Interdisciplinary Applied Mathematics

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but comparable to the mean free path, then the no-slip condition is satisfied. Otherwise, significant slip at the wall is present, which for atomically smooth walls occurs if the global Knudsen number, i.e., the ratio A/h (with h the    channel    height),    is finite.    In    summary,    it    was    concluded    that:


For a microchannel flow with atomically smooth walls, if the global Knudsen number Kng = X/h is less than 0.01, then the no-slip condition at the walls is valid (h is the channel height).


For a microchannel flow with atomically rough walls, if the local Knudsen number Kn = X/A is of order unity, then the no-slip condition at the walls is valid (A is the roughness height).


Otherwise, in both smooth or rough walls, there is significant velocity slip at the walls.


In another study, (Li et al., 2002) considered surface roughness effects on gas flows through long microtubes. They treated the rough surface as a porous film covering an impermeable surface. In the porous film region they used the Brinkman-extended Darcy model, and they employed a high-order slip model in the core region of the microtubes. Solutions in these two different regions of the tube were combined by matching the velocity slip and the shear stress at the porous-core flow interface. This enabled derivation of expressions for the pressure distribution in microtubes, including the slip effects.

2.3 High-Order Models


The conservation equations (2.16) are still valid for larger deviations from the equilibrium conditions; however, the stress tensor (and heat flux vector) have to be corrected for high-order rarefaction effects. The general tensor expression of the Burnett level stress tensor is

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