Abstract
It is proposed a procedure for constructing a non-integrable variational form, allowing to extend the procedure of formal construction of variational models of mechanics of deformable media to irreversible deformation processes. We introduce the non-integrable variational forms that determine possible dissipation channels depending on the list of generalized variables for a particular model of media. The problems of filtering of Biot models are considered and it is proved that non-integrable variational forms allow us to construct variational hydrodynamic models of Darcy, Navier-Stokes and Navier-Stokes-Darcy. It is shown that the equations of the Navier—Stokes—Darcy model contain the Helmholtz operator, which will allow to take into account the scale effects associated with the boundary layers.
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Funding
This work was carried out with the support of the Russian Foundation for Basic Research (project 18-01-00553-a) and particular by the Russian Government Foundation to the Institute of Applied Mechanics of the Russian Academy of Sciences (AAAA-A17-117032010137-0).
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Belov, P.A., Lurie, S.A. On Variation Models of the Irreversible Processes in Mechanics of Solids and Generalized Hydrodynamics. Lobachevskii J Math 40, 896–910 (2019). https://doi.org/10.1134/S1995080219070060
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DOI: https://doi.org/10.1134/S1995080219070060