We present here the homogenization of the equations for the initial modulational (large-scale) perturbations of stationary solutions of the two-dimensional Navier-Stokes equations with a time-independent periodic rapidly oscillating forcing. The stationary solutions are cellular flows and they are determined by the stream function φ = sin x1/ε sin x2 + δ cos x1/ε cos x2/ε, 0 < δ < 1. Two results are given here. For any Reynolds number we prove the homogenization of the linearized equations. For small Reynolds number we prove the homogenization for the fully nonlinear problem. These results show that the modulational stability of cellular flows is determined by the stability of the effective (homogenized) equations.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Physics and Astronomy(all)
- Applied Mathematics