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An Asymptotic Approach for Modeling Multiscale Complex Fluids at the Fast Relaxation Limit

Xuenan Li, Chun Liu, Di Qi

TL;DR

This work targets the computational bottleneck of simulating multiscale viscoelastic fluids by deriving an asymptotic density expansion in the fast-relaxation limit, where the microscopic state rapidly approaches Gibbs equilibrium. By introducing the balance $c = D\gamma^{4}$ and expanding the joint density around $e^{-U/\gamma^{2}}$, the authors obtain explicit higher-order corrections to the equilibrium distribution and derive two hierarchy-based macroscopic closures (Closure1 and Closure2) that couple to the flow through asymptotic microstress terms. These closures preserve an energy-dissipation law, ensuring stability and consistency with the energetic variational principle, and they are validated through numerical experiments with FENE and double-well potentials, showing accurate macroscopic predictions even beyond the strict asymptotic regime. The approach substantially reduces computational costs while maintaining microscopic feedback, providing a flexible framework for simulating a wide class of multiscale complex fluids and enabling future extensions to lattice materials and data-driven strategies. Overall, the paper offers a principled, energetically consistent pathway to efficient micro-macro simulations with robust accuracy across parameter regimes.

Abstract

We present a new asymptotic strategy for general micro-macro models which analyze complex viscoelastic fluids governed by coupled multiscale dynamics. In such models, the elastic stress appearing in the macroscopic continuum equation is derived from the microscopic kinetic theory, which makes direct numerical simulations computationally expensive. To address this challenge, we introduce a formal asymptotic scheme that expands the density function around an equilibrium distribution, thereby reducing the high computational cost associated with the fully coupled microscopic processes while still maintaining the dynamic microscopic feedback in explicit expressions. The proposed asymptotic expansion is based on a detailed physical scaling law which characterizes the multiscale balance at the fast relaxation limit of the microscopic state. An asymptotic closure model for the macroscopic fluid equation is then derived according to the explicit asymptotic density expansion. Furthermore, the resulting closure model preserves the energy-dissipation law inherited from the original fully coupled multiscale system. Numerical experiments are performed to validate the asymptotic density formula and the corresponding flow velocity equations in several micro-macro models. This new asymptotic strategy offers a promising approach for efficient computations of a wide range of multiscale complex fluids.

An Asymptotic Approach for Modeling Multiscale Complex Fluids at the Fast Relaxation Limit

TL;DR

This work targets the computational bottleneck of simulating multiscale viscoelastic fluids by deriving an asymptotic density expansion in the fast-relaxation limit, where the microscopic state rapidly approaches Gibbs equilibrium. By introducing the balance and expanding the joint density around , the authors obtain explicit higher-order corrections to the equilibrium distribution and derive two hierarchy-based macroscopic closures (Closure1 and Closure2) that couple to the flow through asymptotic microstress terms. These closures preserve an energy-dissipation law, ensuring stability and consistency with the energetic variational principle, and they are validated through numerical experiments with FENE and double-well potentials, showing accurate macroscopic predictions even beyond the strict asymptotic regime. The approach substantially reduces computational costs while maintaining microscopic feedback, providing a flexible framework for simulating a wide class of multiscale complex fluids and enabling future extensions to lattice materials and data-driven strategies. Overall, the paper offers a principled, energetically consistent pathway to efficient micro-macro simulations with robust accuracy across parameter regimes.

Abstract

We present a new asymptotic strategy for general micro-macro models which analyze complex viscoelastic fluids governed by coupled multiscale dynamics. In such models, the elastic stress appearing in the macroscopic continuum equation is derived from the microscopic kinetic theory, which makes direct numerical simulations computationally expensive. To address this challenge, we introduce a formal asymptotic scheme that expands the density function around an equilibrium distribution, thereby reducing the high computational cost associated with the fully coupled microscopic processes while still maintaining the dynamic microscopic feedback in explicit expressions. The proposed asymptotic expansion is based on a detailed physical scaling law which characterizes the multiscale balance at the fast relaxation limit of the microscopic state. An asymptotic closure model for the macroscopic fluid equation is then derived according to the explicit asymptotic density expansion. Furthermore, the resulting closure model preserves the energy-dissipation law inherited from the original fully coupled multiscale system. Numerical experiments are performed to validate the asymptotic density formula and the corresponding flow velocity equations in several micro-macro models. This new asymptotic strategy offers a promising approach for efficient computations of a wide range of multiscale complex fluids.

Paper Structure

This paper contains 20 sections, 3 theorems, 87 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Energetic variational formulation of the micro-macro model
  3. The micro-macro model from energetic variational principle
  4. A connection to the deformation tensor in the macroscopic model
  5. Asymptotic solution of the probability density at the fast relaxation limit
  6. Physical scaling law with detailed micro and macro deformation balance
  7. Explicit asymptotic expansion for the multiscale density function
  8. Macroscopic closure models with the asymptotic microscopic feedback
  9. Explicit macroscopic equation with asymptotic microstress feedback
  10. Structure-preserving energy equation for the asymptotic closure models
  11. Numerical results with the asymptotic closure models
  12. Approximation of the density distribution functions
  13. Development of macroscopic flows from microscopic elastic stress
  14. Simulations of fully coupled micro-macro systems
  15. Conclusion and future directions
  16. ...and 5 more sections

Key Result

Proposition 2

If either of the conditions in Assumption assump is satisfied, the probability density function in the following expansion up to $O\left(\gamma^{6}\right)$ gives an asymptotic solution to the Fokker-Planck equation eq:asymp_den_spring at the limit $\gamma\rightarrow 0$, $D\rightarrow\infty$ and $\gamma^4 D=c$, with $C_{\gamma}$ the normalizing constant. In addition, the leading-order macroscopic d

Figures (13)

  • Figure 5.1: Approximation of the microscopic density function in the FENE model. Colors in the 2d scatter plot indicate the density of the MC samples.
  • Figure 5.2: Approximation of the microscopic density function in the double well model. Colors in the 2d scatter plot indicate the density of the MC samples.
  • Figure 5.3: Comparison of the polymer stress $\tau_{11}-\tau_{22}$ and elastic energy $E$ from the MC solution and asymptotic approximations in the FENE model.
  • Figure 5.4: Comparison of the polymer stress $\tau_{11}-\tau_{22}$ and elastic energy $E$ from the MC solution and asymptotic approximations in the double well model.
  • Figure 5.5: Absolute error in the polymer stress $\tau_{11}-\tau_{22}$ as a function of the parameter $\gamma$.
  • ...and 8 more figures

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Proposition 2
  • Proposition 3
  • Remark 3
  • Remark 4
  • Proposition 4