A Stochastic Two-Scale Model for Rarefied Gas Flow in Highly Heterogeneous Porous Media
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A Stochastic Two‑Scale Model for Rarefied Gas Flow in Highly Heterogeneous Porous Media Francesc Pérez‑Ràfols1 · Fredrik Forsberg2 · Gunnar Hellström2 · Andreas Almqvist1 Received: 15 June 2020 / Accepted: 2 September 2020 / Published online: 16 September 2020 © The Author(s) 2020
Abstract This paper presents the development of a model enabling the analysis of rarefied gas flow through highly heterogeneous porous media. To capture the characteristics associated with the global- and the local-scale topology of the permeable phase in a typical porous medium, the heterogeneous multi-scale method, which is a flexible framework for constructing twoscale models, was employed. The rapid spatial variations associated with the local-scale topology are accounted for stochastically, by treating the permeability of different localscale domains as a random variable. The results obtained with the present model show that an increase in the spatial variability in the heterogeneous topology of the porous medium significantly reduces the relevance of rarefaction effects. This clearly shows the necessity of considering a realistic description of the pore topology and questions the applicability of the results obtained for topologies exhibiting regular pore patterns. Although the present model is developed to study low Knudsen number flows, i.e. the slip-flow regime, the same development procedure could be readily adapted for other regimes as well. Keywords Rarefaction · Two-scale · Stochastic · Heterogeneous
The authors acknowledge the support from Luleå University of Technology, under the program “Smart Machines and Materials”. Also, A.A. thanks the Swedish Research Council (Vetenskapsrådet), via the project: Multiscale topological optimisation for lower friction, less wear and leakage, registration number 2019-04293, and G.H. thanks the Swedish Research Council (Vetenskapsrådet), via the project with registration number 2017-04390. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s1124 2-020-01476-z) contains supplementary material, which is available to authorised users. * Francesc Pérez‑Ràfols [email protected] 1
Division of Machine Elements, Luleå University of Technology, 97187 Luleå, Sweden
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Division of Fluid and Experimental Mechanics, Luleå University of Technology, 97187 Luleå, Sweden
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1 Introduction When the density of a gas is very small, or when it is forced through a very small channel, the gas can no longer be described using a continuum representation. In this case, one says that the flow is rarefied. Knudsen (1909) observed this rarefaction effect already in 1909, when he studied the flow of a gas driven through a tube at low pressure. He found that the flow rate differed from the one predicted by the Poiseuille formula, which is based on theory governing flow of continuous media. Another situation where a gas does not behave as a continuum is when it flows through channels where the mean free
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