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

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.

💡 Research Summary

This paper addresses the computational bottleneck inherent in micro‑macro models of complex viscoelastic fluids, where the macroscopic Navier‑Stokes equations are coupled with a high‑dimensional Fokker‑Planck equation for the probability density function f(x,q,t) of microscopic configurations q. Directly solving the coupled system is prohibitively expensive, especially for non‑Gaussian potentials or high‑dimensional configuration spaces.

The authors start from an energetic variational formulation that combines the Least Action Principle (LAP) and the Maximum Dissipation Principle (MDP). The total energy consists of kinetic energy, Helmholtz free energy (including entropy and a microscopic potential U(q)), and two dissipation functionals: one for the macroscopic viscous stress and one for the microscopic drag. Variation of this action yields the macroscopic momentum equation and an explicit expression for the microscopic velocity V = (∇u) q − D ∇_q(γ² log f + U). The resulting micro‑macro system reads

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