborations with (during visits made possible by his AF) have already led to the PI developing strong and varied interests in low Reynolds number swimming to the extent that he is now recognized as part of an international network of workers in this area. For example, while visiting Caltech in 2009, the PI initiated a successful ongoing collaboration with Dr. Y. Or (now working as an academic in Israel) and this has opened doors to interaction with an academic community that was previously unknown to him. The UK has very few mathematical modellers interested in low Reynolds number swimming so the PI's emergence as a worker in this area strengthens the UK's research base. The PI plans to use the proposed exchange trip to bolster his recognition within the international community and further strengthen the UK as a contributing nation in these developments. Educational training: The PI has guided an EPSRC-funded PhD student (O. Samson) to the successful production of an authoritative thesis on low Reynolds number swimming in complicated geometries. This is evidence that skills/knowledge acquired by the PI during exchange visits can be profitably transferred back to influence home-grown research initiatives and training. Samson will soon graduate with a strong background in mathematical modelling and numerical computation as a result of these endeavours and will represent a skilled and highly competent addition to the national workforce. During a visit to MIT in 2006/7, the PI initiated collaborations with several MIT PhD students (A. Surana, P. Buchak, K-Y Yick) with whom he has written published papers. This benefits both the PI and the students with whom he works: the PI's visit to UCSD is expected to similarly result in his interacting with students there and sharing his knowledge and expertise with future generations of scientists. The long term educational and training impacts of these exchanges cannot be exaggerated.


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Crowdy D (2013) Translating hollow vortex pairs in European Journal of Mechanics - B/Fluids

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Crowdy D (2011) Treadmilling swimmers near a no-slip wall at low Reynolds number in International Journal of Non-Linear Mechanics

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Crowdy D (2011) Frictional slip lengths and blockage coefficients in Physics of Fluids

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Davis A (2012) Stresslet asymptotics for a treadmilling swimmer near a two-dimensional corner: hydrodynamic bound states in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

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Davis A (2012) Matched asymptotics for a treadmilling low-Reynolds-number swimmer near a wall in The Quarterly Journal of Mechanics and Applied Mathematics

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Llewellyn Smith S (2011) Structure and stability of hollow vortex equilibria in Journal of Fluid Mechanics

Description Complex variable methods provide a particularly powerful armoury of mathematical techniques and are useful in certain areas of the applied sciences. Fluid dynamics is one such area. Modern day problems in fluid dynamics, where complex variable methods can be applied, include the study of swimming microorganisms in very viscous fluids (e.g the motility of spermatozoa), the design of microstructured surfaces which have ``superhydrophobic'' properties (which means that the frictional forces experienced by flows over them is greatly reduced) and the study of regions of swirling fluid (called vortices). This project involves using complex analysis to study problems in all of these important areas.
Exploitation Route The reduced models we have devised, based on the powerful mathematics of complex analysis, are capable of providing key insights into the underlying physics of the systems we have modelled. Already, the suite of analytical solutions found during the grant are providing guiding insights, and new guiding ideas, that are being explored by other researchers using more sophisticated models. This is especially true of our results in superhydrophobic surface theory, and in low-Reynolds-number swimming.

More recently, in joint work with a PhD student V. Krishnamurthy, methods developed during this award have been extended to the analysis of steady compressible vortex structures in the small Mach number limit. These are considered to be important new results in theoretical fluid dynamics. The PhD thesis, and associated publications, are in preparation.
Sectors Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Other

Description The key idea of the modelling is to make use of simple, but powerful, methods based on complex variable formulations to give insights into physical mechanisms in a variety of scientific contexts and applications. This includes problems of vortex dynamics, and those arising in microfluidics and the study of superhydrophobic surfaces. In all these contexts we were able to provide important benchmark solutions that are providing important new tools for the respective scientific communities.
First Year Of Impact 2011
Sector Security and Diplomacy
Description National Science Foundation
Amount £152,000 (GBP)
Funding ID 970113 
Organisation National Science Foundation (NSF) 
Sector Public
Country United States
Start 10/2010 
End 10/2013