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Trudy Mat. Inst. Steklova, 2020, Volume 310, Pages 309–321 (Mi tm4100)  

Dynamics of Perturbations under Diffusion in a Porous Medium

V. A. Shargatovab, A. T. Il'ichevcd

a Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526 Russia
b National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow, 115409 Russia
c Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
d Bauman Moscow State Technical University, Vtoraya Baumanskaya ul. 5, Moscow, 105005 Russia

Abstract: We consider the dynamics of finite perturbations of a plane phase transition surface in the problem of evaporation of a fluid inside a low-permeability layer of a porous medium. In the case of a nonwettable porous medium, the problem has two stationary solutions, each containing a discontinuity. These discontinuities correspond to plane stationary phase transition surfaces located inside the low-permeability porous layer. One of these surfaces is unstable with respect to long-wavelength perturbations, while the other is stable. We study the evolution of perturbations of the stable plane phase transition surface. It is known that when two phase transition surfaces are located close enough to each other, the dynamics of a weakly nonlinear and weakly unstable wave packet is described by the Kolmogorov–Petrovskii–Piskunov (KPP) diffusion equation. As traveling wave solutions, this equation has heteroclinic solutions with either oscillating or monotonic structure of the front. The boundary value problem in the full statement, which should be considered if the distance between the stable and unstable plane phase transition surfaces is not small, also has similar solutions. We formulate a sufficient condition for the decrease of finite perturbations of the stable plane phase transition surface. This condition depends on their position with respect to the standing wave type and traveling front type solutions of the model equations in the model description when the KPP equation holds.

Funding Agency Grant Number
Russian Science Foundation 16-11-10195
This work is supported by the Russian Science Foundation under grant 16-11-10195.


DOI: https://doi.org/10.4213/tm4100

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English version:
Proceedings of the Steklov Institute of Mathematics, 2020, 310, 291–303

Bibliographic databases:

UDC: 532.59
Received: December 22, 2019
Revised: December 22, 2019
Accepted: April 6, 2020

Citation: V. A. Shargatov, A. T. Il'ichev, “Dynamics of Perturbations under Diffusion in a Porous Medium”, Selected issues of mathematics and mechanics, Collected papers. On the occasion of the 70th birthday of Academician Valery Vasil'evich Kozlov, Trudy Mat. Inst. Steklova, 310, Steklov Math. Inst., Moscow, 2020, 309–321; Proc. Steklov Inst. Math., 310 (2020), 291–303

Citation in format AMSBIB
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\paper Dynamics of Perturbations under Diffusion in a Porous Medium
\inbook Selected issues of mathematics and mechanics
\bookinfo Collected papers. On the occasion of the 70th birthday of Academician Valery Vasil'evich Kozlov
\serial Trudy Mat. Inst. Steklova
\yr 2020
\vol 310
\pages 309--321
\publ Steklov Math. Inst.
\publaddr Moscow
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