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Exact solution to non-linear filtration in heterogeneous porous media
International Journal of Non-Linear Mechanics. 2023. Vol. 150. Article 104363.
L.I. Kuzmina, Osipov, Y. V.
Many technological processes of chemical and environmental engineering are associated with the filtration of fine particles in porous media, specifically in heterogeneous reservoirs. A one-dimensional model of deep bed filtration of suspensions and colloids in a heterogeneous porous medium is derived. The model includes the mass balance equation with variable porosity for the concentrations of suspended and retained particles and the deposit growth equation with the non-linear filtration function depending on the retained concentration and on the spatial coordinate. Using the method of characteristics, a formula for the curvilinear concentration front is obtained. The problem is reduced to one nonlinear equation in partial derivatives of the first order. An exact solution is obtained for the hyperbolic heterogeneity, and a Riemann invariant providing a relation between the solutions on the characteristics, is found. The properties of the solutions are studied. The cases of infinite and finite filtration time are considered.
Смирнов С. В., Миллионщиков Д. В., Уфимский математический журнал 2021 Т. 13 № 2 С. 44–73
В данной работе изучаются характеристические алгебры для систем экспоненциального типа, соответствующих вырожденным матрицам Картана. Эти системы обобщают хорошо известные в теории интегрируемых систем гиперболические уравнения синус-Гордон и Цицейки. Для таких систем, соответствующих матрицам Картана ранга 2, характеристические алгебры описаны явно в терминах образующих и соотношений, и доказано, что они имеют линейный рост. Исследуется связь между ...
Added: September 11, 2026
Смирнов С. В., Glasgow Mathematical Journal 2026 Vol. 68 No. 2 P. 299–316
In the second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three ...
Added: September 11, 2026
Смирнов С. В., Journal of Physics A: Mathematical and Theoretical 2023 Vol. 56 No. 26 Article 265204
There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case ...
Added: September 11, 2026
Enatskaya N., Труды Карельского научного центра РАН. Серия 10: Математическое моделирование и информационные технологии 2026 № 6 С. 132–138
In schemes S for placing r particles in n distinguishable cells, their placement in selected m cells – schemes S∗ is studied, for which the analysis is carried out by a new enumerative method (EM) in extended areas of enumerative combinatorics: finding the number of outcomes and, based on the construction of a model of ...
Added: September 11, 2026
Enatskaya N., Труды Карельского научного центра РАН. Серия 10: Математическое моделирование и информационные технологии 2026 № 6 С. 139–147
The class of particle cell arrangements under consideration is characterized by the introduction of an upper limit on the filling levels of the cells, which must be achieved in at least one cell of each outcome of each arrangement. The circuits are distinguished by the paired qualities of their constituent elements (cells and particles) according ...
Added: September 11, 2026
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 8 С. 110–123
Для многокритериальных задач принятия решений по аналогии с качественной вероятностью введены понятия полной и частичной качественной важности как бинарных отношений, обладающих постулируемыми
свойствами. Предложено новое определение отношения нестрогого предпочтения на множестве вариантов решений, порождаемое качественной
важностью. Исследованы его свойства. Указаны аналитические правила,
позволяющие попарно сравнивать варианты по предпочтительности. Проведено сравнение новых отношений предпочтения с разработанными ранее для задач, ...
Added: September 9, 2026
Podinovskiy V. V., Нелюбин А. П., Автоматика и телемеханика 2026 № 7 С. 113–126
Рассматриваются задачи принятия решений, когда предпочтения оцениваются в порядковой шкале, а возможности реализации значений
неопределенного фактора описываются качественной вероятностью (полной или только частичной). Вводятся определения отношений предпочтения и безразличий на множестве стратегий. Предлагаются простые
решающие правила, позволяющие сравнивать стратегии по предпочтительности, и приводятся иллюстративные примеры. ...
Added: September 9, 2026
Chemical Papers 2026
Benzenoid hydrocarbons are structurally regular aromatic compounds, which makes them well suited for quantitative structure–property relationship (QSPR) studies. This study presents a QSPR analysis of nineteen benzenoid hydrocarbon species using degree-based topological co-indices. We developed a Python program to compute eleven degree-based topological co-indices from molecular graphs and evaluated their ability to explain variations in ...
Added: September 8, 2026
Glutsyuk A., / Series arXiv "math". 2026.
B.Josephson (Nobel Prize, 1973) predicted a tunnelling effect for a system of two superconductors separated by a narrow dielectric (such a system is called Josephson junction): existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modeled by the family of differential equations on the 2-torus, dθdτ=1ω(cosθ+B+Acosτ), which is known as ...
Added: September 8, 2026
Glutsyuk A., / Series arXiv "math". 2026.
A planar dual billiard is a planar curve γ equipped with a family (σP)|P∈γ of projective involutions of the projective lines LP tangent to γ at P that fix P. A dual billiard is called rationally integrable, if there exists a rational function R(x,y) of two variables (called first integral) whose restriction to each tangent ...
Added: September 8, 2026
Alexandrov A., Glutsyuk A., Journal of Differential Equations 2026 Vol. 465 Article 114178
The overdamped Josephson junction in superconductivity theory can be modeled by the family of dynamical systems on the torus, which is known as the RSJ model. This family admits an equivalent description by a family of second-order differential equations: special double confluent Heun equations. In the present paper, we construct two new families of dynamical systems on ...
Added: September 8, 2026
Glutsyuk A., Inventiones Mathematicae 2026 Vol. 245 P. 977–1058
A caustic of a strictly convex planar bounded billiard is a smooth curve whose tangent lines are reflected from the billiard boundary to its tangent lines. The famous Birkhoff Conjecture, studied by many mathematicians, states that if the billiard boundary has an inner neighborhood foliated by closed caustics, then it is an ellipse. In the paper we study ...
Added: September 8, 2026
Prikhodko Artem, Kubrak D., Compositio Mathematica 2026 Vol. 162 No. 6 P. 1377–1438
In this follow-up paper we show that smooth Hodge-proper stacks over O𝐾 are ℚ𝑝-locally acyclic: namely the natural map between étale ℚ𝑝-cohomology of the algebraic and Raynaud generic fibers is an equivalence. This establishes the ℚ𝑝-case of general conjectures made in D. Kubrak and A. Prikhodko [p-adic Hodge theory for Artin stacks, Mem. Amer. Math. ...
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Ismailov A., Constructive Approximation 2026
The measure of the positivity set {x ∈ [0; 2π] | f (x) > 0} of a trigonometric polynomial f is bounded from below by the Motzkin density.
We generalize the bound to polynomials in several variables and almost periodic functions.
We then use these generalizations to extend known results on Taikov’s problem. ...
Added: September 7, 2026
Kelbert M., Statistics 2026 Vol. 60
We present variational representations for the weighted divergencies
and exponential error function, and discuss implications for the
statistical inference and entropic optimal transport. ...
Added: September 7, 2026
Kucheryavyy P., Математические заметки 2026 Т. 2026 № 120 С. 380–401
В работе изучаются перестановки, возникающие при упорядочивании по возрастанию дробных долей произведений элементов фиксированной целочисленной последовательности на вещественный параметр. Исследуется количество различных перестановок, которые можно получить таким образом при изменении этого параметра от нуля до единицы. ...
Added: September 7, 2026
Осипов Д.В., Математический сборник 2026 Т. 217 № 9 С. 130–146
Изучаются законы взаимности, связанные с комплексными линейными расслоениями на расслоениях на ориентируемые окружности. В частности, доказывается следующий закон взаимности. Пусть B – комплексное многообразие и πi:Mi→B – расслоение на ориентируемые окружности, где индекс i пробегает конечное множество. Пусть Li и Ni – комплексные линейные расслоения на каждом многообразии Mi. Закон взаимности утверждает, что сумма всех элементов (πi)∗(c1(Li)∪c1(Ni)), где (πi)∗ – ...
Added: September 3, 2026
L.I. Kuzmina, Osipov , Y. V., Kolokoltseva, T. N., Advances in Water Resources 2026 Vol. 213 Article 105335
We study 1D transport of mobilized particles, detached from the solid matrix, in porous media (so-called fines migration). Three types of colloidal-suspension flow models are considered: (i) averaged model for multicomponent colloids with distributed properties; (ii) flow of binary colloids with interacting particles; (iii) discrete system for multicomponent low-concentration colloid. These models account for distributed ...
Added: August 25, 2026
Промыслов П. В., Романов М. С., Юрьева Г. В., / Series arXiv "math". 2026.
Added: July 22, 2026
Liudmila I. Kuzmina, Osipov Y., , in: Proceedings of the 10th International Conference on Civil Engineering.: Singapore: Springer, 2024. P. 585–593.
Filtration problems of suspensions and colloids in porous media are considered when designing tunnels and underground structures. To strengthen weak soil, a liquid solution is injected into the rock, the particles of which are filtered in the pores and distributed far from the well. A deep bed filtration model of 2-particle suspension in a porous ...
Added: October 31, 2024
Liudmila I. Kuzmina, Osipov, Y., Pesterev, A., International Journal of Non-Linear Mechanics 2024 Vol. 167 Article 104905
We consider filtration of a suspension in a homogeneous porous medium which is described by a macroscopic 1-D model including the mass exchange equation and the kinetic equation for deposit growth. The standard model assumes that suspended particles move at the same speed as the carrier fluid. However, in some experiments, a lag between the ...
Added: October 31, 2024
Khazali N., Malgaresi G., Russell T. et al., Chemical Engineering Journal 2024 Vol. 496 Article 153973
Modelling of colloidal and nano-suspension transport in porous media has garnered significant attention due to the prevalence of these processes in many engineering applications. A number of experimental studies have reported retention profiles after coreflooding that are hyper-exponential, a feature that the traditional models for deep bed filtration are unable to capture. The aim of ...
Added: October 31, 2024
Kuzmina L., Osipov, Y. V., Astakhov, M. D., Mathematical Methods in the Applied Sciences 2024 Vol. 47 No. 11 P. 8319–8341
A one‐dimensional deep bed filtration model of a polydisperse suspension or colloid in a porous medium is considered. The model includes a quasilinear system of 2n equations for concentrations of suspended and retained particles of n types. The problem is reduced to a closed 3 × 3 system for total concentrations of suspended and retained ...
Added: June 27, 2024
Liudmila Kuzmina, Osipov Y., , in: BIO Web of Conferences: Volume 107:19th International Conference Water and Wastewater: Transportation, Treatment, Management “Yakovlev Readings” (YRC-2024)Vol. 107.: EDP Sciences, 2024. Ch. 03003.
The formation of grout sediment in the pores of loose rock increases the water resistance of the soil and strengthens the foundation. A one-dimensional model of filtration in a porous medium considers the particles transport by the flow of a carrier fluid and the deposition of particles on the framework of a porous medium. The ...
Added: June 27, 2024