Multiscale analysis and numerical simulation for stability of the incompressible flow of a Maxwell fluid

How to predict the stability of a small-scale flow subject to perturbations is a significant multiscale problem. It is difficult to directly study the stability by the theoretical analysis for the incompressible flow of a Maxwell fluid because of its analytical complexity. Here, we develop the multiscale analysis method based on the mathematical homogenization theory in the stress–stream function formulation. This method is used to derive the homogenized equation which governs the transport of the large-scale perturbations. The linear stabilities of the large-scale perturbations are analyzed theoretically based on the linearized homogenized equation, while the effect of the nonlinear terms on the linear stability results is discussed numerically based on the nonlinear homogenized equation. The agreements between the multiscale predictions and the direct numerical simulations demonstrate the multiscale analysis method is effective and credible to predict stabilities of flows.