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

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.

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.