Interdisciplinary Applied Mathematics

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paramagnetic microparticles. The second category includes problems with hundreds or even thousands of paramagnetic microparticles. It has been observed experimentally that in the presence of an orienting external field, at low concentrations one-dimensional lattices are formed, while at high concentrations two-dimensional lattices of staggered rows of particles are obtained. Both    classes are    of    great    interest    from    the    theoretical    as well    as

the engineering point of view. More specifically, in the first class of problems, paramagnetic microparticles have been used as the active flow-control element to design and then optimize basic elements of a microfluidic network, e.g., pumps, valves, mixers (Hayes et al., 2001). This approach has the potential of avoiding some of the difficulties experienced by other techniques, while the materials can be synthesized in large scales. For example, colloidal silica is easily modified for dispersion in both aqueous and nonaqueous solvents and is biocompatible; see, for example, (Terray et al., 2002). Typically, fewer than one dozen microspheres are involved in these designs. In the second class of problems, we are interested in fundamental understanding of the scaling laws that govern the interaction of many chains or    magnetic    columns    formed    by    hundreds    or    thousands    of    micro

spheres. Self-assembled structures can be used in fabricating magnetically controlled microdevices with complex functionality, for example, actively addressable arrays of microreactors that can react dynamically with fluids for sorting and mixing applications; or self-assembled magnetic matrices for DNA separation chips. A fundamental unresolved question is reversibility in static or dynamic self-assembly processes and its validity as a function of the magnetic field strength and geometry.

Modeling of ER or MR fluids is quite complicated, and in simulating such flows, it is more effective to adopt a hierarchical simulation methodology that performs best in a certain range of parameters in terms of both accuracy and computational complexity. It should include stochastic techniques to represent Brownian noise, geometric roughness or other uncertainties associated with the boundary conditions, particle size, and interaction forces. That is, both continuum and atomistic techniques, as we discuss in Chapters 14 and 16, respectively, should be used.

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