Pub Date : 2024-04-13DOI: 10.26907/2541-7746.2024.1.58-73
R. Okulov, V. A. Krashaninin, B. R. Gelchinsky, A. Rempel
This article considers how the shape of the inner channel in the anode assembly affects plasma flow velocity in a plasma torch. Three different shapes of the anode assembly were analyzed, all with a conical confusor part of 50 mm in length: with a diameter transition from 12 to 6 mm, from 12 to 8 mm, and from 12 to 10 mm. A computer experiment was performed using the finite element method and then validated by the subsequent full-scale experiment on a laboratory plasma unit. The obtained results were verified. The verification outcomes showed a satisfactory convergence and were consistent with the published data. A review of the existing plasma unit designs for powder production, application of functional coatings, and surface modification was carried out. The software packages implementing the finite element method to solve these problems were examined. The study yielded practical recommendations for consumers and developers of plasma equipment and identified the shapes of the anode assembly enabling both supersonic and subsonic plasma flow regimes.
{"title":"Influence of the shape of the anode assembly inner channel on plasma flow velocity","authors":"R. Okulov, V. A. Krashaninin, B. R. Gelchinsky, A. Rempel","doi":"10.26907/2541-7746.2024.1.58-73","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.58-73","url":null,"abstract":" This article considers how the shape of the inner channel in the anode assembly affects plasma flow velocity in a plasma torch. Three different shapes of the anode assembly were analyzed, all with a conical confusor part of 50 mm in length: with a diameter transition from 12 to 6 mm, from 12 to 8 mm, and from 12 to 10 mm. A computer experiment was performed using the finite element method and then validated by the subsequent full-scale experiment on a laboratory plasma unit. The obtained results were verified. The verification outcomes showed a satisfactory convergence and were consistent with the published data. A review of the existing plasma unit designs for powder production, application of functional coatings, and surface modification was carried out. The software packages implementing the finite element method to solve these problems were examined. The study yielded practical recommendations for consumers and developers of plasma equipment and identified the shapes of the anode assembly enabling both supersonic and subsonic plasma flow regimes.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"70 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140707892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-13DOI: 10.26907/2541-7746.2024.1.74-91
N. Pleshchinskii
Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.
推导了在研究拉夫连季耶夫-比萨泽方程的特里科米问题时出现的积分方程的反演公式。利用格林函数法找到了混合域椭圆部分辅助超定问题的可解性条件。以积分方程相互倒置的形式，在迪里夏特问题的格林函数和拉普拉斯方程的问题 N 之间建立了联系。考虑了各种积分方程，包括可明确求解的积分方程，Tricomi 问题可以简化为这些方程。在不涉及解析函数边界值问题理论的情况下，获得了具有考奇内核的特征奇异方程的显式解。
{"title":"Tricomi problem and integral equations","authors":"N. Pleshchinskii","doi":"10.26907/2541-7746.2024.1.74-91","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.74-91","url":null,"abstract":" Formulas for inverting integral equations that arise when studying the Tricomi problem for the Lavrentyev–Bitsadze equation were derived. Solvability conditions of an auxiliary overdetermined problem in the elliptic part of the mixed domain were found using the Green function method. A connection was established between the Green functions of the Dirichlet problem and problem N for the Laplace equation in the form of integral equations mutually inverting each other. Various integral equations were considered, including explicitly solvable ones, to which the Tricomi problem can be reduced. An explicit solution of the characteristic singular equation with a Cauchy kernel was obtained without involving the theory of boundary value problems for analytic functions.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"26 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140707549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-13DOI: 10.26907/2541-7746.2024.1.99-110
V. L. Sennitskii
The problem of the non-stationary flow of a viscous liquid with an external free boundary around a moving solid cylindrical body was formulated and solved. The liquid is subject to periodic impacts with or without the predominant direction in space. To formulate the problem, the Navier—Stokes equation, the continuity equation, and the equation of conditions at both the solid and free boundaries of the liquid were used. New hydro-mechanical effects were discovered.
{"title":"On the motion of a viscous liquid with a free boundary","authors":"V. L. Sennitskii","doi":"10.26907/2541-7746.2024.1.99-110","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.99-110","url":null,"abstract":" The problem of the non-stationary flow of a viscous liquid with an external free boundary around a moving solid cylindrical body was formulated and solved. The liquid is subject to periodic impacts with or without the predominant direction in space. To formulate the problem, the Navier—Stokes equation, the continuity equation, and the equation of conditions at both the solid and free boundaries of the liquid were used. New hydro-mechanical effects were discovered.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"66 10","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140707744","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-13DOI: 10.26907/2541-7746.2024.1.92-98
M. A. Sevodin
The possibilities of combining known techniques for optimizing the securities portfolio (SP) structure were studied. A method was introduced that enables the simultaneous use of both passive and active approaches to managing the SP structure. The combined application of these methods is based on techniques for SP diversification and searching for an SP structure that mirrors the SP structure of an index fund. The objective function was modified in order to optimize the SP structure according to the traditional “return–risk” approach. The proposed objective function, along with the security risk, describes the degree to which the desired distribution of SP shares coincides with the distribution generated using an index fund. It was established that the main properties of optimal SPs obtained with the “return–risk” approach also occur in the case under consideration.
{"title":"Combined strategies for managing the securities portfolio structure","authors":"M. A. Sevodin","doi":"10.26907/2541-7746.2024.1.92-98","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.92-98","url":null,"abstract":" The possibilities of combining known techniques for optimizing the securities portfolio (SP) structure were studied. A method was introduced that enables the simultaneous use of both passive and active approaches to managing the SP structure. The combined application of these methods is based on techniques for SP diversification and searching for an SP structure that mirrors the SP structure of an index fund. The objective function was modified in order to optimize the SP structure according to the traditional “return–risk” approach. The proposed objective function, along with the security risk, describes the degree to which the desired distribution of SP shares coincides with the distribution generated using an index fund. It was established that the main properties of optimal SPs obtained with the “return–risk” approach also occur in the case under consideration.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"11 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140708824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-13DOI: 10.26907/2541-7746.2024.1.52-57
M. F. Nasrutdinov
A key exchange protocol over a special class of formal matrices Bn(R, P) was proposed. The potential of this design for constructing key exchange protocols using suitable associative rings and ideals over them was shown.
{"title":"A key exchange protocol based on the ring Bn(R, P)","authors":"M. F. Nasrutdinov","doi":"10.26907/2541-7746.2024.1.52-57","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.52-57","url":null,"abstract":" A key exchange protocol over a special class of formal matrices Bn(R, P) was proposed. The potential of this design for constructing key exchange protocols using suitable associative rings and ideals over them was shown.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"51 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140708562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-13DOI: 10.26907/2541-7746.2024.1.111-122
P. Shabalin, R. Faizov
This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.
{"title":"The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis","authors":"P. Shabalin, R. Faizov","doi":"10.26907/2541-7746.2024.1.111-122","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.111-122","url":null,"abstract":" This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"32 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140708630","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-12DOI: 10.26907/2541-7746.2024.1.22-35
S. Baizaev, R. Barotov
This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.
本文探讨了一个包含 n 个方程的椭圆系统，其中主要部分是比萨泽算子（考奇-黎曼算子的平方），下部项是给定矩阵函数与所需矢量函数共轭的乘积。该系统是在整个复平面上有界且均匀 H¨older 连续的矢量函数的巴拿赫空间中分析的。结果发现，在指定空间中求解系统的问题可能不是诺特问题。举例说明了具有无限多个线性独立解的均质系统。众所周知，对于许多类别的椭圆系统，紧凑域中边界值问题的 Noetherian 性等同于相应空间中先验估计的存在。因此，研究在上述空间中为所考虑的系统建立先验估计的相关问题非常重要。在系数在无穷大处弱振荡的情况下，找到了先验估计有效性的必要条件和充分条件。这些条件是用系数矩阵的部分极限在无穷大时形成的极限矩阵谱语言写出来的。我们还提供了具体的例子来说明如何构造极限矩阵以及上述条件是什么样的。
{"title":"Some estimates for elliptic systems generalizing the Bitsadze system of equations","authors":"S. Baizaev, R. Barotov","doi":"10.26907/2541-7746.2024.1.22-35","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.22-35","url":null,"abstract":" This article explores an elliptic system of n equations where the main part is the Bitsadze operator (the square of the Cauchy–Riemann operator) and the lower term is the product of a given matrix function by the conjugate of the desired vector function. The system was analyzed in the Banach space of vector functions that are bounded and uniformly H¨older continuous in the entire complex plane. It was revealed that the problem of solving the system in the specified space may not be Noetherian. An example of a homogeneous system with an infinite number of linearly independent solutions was given. As is known, for many classes of elliptic systems, the Noetherianity of boundary value problems in a compact domain is equivalent to the presence of a priori estimates in the corresponding spaces. In this regard, it is important to study the issues related to the establishment of a priori estimates for the system under consideration in the above space. In the case of coefficients weakly oscillating at infinity, necessary and sufficient conditions for the validity of the a priori estimate were found. These conditions were written out in the language of the spectrum of limit matrices formed by the partial limits of the coefficient matrix at infinity. Specific examples were provided to illustrate how the limit matrices are constructed and what the above conditions look like.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"7 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140709974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-12DOI: 10.26907/2541-7746.2024.1.36-51
Y. Gubarev, M. S. Kotelnikova
The one-dimensional problem of the linear stability of dynamic states of local thermodynamic equilibria with respect to small perturbations was studied for the case when the Vlasov–Poisson electron gas contains electrons with a stationary distribution function that is constant in physical space and variable in a continuum of velocities. The absolute instability of all considered one-dimensional dynamic states of any local thermodynamic equilibrium was proved using the direct Lyapunov method. The scope of applicability of the Newcomb–Gardner–Rosenbluth sufficient condition for linear stability was outlined. An a priori exponential estimation was obtained for the rise of small one-dimensional perturbations from below. Analytic counterexamples to the spectral Newсomb–Gardner theorem and the Penrose criterion were constructed. Earnshaw’s theorem was extended from classical mechanics tostatistical one.
{"title":"On the stability of a particular class of one-dimensional states of dynamic equilibrium of the Vlasov–Poisson electron gas","authors":"Y. Gubarev, M. S. Kotelnikova","doi":"10.26907/2541-7746.2024.1.36-51","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.36-51","url":null,"abstract":" The one-dimensional problem of the linear stability of dynamic states of local thermodynamic equilibria with respect to small perturbations was studied for the case when the Vlasov–Poisson electron gas contains electrons with a stationary distribution function that is constant in physical space and variable in a continuum of velocities. The absolute instability of all considered one-dimensional dynamic states of any local thermodynamic equilibrium was proved using the direct Lyapunov method. The scope of applicability of the Newcomb–Gardner–Rosenbluth sufficient condition for linear stability was outlined. An a priori exponential estimation was obtained for the rise of small one-dimensional perturbations from below. Analytic counterexamples to the spectral Newсomb–Gardner theorem and the Penrose criterion were constructed. Earnshaw’s theorem was extended from classical mechanics tostatistical one.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"15 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140710019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-11DOI: 10.26907/2541-7746.2024.1.5-21
V. K. Andreev, I. V. Stepanova
This article considers an initial-boundary value problem for a system of parabolic equations, which arises when studying the flow of a binary mixture in a horizontal channel with walls heated non-uniformly. The problem was reduced to a sequence of initial-boundary value problems with Dirichlet or Neumann conditions. Among them, an inverse problem with a non-local overdetermination condition was distinguished. The solution was constructed using the Fourier method and validated as classical. The behavior of the non-stationary solution at large times was discussed. It was shown that certain functions within the solution tend to their stationary analogs exponentially at large times. For some functions, only boundedness was proved. The problem and its solution are relevant for modeling the thermal modes associated with the separation of liquid mixtures.
{"title":"A priori and a posteriori estimates for solving one evolutionary inverse problem","authors":"V. K. Andreev, I. V. Stepanova","doi":"10.26907/2541-7746.2024.1.5-21","DOIUrl":"https://doi.org/10.26907/2541-7746.2024.1.5-21","url":null,"abstract":" This article considers an initial-boundary value problem for a system of parabolic equations, which arises when studying the flow of a binary mixture in a horizontal channel with walls heated non-uniformly. The problem was reduced to a sequence of initial-boundary value problems with Dirichlet or Neumann conditions. Among them, an inverse problem with a non-local overdetermination condition was distinguished. The solution was constructed using the Fourier method and validated as classical. The behavior of the non-stationary solution at large times was discussed. It was shown that certain functions within the solution tend to their stationary analogs exponentially at large times. For some functions, only boundedness was proved. The problem and its solution are relevant for modeling the thermal modes associated with the separation of liquid mixtures.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"14 8","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140716213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-02-18DOI: 10.26907/2541-7746.2023.4.326-343
A. A. Azanov, E. Lemeshkova
The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.
{"title":"Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection","authors":"A. A. Azanov, E. Lemeshkova","doi":"10.26907/2541-7746.2023.4.326-343","DOIUrl":"https://doi.org/10.26907/2541-7746.2023.4.326-343","url":null,"abstract":"The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.","PeriodicalId":516762,"journal":{"name":"Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki","volume":"42 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140452691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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