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The Diffusion of Passive Tracers in Laminar Shear Flow
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Diffusion in narrow channels on curved manifolds.

Guillermo Chacón-Acosta1, Inti Pineda, Leonardo Dagdug

  • 1Department of Applied Mathematics and Systems, Universidad Autónoma Metropolitana-Cuajimalpa, Artificios 40, México D. F. 01120, Mexico.

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We developed a new method to calculate diffusion in curved channels. This effective diffusion coefficient accounts for channel asymmetry and surface curvature, improving predictions for particle movement.

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Area of Science:

  • Statistical Mechanics
  • Physical Chemistry
  • Mathematical Physics

Background:

  • Two-dimensional (2D) diffusion is crucial in various physical and chemical processes.
  • Describing diffusion in confined geometries, especially on curved surfaces with asymmetry, presents significant challenges.
  • Existing models often simplify channel geometry or surface curvature, limiting their applicability.

Purpose of the Study:

  • To derive a general effective diffusion coefficient for 2D diffusion in narrow, asymmetric channels on curved surfaces.
  • To extend the Kalinay-Percus projection method to account for surface curvature and channel asymmetry.
  • To provide a modified one-dimensional generalized Fick-Jacobs equation incorporating these geometric factors.

Main Methods:

  • Extension of the Kalinay-Percus projection method for asymmetric channels.
  • Projection of the anisotropic 2D diffusion equation onto a curved manifold.
  • Development of a perturbation series for marginal concentration and derivation of the effective diffusion coefficient.

Main Results:

  • A modified generalized Fick-Jacobs equation is derived, including a curvature-dependent term.
  • The first-order correction for invariant effective concentration is explicitly obtained.
  • A general expression for the effective diffusion coefficient is derived as a function of surface metric elements and channel geometry.

Conclusions:

  • The derived effective diffusion coefficient accurately describes 2D diffusion in narrow, asymmetric channels on curved surfaces.
  • The method provides a more refined understanding of particle transport in complex geometries.
  • The study validates the approach using spherical and cylindrical surfaces with specific channel configurations.