# Interdisciplinary Applied Mathematics

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In Chapter 8 we consider surface tension-driven flows and capillary phenomena involving wetting and spreading of liquid thin films and droplets. For microfluidic delivery on open surfaces, electrowetting and thermocapillary along with dielectrophoresis have been employed to move continuous and discrete streams of fluid. A new method of actuation exploits optical beams and photoconductor materials in conjunction with electrowetting. Such electrically or chemically defined paths can be reconfigured dynamically using electronically addressable arrays that respond to electric potential, temperature, or laser beams and control the direction, timing, and speed of fluid droplets. In addition to the above themes, we also study bubble transport in capillaries including both classical theoretical results and more recent theoretical and experimental results for electrokinetic flows.

In Chapter 9 we consider micromixers and chaotic advection. In microchannels the flow is laminar and steady, so diffusion is controlled solely by the diffusivity coefficient of the medium, thus requiring excessive amounts of time for complete mixing. To this end, chaotic advection has been exploited in applications to accelerate mixing at very low speeds. Here, we present the basic ideas behind chaotic advection, and discuss examples of passive and active mixers that have been used in microfluidic applications. We also provide effective quantitative measures of characterizing mixing.

In Chapter 10 we consider simple liquids in nanochannels described by standard Lennard-Jones potentials. A key difference between the simulation of the fluidic transport in confined nanochannels and at macroscopic scales is that the well-established continuum theories based on Navier-Stokes equations may not be valid in confined nanochannels. Therefore, atomistic scale simulations are required to shed fundamental insight on fluid transport. Here we discuss density distribution, diffusion transport, and validity of the Navier-Stokes equations. In the last section we discuss in detail the slip condition at solid-liquid interfaces, and present experimental and computational results as well as conceptual models of slip. We also revisit the lubrication problem and present the Reynolds-Vinogradova theory for hydrophobic surfaces.

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