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Stochastic Trajectories and Spectral Boundary Conditions for Enhanced Diffusion in Immersed Boundary Problems

Rômulo Damasclin Chaves dos Santos, Jorge Henrique de Oliveira Sales

Abstract

This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using semi-Lagrangian schemes, this approach captures complex diffusion dynamics at moving interfaces, integrating probabilistic methods that reflect multi-scale fluctuations. In addition to a rigorous mathematical foundation that includes stability proofs, this model exhibits reduced numerical diffusion errors and improved stability in long-term simulations. Comparative studies highlight its effectiveness in multi-scale scenarios that require precision in interface dynamics. Focusing on various shear and circular flows, including those with Hölder and Lipschitz regularities and critical points, we establish sharp bounds on effective diffusion rates using specific initial data examples. This dual exploration in enhanced diffusion highlights how flow regularity and critical points influence dissipation. These findings advance both the theoretical understanding and practical applications of enhanced diffusion in fluid dynamics, offering new insights into diffusion rate optimization through interface dynamics and flow structure regularities. Future research can further refine the IBM framework by exploring alternative probabilistic methods to improve interface accuracy, opening up the potential for improved modeling in applications that require precise control over mixing rates and dissipation processes.

Stochastic Trajectories and Spectral Boundary Conditions for Enhanced Diffusion in Immersed Boundary Problems

Abstract

This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using semi-Lagrangian schemes, this approach captures complex diffusion dynamics at moving interfaces, integrating probabilistic methods that reflect multi-scale fluctuations. In addition to a rigorous mathematical foundation that includes stability proofs, this model exhibits reduced numerical diffusion errors and improved stability in long-term simulations. Comparative studies highlight its effectiveness in multi-scale scenarios that require precision in interface dynamics. Focusing on various shear and circular flows, including those with Hölder and Lipschitz regularities and critical points, we establish sharp bounds on effective diffusion rates using specific initial data examples. This dual exploration in enhanced diffusion highlights how flow regularity and critical points influence dissipation. These findings advance both the theoretical understanding and practical applications of enhanced diffusion in fluid dynamics, offering new insights into diffusion rate optimization through interface dynamics and flow structure regularities. Future research can further refine the IBM framework by exploring alternative probabilistic methods to improve interface accuracy, opening up the potential for improved modeling in applications that require precise control over mixing rates and dissipation processes.

Paper Structure

This paper contains 28 sections, 1 theorem, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Mathematical Theory
  3. Proof of Enhanced Dissipation via Stochastic Trajectories
  4. Definition of Stochastic Trajectories
  5. Calculation of Energy Dissipation
  6. Estimate of Dissipation via Variance
  7. Feynman-Kac Relation
  8. Variance and Dissipation
  9. Applying Itô's Formula and Jensen's Inequality
  10. Relation to Spatial Integral
  11. Derivation of the Formulation
  12. Role of the Velocity Field and Stochastic Trajectories
  13. Building the Inequality
  14. Physical Interpretation
  15. Regular Shear Flows with Critical Points
  16. ...and 13 more sections

Key Result

Theorem 1

Consider a generalized circular flow given by coupled with an advection-diffusion model, where the diffusion coefficient varies anisotropically with radial distance. Let the initial density distribution $\rho_0(r, \theta)$ be supported on the annular ring with diffusion anisotropy scaling as where $\gamma \in [0, 1]$. Then, the time scale $T(\kappa)$ for enhanced mixing is expressed as: This r

Figures (2)

  • Figure 1: 3D Representation of Anisotropic Diffusion in Circular Flows
  • Figure :

Theorems & Definitions (1)

  • Theorem 1: Anisotropic Enhanced Diffusion in Circular Flows