Nonscaling calculation of the effective diffusion coefficient in periodic channels

Area of Science:

• Physics
• Applied Mathematics
• Chemical Engineering

Background:

• Diffusion processes are crucial in various scientific fields.
• Accurate calculation of the effective diffusion coefficient (D(x)) is essential for modeling transport phenomena.
• Existing methods for calculating D(x) in channels with varying cross-sections have limitations.

Purpose of the Study:

• To present a novel algorithm for calculating the effective diffusion coefficient D(x) in 2D and 3D channels.
• To develop a method that is not reliant on scaling transverse coordinates or assuming narrow channels.
• To provide a robust approach applicable to channels with complex geometries, including cusps and jumps.

Main Methods:

• The algorithm calculates D(x) as an integral of local contributions from neighboring positions.
• It avoids the need for higher-order derivatives of the channel's shaping function h(x).
• The method is validated for its applicability to wide channels and the limit of infinite width.

Main Results:

• The developed algorithm provides an accurate calculation of D(x) for channels with periodically varying cross-sections.
• The method demonstrates suitability for channels with non-smooth variations in width, such as cusps or jumps.
• The algorithm correctly predicts D(x) in wide channels, matching theoretical expectations for unconfined diffusion.

Conclusions:

• The presented algorithm offers a significant advancement in calculating effective diffusion coefficients in complex channel geometries.
• This method provides a more general and robust tool for researchers studying diffusion in confined and non-uniformly shaped systems.
• The algorithm's ability to handle complex geometries and wide channels expands its applicability in various scientific and engineering domains.