Effective diffusion in the region between two surfaces

Area of Science:

• Physics
• Applied Mathematics

Background:

• Diffusion processes are fundamental in various scientific fields.
• Modeling diffusion in complex geometries often requires dimensionality reduction.
• Understanding effective operators is crucial for simplifying complex physical phenomena.

Purpose of the Study:

• To derive a formula for the effective diffusion operator.
• To project a three-dimensional diffusion equation onto a two-dimensional plane.
• To incorporate reflective boundary conditions and transversal stabilization rates into the model.

Main Methods:

• Mathematical projection of the 3D diffusion equation onto a 2D plane.
• Application of reflective boundary conditions at specified surfaces.
• Inclusion of a finite transversal stabilization rate in the derivation.

Main Results:

• A novel formula for the effective diffusion operator in 2D is established.
• The derived operator accurately represents the projected 3D diffusion process.
• The influence of boundary conditions and stabilization rate on the operator is quantified.

Conclusions:

• The developed formula provides a simplified yet accurate model for diffusion in reduced dimensions.
• This work offers a valuable tool for analyzing diffusion phenomena with specific boundary constraints.
• The findings have implications for simulations and theoretical studies involving diffusion processes.