= Homogenization of a periodic advection-diffusion equation
At long wavelengths and on a <diffusive time scale>, a rapidly varying periodic <advection-diffusion equation> can have a constant effective <diffusivity>. A <multiple-scale expansion> solves periodic cell equations for the fast variables. The <solvability condition> at the next order gives the macroscopic equation. For a periodic sinusoidal shear this is the <effective diffusivity of a periodic sinusoidal shear>; its transverse zero-mean cell modes decay, leaving the slow conserved concentration. This use of homogenization concerns differential equations, rather than polynomial <homogenization>.
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