Next Article in Journal
Situatedness and Embodiment of Computational Systems
Previous Article in Journal
Consistent Estimation of Partition Markov Models
Article Menu
Issue 4 (April) cover image

Export Article

Open AccessArticle
Entropy 2017, 19(4), 161; doi:10.3390/e19040161

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)

1
Centro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico (UNAM), Blvd. Juriquilla 3001, Queretaro 76230, Mexico
2
International Center for Mathematical Modelling in Physics and Cognitive Sciences, Mathematical Institute, Linnaeus University, Vaxjo SE-351 95, Sweden
3
Coordinación 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, Mexico
*
Author to whom correspondence should be addressed.
Academic Editor: Kevin H. Knuth
Received: 15 March 2017 / Revised: 24 March 2017 / Accepted: 28 March 2017 / Published: 7 April 2017
View Full-Text   |   Download PDF [3677 KB, uploaded 7 April 2017]   |  

Abstract

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. View Full-Text
Keywords: 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 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
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

Scifeed alert for new publications

Never miss any articles matching your research from any publisher
  • Get alerts for new papers matching your research
  • Find out the new papers from selected authors
  • Updated daily for 49'000+ journals and 6000+ publishers
  • Define your Scifeed now

SciFeed Share & Cite This Article

MDPI and ACS Style

Oleschko, K.; Khrennikov, A.; Correa López, M.J. 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 2017, 19, 161.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Entropy EISSN 1099-4300 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top