Abstract
We derive a new homogenized model for heterogeneous porous media driven by inhomogeneous body forces. We assume that the <i>fine scale</i>, characterizing the heterogeneities in the medium, is larger than the <i>pore scale</i>, but nonetheless much smaller than the size of the material (<i>the coarse scale</i>). We decouple spatial variations and assume periodicity on the fine scale. Fine scale variations are formally reflected in a <i>locally unbounded</i> source for the arising system of partial differential equations. We apply the asymptotic homogenization technique to obtain a well-defined coarse scale Darcy-type model. The resulting problem is driven by an effective source which comprises both the coarse scale divergence of the average body force, and additional contributions which are to be computed solving a well-defined diffusion-type cell problem which is driven solely by fine scale variations of the given force. The present model can be used to predict the effect of externally applied magnetic (or electric) fields on ferrofluids (or electrolytes) flowing in porous media. This work can, in perspective, pave the way for investigations of the effect of applied forces on complex and heterogeneous hierarchical materials, such as systems of fractures or cancerous biological tissues.
Highlights
The flow of Netwonian fluids slowly percolating through a rigid porous matrix can be macroscopically described by the Darcy law, which linearly relates the fluid discharge to the pressure gradient and externally applied volume loads
We derive a new homogenized model for heterogeneous porous media driven by inhomogeneous body forces
Such a problem describes the homogenized fluid mechanics, i.e., where pore-scale inhomogeneities are smoothed out, the hydraulic conductivity and the effective body force can in turn exhibit heterogeneities on well-separated fine and coarse spatial scales, the former being much larger than the pore scale and, at the same time, much smaller than the coarse one characterizing the size of the whole domain
Summary
The flow of Netwonian fluids slowly percolating through a rigid porous matrix can be macroscopically described by the Darcy law, which linearly relates the fluid discharge to the pressure gradient (via the hydraulic conductivity tensor) and externally applied volume loads. Darcy’s law can be derived by means of suitable upscaling techniques It can be viewed as an approximation of the balance equations of linear momentum for a viscous fluid flowing in the pores in the context of the mixture theory, see, e.g., [29], as well as the two-scale asymptotic limit (when the average radius of the pores approaches zero sufficiently faster with respect to the whole domain’s size) of the Stokes’ problem for low Reynolds number fluids. The system of partial differential equations (PDEs) under consideration is a diffusion-type problem in terms of the pressure driven by an inhomogeneous volume source given by the divergence of the body force Whenever both fine and coarse scale variations of the fields are considered, the latter source is locally unbounded when the ratio between scales approaches zero.
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