Multiscale Analysis and Numerical Simulations for Stability of Incompressible Periodic Flow of Maxwell Fluid

For incompressible small-scale flow of Maxwell fluid subject to forcing periodic in space and time, the mean-field equations which govern the transport of the large-scale perturbations, were obtained by the multiscale asymptotic analysis. A general mathematical formalism was developed to determine the effective tensor. And in general the effective tensor is a fourth-order tensor, for which a compact representation was provided. The mean-field equations were derived in detail and the exact explicit expressions of the effective tensor were given for the parallel time-independent flow. For Kolmogorov flow, the critical value of the viscosity for stability of the large scale perturbations was obtained by a theoretic analysis of eigenvalues of the homogenized operator appearing in the mean-field equations. And then the mean-field equations and the original linearized equations for different parameters and initial conditions were simulated by using SIMPLEC(Semi-Implicit Method for Pressure Linked Equations, Consistent) method in the collocated grid system. The comparisons between the results of direct numerical simulations and the analytical predictions of the multiscale analysis demonstrate the multiscale asymptotic analysis and the numerical algorithm used in this paper are effective and credible.