Interdisciplinary Applied Mathematics

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3. The effective viscosity converges toward the bulk viscosity as the pore width increases.


The first observation originates from the definition of the effective viscosity for the flow. The second observation is caused by the fluid layering near the pore    surface.    As    the    pore    width    decreases,    more    fluid    molecules    are


observed within the fluid layers near the pore wall, where the movement of the fluid molecules is highly confined, thus leading to a higher effective viscosity.


Over the past decade, Pozhar has developed a rigorous statistical-mechanics approach to nonequilibrium phenomena in strongly inhomogeneous fluids (Pozhar, 1994; Pozhar, 2001; Pozhar, 2000). The approach relies on the rigorous generalization of the Mori-Zwanzig projection operator technique developed in    the    framework    of    the    theory    of    dynamical    systems.    This    ap


proach, unlike the LADM method, has been realized for the most general case of strongly inhomogeneous fluids. However, due to the complicated nature of the theory, there have been very few applications of this method for the analysis of fluid flow. Therefore, the accuracy of the theory has not yet been rigorously established.

FIGURE 10.17. Effective viscosities for Couette and Poiseuille flows versus pore width. The viscosity of a homogeneous fluid at the pore average density is also shown. The inset shows the variation of the pore-averaged self-diffusion coefficients. (Courtesy of H.T. Davis.)

10.5 Boundary Conditions at Solid-Liquid Interfaces


In this section we revisit the question of slip at solid-liquid interfaces and present an in-depth review of the experimental work as well as the conceptual models derived from these findings. We then present different mathematical models that correct the no-slip condition and have been found to represent the available experimental data reasonably well.

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