The design of robust networks for massive parallel micro-fluidic devices.

John Melrose, Unilever

pdf version of problem description

Micro-fluidic devices consist of etched or printed channels and nodes in thin wafers through which fluids are pumped to mix, react and structure (e.g. form drops). Applications of these range from sensors, chemical analysis, fast throughput screening, fast throughput chemistry and manufacturing. In manufacturing applications it would be desirable to have many such devices coupled to feed-streams and giving a product output. In the particular aspect of manufacturing we would like to consider is two feed-streams oil and water and the function of the devices is to produce drops of controlled size of one phase in another. Some channels would be pure water, others pure oil and others slugs of oil in water switching from one to other would occur at drop generation junctions. The study group may also wish to formulate some of the other applications.

The devices are a network of channels, junctions and nodes. The problem is to understand the steady-state and dynamic response of such fluidic networks. We have observed instabilities in the flow pattern where the flow of components ceases in some channels, dominates in others and the overall flow pattern breaks symmetries of the network. Also channels may foul and block, and this need occur with out catastrophic consequences for the function of the rest of the network. We wish to establish design principles for robust networks and to have a mathematical tool-kit to analyse proposed designs. The following questions/issues occur to us:

• Is there a classification on general grounds of the possible instabilities?

• What is the sensitivity of the instabilities to the tolerances of the chip manufacture (e.g. channel width, junction geometry); turning this around to what tolerances do devices need be made to limit instabilities?

• Are there general design principles (e.g. feedback loops, re- routing, diodes) etc that will give robustness for given tolerance and robust against blocking?

• Where to place monitors, would there be any signatures of the onset of problems?

There are a hierarchy of physical and mathematical issues.

• For the steady states, the generalisation and analysis of Kirchhoffs laws of mass conservation to multi-component flow of conserved components and immiscible components in complex networks of channels and nodes (although we imagine flow in channels involving continuum of one phase and slugs of the other, we anticipate a continuum formulation based on volume fractions).

• General theorems of such network equations, in particular stability.

• The formulation of physical inspired suitable (again continuum) dynamic equations based on conservation of mass and momentum that will allow prediction of dynamic response.

• Statistical generalisations of the above including variations in channel geometry, widths etc and giving relationships between tolerance and instability.

• Discrete formulations, simulations of discrete slugs travelling through such networks.

We imagine there are analogous problems in traffic flow, blood flow and chemical plant design, flow in porous media. In addition at the finer scale there are a host of more traditional fluid-mechanics and physics problems: 3D modelling of multiphase flow in confined geometry especially a T-junctions and converging nozzles; dynamic contact angles; dynamics of three phase contact lines; inclusion of interface dynamic with Marangoni effects; wall boundary conditions; electrophoresis in confined geometries; phase changes during flow. Much of these however become heavy finite-element work or questions of physics input. We would prefer emphasis on the larger scale problems, but do not mind if some want to stray/advise on these latter areas.

Further material will be available at the Study Group.