P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)
Recently p-adic (and, more generally, ultrametric) spaces representing tree-like networks of percolation, and as a special case of capillary patterns in porous media, started to be used to model the propagation of fluids (e.g., oil, water, oil-in-water, and water-in-oil emulsion). The aim of this no...
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MDPI AG
2017-04-01
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| author | Klaudia Oleschko Andrei Khrennikov María de Jesús Correa López |
| author_facet | Klaudia Oleschko Andrei Khrennikov María de Jesús Correa López |
| author_sort | Klaudia Oleschko |
| collection | DOAJ |
| description | Recently p-adic (and, more generally, ultrametric) spaces representing tree-like networks of percolation, and as a special case of capillary patterns in porous media, started to be used to model the propagation of fluids (e.g., oil, water, oil-in-water, and water-in-oil emulsion). The aim of this note is to derive p-adic dynamics described by fractional differential operators (Vladimirov operators) starting with discrete dynamics based on hierarchically-structured interactions between the fluids’ volumes concentrated at different levels of the percolation tree and coming to the multiscale universal topology of the percolating nets. Similar systems of discrete hierarchic equations were widely applied to modeling of turbulence. However, in the present work this similarity is only formal since, in our model, the trees are real physical patterns with a tree-like topology of capillaries (or fractures) in random porous media (not cascade trees, as in the case of turbulence, which we will be discussed elsewhere for the spinner flowmeter commonly used in the petroleum industry). By going to the “continuous limit” (with respect to the p-adic topology) we represent the dynamics on the tree-like configuration space as an evolutionary nonlinear p-adic fractional (pseudo-) differential equation, the tree-like analog of the Navier–Stokes equation. We hope that our work helps to come closer to a nonlinear equation solution, taking into account the scaling, hierarchies, and formal derivations, imprinted from the similar properties of the real physical world. Once this coupling is resolved, the more problematic question of information scaling in industrial applications will be achieved. |
| first_indexed | 2024-04-11T12:40:25Z |
| format | Article |
| id | doaj.art-c860435a77064a2c8b1e069568af0987 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T12:40:25Z |
| publishDate | 2017-04-01 |
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| record_format | Article |
| series | Entropy |
| spelling | doaj.art-c860435a77064a2c8b1e069568af09872022-12-22T04:23:31ZengMDPI AGEntropy1099-43002017-04-0119416110.3390/e19040161e19040161P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)Klaudia Oleschko0Andrei Khrennikov1María de Jesús Correa López2Centro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico (UNAM), Blvd. Juriquilla 3001, Queretaro 76230, MexicoInternational Center for Mathematical Modelling in Physics and Cognitive Sciences, Mathematical Institute, Linnaeus University, Vaxjo SE-351 95, SwedenCoordinación del Grupo Multidisciplinario de Especialistas Técnicos de Diseño de Proyectos. Caracterizacion de Yacimientos, Activo de Produccion Ku-Maloob-Zaap, Ed. Kaxan, Av. Contadores, Carretera Carmen Puerto Real, Cd. Del Carmen, Campeche 24150, MexicoRecently p-adic (and, more generally, ultrametric) spaces representing tree-like networks of percolation, and as a special case of capillary patterns in porous media, started to be used to model the propagation of fluids (e.g., oil, water, oil-in-water, and water-in-oil emulsion). The aim of this note is to derive p-adic dynamics described by fractional differential operators (Vladimirov operators) starting with discrete dynamics based on hierarchically-structured interactions between the fluids’ volumes concentrated at different levels of the percolation tree and coming to the multiscale universal topology of the percolating nets. Similar systems of discrete hierarchic equations were widely applied to modeling of turbulence. However, in the present work this similarity is only formal since, in our model, the trees are real physical patterns with a tree-like topology of capillaries (or fractures) in random porous media (not cascade trees, as in the case of turbulence, which we will be discussed elsewhere for the spinner flowmeter commonly used in the petroleum industry). By going to the “continuous limit” (with respect to the p-adic topology) we represent the dynamics on the tree-like configuration space as an evolutionary nonlinear p-adic fractional (pseudo-) differential equation, the tree-like analog of the Navier–Stokes equation. We hope that our work helps to come closer to a nonlinear equation solution, taking into account the scaling, hierarchies, and formal derivations, imprinted from the similar properties of the real physical world. Once this coupling is resolved, the more problematic question of information scaling in industrial applications will be achieved.http://www.mdpi.com/1099-4300/19/4/161capillary network in porous mediafluid flowtree-like structureutrametric spacep-adic numbersp-adic pseudo-differential operatorsdiscrete dynamics with tree-like hierarchycontinuous limitthe tree-like analog of Navier–Stokes equation |
| spellingShingle | Klaudia Oleschko Andrei Khrennikov María de Jesús Correa López P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model) Entropy capillary network in porous media fluid flow tree-like structure utrametric space p-adic numbers p-adic pseudo-differential operators discrete dynamics with tree-like hierarchy continuous limit the tree-like analog of Navier–Stokes equation |
| title | P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model) |
| title_full | P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model) |
| title_fullStr | P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model) |
| title_full_unstemmed | P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model) |
| title_short | P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model) |
| title_sort | p adic analog of navier stokes equations dynamics of fluid s flow in percolation networks from discrete dynamics with hierarchic interactions to continuous universal scaling model |
| topic | capillary network in porous media fluid flow tree-like structure utrametric space p-adic numbers p-adic pseudo-differential operators discrete dynamics with tree-like hierarchy continuous limit the tree-like analog of Navier–Stokes equation |
| url | http://www.mdpi.com/1099-4300/19/4/161 |
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