Abstract
We describe the procedure of continuous upscaling of the flow equations. Multiscale averaging is a necessary operation in many applications. Under the introduction of the continuous line of scales, upscaling becomes similar to a Markov process described by a partial differential equation of the Ornstein-Uhlenbeck type. While previously [Shapiro, Chem. Eng. Sci. 234, 116454] the procedure of continuous upscaling was formulated for 1D flows, here we generalize it onto 3D processes. For upscaling 3D fluxes, new laws are formulated. The procedure is applied to upscale the steady-state diffusion (or heat conduction, or the pressure equation) in a heterogeneous medium. Rules for upscaling the diffusion coefficient are derived. In many cases, the upscaling of the diffusion coefficient may depend on the chosen class of solutions. It is studied numerically. The lowest values of the coefficient and the zones of its sharp variation contribute most to upscaling.
Highlights
We describe the procedure of continuous upscaling of the flow equations
Can the diffusion coefficients be upscaled without any knowledge about a solution for the diffusion equation? This study gives a double answer to this question
The multiplier es is introduced to make the resulting expression tending to a constant as s increases. This tendency follows from the dimension of the diffusion coefficient, cf. the Eq (34) and the discussion above it. According to this dimensional analysis, if x and t are scaled by eÀs, a constant diffusion coefficient should be scaled by eÀs
Summary
Upscaling is a common operation in many areas of fundamental and applied science involving multiple scales. It is applied in the theory of turbulence (Monin and Yaglom, 1971; Germano, 1992; Catton, 2006), microfluids and colloids (Eringen, 1964; Shapiro, 1996, Krehel et al, 2015), multiphase flows (Cushman, 1985; Faghri and Zhang, 2020), flows in porous media (Gray and Miller, 2014). The goal of upscaling is to derive the flow equations and expressions for transport coefficients on a coarser scale, starting from the description of the flow and geometry of the medium on a finer scale. The goal is to establish the connection between, seemingly, different physical descriptions of the processes on the different scales and, by that, to validate the application of a simpler and more practically useful macroscale description
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