The Mathematics of Multilayer Microfluidics: analysis, hybrid modelling and novel simulations underpinning new technologies at the microscale

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


One of the widest scientific revolutions currently taking place is the quest towards miniaturization and manufacture of tiny devices that can perform tasks (such as fluid handling and processing) on the micro-scale. In many cases the manipulation can be done rapidly and accurately and the automation of such processes is expected to have a huge impact in areas such as drug development and delivery (e.g. ``lab-on-chip" technologies). Small volumes of fluid imply large surface to volume ratios, and such geometries enhance the effects of mechanisms that are absent in larger scale devices. Many applications involve processes that utilise more than one immiscible fluid - such fluids do not mix (e.g. water and oil) and more importantly they have separating interface(s) that is free to move under the action of surface tension, flow and any other imposed external effects such as electric fields or gravity. Consequently, a process can be made successful and robust if we can understand how the interface between the different fluids (or phases) evolves. Such understanding opens the way for introducing flow controls. These can be either passive, as for example by building fixed structures such as bumps or rivulets on surfaces over which the fluids flow, or, active as in the case of switching an electric field on and off in a way determined by the evolving flow characteristics. One of the main mechanisms affecting multilayer microfluidic flows is surface tension. Its presence makes the mathematical problems highly challenging both analytically and computationally due to the intrinsically nonlinear nature of the resulting boundary conditions on unknown moving interfaces. The interfacial configuration affects the flow and the flow in turn affects the interfacial position - they need to be solved together and the instability mechanisms present need to be identified and followed into the nonlinear regime where complex dynamics can emerge.

Producing interfaces in multi-fluid flows and controlling their configurations and spatio-temporal dynamics is also of vast importance to state-of-the-art materials science - known as Origami engineering, a mostly experimental research field. Interfaces act as the fabric where particles can self-assemble to produce homogeneous or pre-designed inhomogeneous material membranes to be manipulated and folded for desired engineering purposes.
Our goal is to identify, control and manipulate nonlinear interfacial instabilities in multifluid flows to produce desirable surfaces that could be used for
the directed self-assembly of nano- and micro-particles to create smart films with exotic elastic properties, or that can host mammalian cells for tissue engineering.

To achieve an extensive theoretical knowledge of fluid-surface interactions we consider three canonical models to describe some of the "designer" substrates currently used experimentally: (i) topographical structures (bumps and indentations), (ii) stick-slip superhydrophobic surfaces, and (iii) etched electrode networks that produce non-uniform electric fields. Within channels made up of such surfaces we have multilayer flows with several fluid-fluid interfaces. The resulting instabilities are complicated and include resonance, shear-induced stability or instability, and electrohydrodynamic instability to mention some. An additional challenge addressed by the present proposal is three-dimensionality. The computational challenges are enormous and will be addressed at least partially. We will make analytical progress by deriving reduced model equations to produce coupled systems of nonlinear partial differential equations depending on time and two spatial variables. These will be studied fully, both analytically and computationally, and compared with direct numerical simulations. Emphasis will be given to new solutions and mathematical structures but also on the phenomena that they describe and the underlying mechanisms that produce complex dynamics.

Planned Impact

We will discuss three areas where the proposed research will have impact: (i) societal, (ii) exploitation, and (iii) human resources.

Societal impact: Microfluidics is a vitally important sector in the UK's scientific and technological portfolio. The proposed research
will lead to significant advances in our understanding of the fundamentals of complex interfacial phenomena at
the micro-scale and will contribute significantly in sustaining and growing the UK's position in the international microfluidics arena.
The UK is committed to advanced technologies and advanced manufacturing, and is home to numerous microfluidics companies
and research laboratories. The proposed work will have a unique impact on these activities because it brings a theoretical and
computational dimension to the field that is typically lacking due to the mostly applied engineering nature of the sector.
The quantitative tools that will be developed will provide a number of advanced analytical and computational solution avenues to
multilayer microfluidics problems in complex-structured vessels, providing industry with exploration-friendly "virtual laboratories"
to be used to simulate realistic flows found in applications.

Exploitation: EPSRC has identified microfluidics as a strategic research area. Of relevance to this proposal is the EPSRC "Grand Challenges in Microelectronic
Design" paper, discussing microelectronics design (lab on chip, MEMS, electrodes, microactuators, microfluidics). As a result EPSRC
funding has been dedicated to this area has led to several university spin-offs (Microsaic Systems, Imperial College; Microstencil Ltd, Heriot-Watt;
Perpetuum, Southampton). The proposed research is of fundamental relevance to the parent research base of such spin-offs
(e.g. the Optical and Semiconductor Devices group at Imperial College who use self-assembly techniques utilising the interfacial tension of molten
solder to fabricate microwave inductors). It is also of relevance to other groups such as the Polymers and Microfluidics research lab
led by Dr Cabral at Imperial. We will seek interaction with such groups in order to disseminate our results and present them with our analytical
and computational capabilities to be developed in the proposed research.

Human Resources: The PDRAs trained under the proposed research will be uniquely qualified to enter either academia or industry and
will be equally at ease in working in applied mathematics groups or other scientific teams working on fluid mechanics. The PDRAs will
work in a collaborative research environment that includes Imperial College, Loughborough University and the University of East Anglia,
but with strong links also with engineers performing experiments (e.g. Prof. Stebe at U. Pennsylvania, Prof. Matar and Dr Canral at Imperial).
In addition, the PDRAs will be exposed to a wide range of seminar activities (the Applied Section alone at Imperial runs 5 weekly seminars -
Fluid Dynamics, Biomathematics, Applied Mathematics Colloquium, PDE and Applied Analysis, Dynamical Systems; there are also several
fluid dynamics seminars in engineering departments), and an unsurpassed scholarly mathematics environment. This will provide them
with considerable skills and advanced knowledge in the field of fluid dynamics and applied mathematics in general.


10 25 50
Description Have developed (i) computational technologies that can be used to inform design issues in liquid film and droplet aerodynamics with applications to icing; modelling of superhydrophobic surfaces in such applications. (ii) electrostatic control of multilayer flow morphologies with applications in micromanufacturing.
Exploitation Route The work is a starting point for engineering solutions. We put together a team of engineers and mathematicians from the UK and the US to take the project further by directly applying the theory to experiments (the US team would have built the experiments), in order to develop novel scalable cooling technologies for microprocessor thermal management systems. A proposal was submitted to the EPSRC (joint with NSF-CBET) but was unfortunately declined. We are now developing these ideas with our US partners (the research is based there) until we are successful in follow-up funding in the UK. We hope to succeed and use the findings to the benefit of UK thermal management and sustainable energy harvesting initiatives.
Sectors Aerospace, Defence and Marine,Electronics,Manufacturing, including Industrial Biotechology

Description Some of the computational technology developed during the award (Papageorgiou, Cimpeanu) was used to calculate droplet impact dynamics of relevance to high speed flow and ice formation on wings and jet engine intakes. Most of the work was funded by another project but the fundamentals developed for this award played an important role. A patent has been submitted through Imperial Innovations, and is currently undergoing review by the Patent Office. In addition, Prof. Crowdy's work on this award also assisted in his SBIR initiative with the National Science Foundation in the US to develop optical fibre manufacturing.
First Year Of Impact 2017
Sector Aerospace, Defence and Marine,Manufacturing, including Industrial Biotechology
Impact Types Societal,Economic,Policy & public services

Description Imperial - Tufts structured channel flows collaboration 
Organisation Tufts University
Department Department of Mechanical Engineering
Country United States 
Sector Academic/University 
PI Contribution Imperial team has developed theory and computations to model superhydrophobic channel flows with applications in enhanced heat transfer technologies in microchannels as well as use of such technologies in energy sustainability.
Collaborator Contribution Worked closely with Prof. Hodes (Mech. Eng., Tufts) and his students. PhD students from Imperial visited Tufts and worked on common problems. The Tufts team helped with the identification of applications as well as modelling of complex processes.
Impact In addition to several publications (2 in print or accepted and several in preparation; these are listed in the publications section), we have organised two joint events: (i) a Imperial-Tufts 2-day workshop in February 2016 where the two groups got together and identified research directions and projects (the team at Imperial has 2 staff members - Papageorgiou & Crowdy - one postdoc funded by the grant, and 4 PhD students); (ii) a two-day international workshop: "The "Red Lotus Project" Workshop: an interdisciplinary meeting on heat transfer effects in superhydrophobic surface theory and related areas", funded by the Royal Society and held at Chicheley in November 1-2, 2016. These events have enabled us to identify future directions and we are currently preparing a joint EPSRC-NFS CBET proposal with Imperial College the lead partner. We aim to submit this in the very near future (the Expression of Intent has been submitted and approved).
Start Year 2015