Interdisciplinary Applied Mathematics

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^=vy2u + f.    (2.13)


This equation expresses unsteady diffusion and includes volumetric source terms. If we instead drop all terms on the right-hand side of (2.10), as well as the divergence-free constraint, we obtain a nonlinear advection equation. Finally, by dropping the time derivative in the parabolic equation (2.13), we obtain the Poisson equation,


—v У2и = f,


which is encountered often in MEMS (micro electro mechanical systems), e.g., in electrostatics.

2.2 Compressible Flow


The flow regime for Kn < 0.01 is known as the continuum regime, where the Navier-Stokes equations with no-slip boundary conditions govern the flow. In the slip flow regime (0.01 < Kn < 0.1) the often-assumed noslip boundary conditions seem to fail, and a sublayer on the order of one mean free path, known as the Knudsen layer, starts to become dominant between the bulk of the fluid and the wall surface. The flow, in the Knudsen layer cannot be analyzed with the Navier-Stokes equations, and it requires special solutions of the Boltzmann equation (see Section 15.4 and also (Sone,    2002)).    However,    for    Kn    < 0.1,    the    Knudsen    layer    covers    less


than 10% of the channel height (or the boundary layer thickness for external flows), and this layer can be neglected by extrapolating the bulk gas flow towards the walls. This results in a finite velocity slip value at the wall, and the corresponding flow regime is known as the slip flow regime (i.e., 0.01 < Kn < 0.1). In the slip flow regime the flow is governed by the Navier-Stokes equations, and rarefaction effects are modeled through the partial slip at the wall using Maxwell’s velocity slip and von Smoluchowski’s temperature jump boundary conditions (Kennard, 1938).

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