Interdisciplinary Applied Mathematics

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14.1.1 Incompressible Flows

The most efficient way of solving the incompressible (unsteady) Navier-Stokes equations is based on the fractional step method; see Chapter 8 in (Karniadakis and Sherwin, 1999). It is based on the projection of the velocity field obtained from the momentum equation into a divergence-free space. The latter involves the pressure equation and corresponding boundary conditions. However, the fractional step method was first proposed for high Reynolds number flows, and therefore it should be corrected for strong viscous effects in microflows. To this end, a consistent pressure boundary condition should be employed, as we demonstrate next. We follow here the work of (Karniadakis et al., 1991), where the correct pressure boundary condition was employed, leading to a consistent Poisson equation with a proper Neumann condition for the pressure. In two substeps the time-discrete scheme is as follows: First we solve

which is an inviscid-type boundary condition. This can be used for high Reynolds number flows but not for microflows. We assume here that the

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