Physics
Scientific paper
Jan 1992
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1992phdt........33f&link_type=abstract
Thesis (PH.D.)--NEW YORK UNIVERSITY, 1992.Source: Dissertation Abstracts International, Volume: 53-08, Section: B, page: 4144.
Physics
Scientific paper
I study the influence of periodic as well as random convection flows on the effective diffusivity of a passive scalar transported by the fluid. The flows are two-dimensional, steady, divergence-free velocity fields formed by overlapping or nonoverlapping vortices. I found two classes of results: (1) When the flows have a non-zero mean drift, the effective diffusivity is geometry-independent and often extremal in the sense that either varepsilon or 1/varepsilon is achieved. (2) When the flows have a zero mean drift, the effective diffusivity is very sensitive to the streamlines' overall geometry. I show that the effective diffusivity can be obtained from a pair of variational principles. They yield sharp upper and lower bounds when appropriate trial temperature fields are used. Let varepsilon > 0 denote the molecular diffusivity. I prove that: In the limit varepsilon downarrow 0, the effective diffusivity sigma_spɛ{*} obeys various scaling laws sigma_spɛ{*} = C^{*}varepsilon ^alpha+cdots with exponent in the range -1 <= alpha <= 1 as well as the logarithmic law sigma_sp ɛ{*}~varepsilon log(1/varepsilon). The exponent alpha depends on the geometry of the flow and when alpha < 1 we have enhancement of the effective diffusivity. I show also that in certain cases the constant factor C^{*} can be explicitly calculated. Probabilistically, diffusion of a passive scalar in a flow can be represented by a diffusion process with drift equal to the velocity field and diffusion constant equal to varepsilon, the molecular diffusivity. This diffusion process behaves on long time scales like Brownian motion with diffusion constant equal to sigma_spɛ{*}. My results about the behavior of sigma_spɛ{*} as varepsilon downarrow 0 are closely related to the behavior under small random perturbations of the motion of a particle in the flow.
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