Homogenization theory for advection-diffusion equations with mean flow

The problem of periodic homogenization for advection-diffusion equations is considered in this thesis. We study the problem for velocity field which consist of two parts, a spatiotemporally dependent mean flow and a periodically fluctuating part. Under the assumption of scale separation between the characteristic length and time scales of the mean flow and the fluctuations we derive an effective equation which governs the evolution of the passive scalar field at the length and time scales of the mean flow. We are mostly interested in understanding the effect of the mean flow upon the homogenized transport of the passive scalar field.; We show rigorously that for mean flows which are either weak or equal in strength with the fluctuations the effective equation is an advection-diffusion equation with an effective diffusion tensor which is computed through the solution of an auxiliary partial differential equation with periodic boundary conditions, the cell problem. We show that the structure of the cell problem depends on the temporal period of oscillations of the fluctuations in the velocity field. A very efficient algorithm of the solution of the cell problem is also developed.; For weak mean flows and in the absence of slow modulations in the fluctuations the effective diffusion tensor is constant, independent of the mean flow. When fluctuations and mean flow are of equal strength the effective diffusion tensor is a function of space and time, with values depending upon the mean flow as well as the slow modulations in the fluctuations. When the mean flow is stronger than the fluctuations one cannot in general obtain an effective equation which is independent of the fast variables. In this regime greater variability of the effective diffusivity can occur, depending upon the specific properties of the mean flow: from no enhacement in the diffusivity to the appearance of resonant enhanced diffusion phenomena that boost the diffusivity far above its bare molecular value. The problem is studied through a combination of formal asymptotic analysis of the cell problem, numerical experiments and rigorous analysis using the method of two-scale convergence.; The symmetry properties of the effective diffusion tensor are also studied. Necessary and sufficient conditions for the symmetry of the effective diffusivity are derived for steady velocity fields and the dependence of the antisymmetric part of the diffusivity on the Peclet number is analyzed. Numerical examples for both steady and time dependent velocity field are also presented.; Finally, we propose a systematic way of studying higher order homogenization using the method of two-scale convergence. Our technique enables us to rigorously obtain higher order effective equations in cases where the multiple scales technique breaks down.