Interdisciplinary Applied Mathematics

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4-1-4 Effects of Roughness


Surface roughness in the slip flow regime has been studied in (Li et al., 2002), where the rough surface was represented as a porous film based on the Brinkman-extended Darcy model, and the core region of the flow utilized velocity slip to model the rarefcation effects. Then the velocity solution and the shear stress from the core and the porous regions of the flow were matched at the pore-region/flow interface, enabling derivation of expressions for the pressure distribution in microtubes, including rarefaction effects.


In the following we describe a different approach to modeling surface roughness. The pressure drop in channels with rough walls depends critically on the shape and size of the roughness. For random roughness, the pressure drop should depend on the statistical characteristics of the walls, which are expressed by the correlation function of surface inhomogeneities. A particularly effective method has been developed by Meyerovich and his group on how to extract this dependence at the limit of very large Knudsen numbers (Meyerovich and Stepaniants, 1994; Meyerovich and Stepaniants, 1997). In this description, the transport coefficients, such as mobility, diffusion, effective relaxation time, and mean free path along the walls, are expressed directly via the parameters of the correlation function of surface roughness. The main idea is to perform a nonlinear coordinate transformation, assuming that the boundary can be described by a single-valued function. The transformed equation, e.g., Stokes flow equation, acquires extra random terms, which depend on the boundary roughness. This is demonstrated in Figure 4.12 for a microchannel with rough walls. All variables are transformed in the new coordinate system, including the transport coefficients (here the kinematic viscosity), which are renormalized. In this particular example, L is the channel height, and the inhomogeneities are small and random and are described by Ci(x,z) and £2(x, z) for the lower and    upper    wall,    respectively.    Also,    ^1,^2    ^ L    have    zero    mean, i.e.,

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