Multiscale analysis for convection dominated transport equations
In this paper, we perform a systematic multiscale analysis for convection dominated transport equations with a weak diffusion and a highly oscillatory velocity field. The paper primarily focuses on upscaling linear transport equations. But we also discuss briefly how to upscale two-phase miscible flows, in which case the concentration equation is coupled to the pressure equation in a nonlinear fashion. For the problem we consider here, the local Peclet number is of O(ε^(-m+1)) with m is an element of [2, infinity] being any integer, where ε characterizes the small scale in the heterogeneous media. Due to the presence of the nonlocal memory effect, upscaling a convection dominated transport equation is known to be very difficult. One of the key ideas in deriving a well-posed homogenized equation for the convection dominated transport equation is to introduce a projection operator which projects the fluctuation onto a suitable subspace. This projection operator corresponds to averaging along the streamlines of the flow. In the case of linear convection dominated transport equations, we prove the well-posedness of the homogenized equations and establish rigorous error estimates for our multiscale expansion.
Copyright © AIMS 2008. Received February 2008; revised July 2008. The authors thank the referees for their valuable comments and suggestions which have helped to improve the paper greatly. The work of T.Y. Hou was partly supported by NSF through the grants DMS-0713670, ACI-0204932, and by DOE through the grant DE-FG02-06ER25727. The work of D. Liang was supported by the Natural Sciences and Engineering Research Council of Canada.
Published - HOUdcds09.pdf