Interdisciplinary Applied Mathematics

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The main differences between fluid mechanics at microscales and in the macrodomain can be broadly classified into four areas:

   Noncontinuum effects,

   surface-dominated effects,

   low Reynolds number effects, and

   multiscale and multiphysics effects.

Some of these effects can be simulated with relatively simple modifications of the standard numerical procedures of computational fluid dynamics. However, others require new simulation approaches not used typically in the macrodomain, based on multiscale algorithms. For gas microflows, compressibility effects are very important because of relatively large density gradients, although the Mach number is typically low. Depending on the degree of rarefaction, corrections at the boundary or everywhere in the domain need to be incorporated. Increased rarefaction effects may make the constitutive models for the stress tensor and the heat flux vector in the Navier-Stokes equations invalid. On the other hand, working with the Boltzmann equation or with molecular dynamics implementation of Newton’s law directly is computationally prohibitive for complex microgeometries. The same is true for liquids, since atomistic simulation based on Newton’s law for individual atoms is restricted to extremely small volumes. Therefore, mesoscopic and hybrid atomistic-continuum methods need to be employed for both gas and liquid microflows to deal effectively with deviations from the continuum and to provide a link with the large domain sizes. Most important, microflows occur in devices that involve simultaneous action in the flow, electrical, mechanical, thermal, and other domains. This, in turn, implies that fast and flexible algorithms and low-dimensional modeling are required to make full-system simulation feasible, similar to the achievements of the 1980s in VLSI simulation.

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