Interdisciplinary Applied Mathematics

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distance (Ay)    away    from    the    wall,    it    is    possible    to approximate    (to first-

order accuracy and for 7V = 1) the velocity slip Us as

Kn

Us — Uo

Ui-U0

Ay

For no change in the sign of vorticity at the wall, we require that (Uo— U1) > (Uq — Us) = — Kn Ul; this is satisfied if Ay > Kn (in nondimensional form). This limit is a significant restriction, especially for spectral-based methods    such as    the    one    we    use    in    our    discretization,    where    collocation

points are clustered very rapidly close to the boundaries. Therefore, spectral and other high-order methods that typically provide high-order accuracy are subject to numerical instabilities of this form.

We have found that the new boundary conditions (equations (2.26) and (2.31)) are numerically stable for relatively higher values of Knudsen number. Their applications are usually restricted by the flow geometry. For example, the limit of applicability of (2.26) and (2.31) in a channel flow is Kn = 0.5. Since these boundary conditions obtain the slip information, one mean free path away from the surface, meaningful results are achieved when A < h/2, where h is the channel thickness.

14.1.3 Verification Example: Resolution of the Electric Double Layer

As an example of how to verify resolution-independence with the spectral element discretization, we consider the numerical solution of the Poisson-Boltzmann equation (7.4) and the incompressible Navier-Stokes equations (7.12); see Section 7.1. The weak (variational) form of equation (7.4) is solved via a Galerkin projection. A Newton iteration strategy for a variable-coefficient Helmholtz equation is employed to treat the exponential nonlinearity in the following form:

[V2аф cosh(a (ф* )n)] (ф* )n+1 = в sinh(a (ф*)п)

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