We focus on the characterization of dispersion processes in microchannels with fractal boundaries (and translational symmetry in the longitudinal direction) in the presence of laminar axial velocity field. This article extends the theory of laminar dispersion in finite-length channel flows at high Peclet numbers by analyzing the role of the fractal cross-section in the convection-dominated transport regime. In this regime, the properties of the dispersion boundary layer and the values of the scaling exponents controlling the dependence of the moment hierarchy on the Peclet number are determined by the local near-wall behavior of the axial velocity. Specifically, different scaling laws in the behavior of the moment hierarchy occur, depending whether the cross-sectional boundary is smooth or nonsmooth (e.g., presenting corner points or cusps). The limit case of a fractal boundary is analyzed in detail. Analytical and numerical results are presented for two fractal cross-sections (the classical Koch curve and the Koch snowflake) in the Stokes regime.
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This can be simply viewed by observing that for , i.e., within the region of occurrence of the Taylor–Aris regime, particle trajectories explore generically the entire cross-section of the channel. Conversely, for higher values of , a generic particle starting its trajectory at the inlet section does not explore the entire channel cross-section in its flight up to the outlet. This is a fortiori true for solute particles starting from close to the walls which correspond to the stagnation points of the axial velocity field. As increases, these particles tend to be confined in a thin layer close to the walls, and this “kinematic” localization close to the stagnation points determines the long-time properties of the outlet concentration profile, and ultimately the divergent scaling of the moment hierarchy.
25.
The value of has been computed by solving the equation for the auxiliary variable equipped with zero flux boundary conditions at , and then evaluating the spatial average over the cross-section .
© 2011 American Institute of Physics.
2011
American Institute of Physics
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