Integral representation of hydraulic permeability

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Author(s)Bi, Chuan
Author(s)Ou, Miao-Jung Yvonne
Author(s)Zhang, Shangyou
Date Accessioned2023-10-10T20:23:02Z
Date Available2023-10-10T20:23:02Z
Publication Date2022-05-06
DescriptionThis article was originally published in Proceedings of the Royal Society of Edinburgh Section A: Mathematics. The version of record is available at: https://doi.org/10.1017/prm.2022.25. Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
AbstractIn this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda (Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is zμ1 and the viscosity of the hosting fluid is μ1∈R+ , z∈C . The proof is carried out by the construction of solutions for large |z| and small |z| with an iteration process similar to the one used in Bruno and Leo (Arch. Ration. Mech. Anal. 121 (1993), 303–338) and Golden and Papanicolaou (Commun. Math. Phys. 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter s:=1/(z−1) , as long as s is outside the interval [−2E221+2E22,−11+2E21] , where the positive constants E1 and E2 are the extension constants that depend only on the geometry of the periodic pore space of the material.
SponsorThe work of CB and MYO was partially sponsored by the US National Science foundation via grants NSF-DMS-1413039 and NSF-DMS-1821857.
CitationBi, Chuan, Miao-Jung Yvonne Ou, and Shangyou Zhang. “Integral Representation of Hydraulic Permeability.” Proceedings of the Royal Society of Edinburgh Section A: Mathematics 153, no. 3 (2023): 907–36. doi:10.1017/prm.2022.25.
ISSN1473-7124
URLhttps://udspace.udel.edu/handle/19716/33524
Languageen_US
PublisherProceedings of the Royal Society of Edinburgh Section A: Mathematics
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
Keywordshydraulic permeability
Keywordsstokes equations
Keywordscomposite materials
Keywordsintegral representation formula
KeywordsStieltjes class
TitleIntegral representation of hydraulic permeability
TypeArticle
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