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

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.