Some free boundary problems arising in rock mechanics
Meirmanov A. Yerzhanov K.
2024TUBITAK
Turkish Journal of Mathematics
2024#48Issue 61089 - 1109 pp.
An initial boundary value problem for the in situ leaching is considered. We describe physical processes at the microscopic level with a pore size ε << 1 by model Aε, where dynamics of the incompressible solid skeleton is described by the Lamé equations and the physical process in the pore space by the Stokes equations for the incompressible fluid with diffusion equations for the concentration of acid and product of chemical reactions. Since the solid skeleton changes its geometry upon dissolution, the “pore space – solid skeleton” boundary is a free boundary. The goal of the present manuscript is a model H, which is the homogenization of the model Aε. That is, the limit as ε tend to zero, of the model Aε. As usual, free boundary problems are only solvable locally in time. On the other hand, in situ leaching has a very long process duration and there is still no correct microscopic model that describes this process for an arbitrary time interval. To avoid this contradiction, we propose correct approximate microscopic models Bε(r) for this process with a given solid skeleton structure depending on some function r from the set M(0, T ). Problem Bε(r) is the model Aε without an additional boundary condition at the free boundary that defines this boundary, but with some additional terms in the Stokes and Lame equations that depend linearly on the velocities and disappear upon homogenization. To derive a macroscopic mathematical model H(r) and separately the additional boundary condition at free boundary we use Nguenseng’s two-scale convergence method as ε tends to zero. As a result, we obtain a homogenized model H(r) and an additional equation, possesses construct an operator, which fixed point uniquely defines function r∗ from the set M(0, T ) and prove the existence and uniqueness theorem for the macroscopic mathematical model H.
fixed point theorem , Free boundary problems , homogenization , structures with special periodicity
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Physic and Technical Faculty, L.N.Gumilyov Eurasian National University, Astana, Kazakhstan
Physic and Technical Faculty
10 лет помогаем публиковать статьи Международный издатель
Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026