Pub Date : 2023-12-20DOI: 10.1134/s0081543823040223
M. E. Eglit, Yu. A. Drozdova, I. N. Usachev, A. V. Drozdov
Abstract
Laminar flows of fluids with a yield strength in pipes under the action of a periodically changing pressure drop are considered. The Herschel–Bulkley model is adopted to describe the rheological properties of moving fluids. The effect of pressure drop fluctuations on velocity profiles, as well as on the mean flow rates, friction on the pipe walls, and the thickness of the “quasi-solid” core, is studied numerically, depending on the amplitude and frequency of pressure drop fluctuations, the generalized Bingham number, and the fluid power index. It is shown that in flows of viscoplastic fluids with a power index greater than one, the effect of pressure drop fluctuations is qualitatively different in different ranges of shear rates. Additionally, flows are investigated in which the relative amplitudes of pressure drop oscillations are large. At large amplitudes and low frequencies of pressure drop oscillations, counter-flows periodically arise in the flow and two (rather than one) zones of “quasi-solid” flow are formed; moreover, finite time intervals periodically appear in which the flow rate is zero.
{"title":"Flows of Liquids with a Yield Strength in Pipes under a Pulsating Pressure Drop","authors":"M. E. Eglit, Yu. A. Drozdova, I. N. Usachev, A. V. Drozdov","doi":"10.1134/s0081543823040223","DOIUrl":"https://doi.org/10.1134/s0081543823040223","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Laminar flows of fluids with a yield strength in pipes under the action of a periodically changing pressure drop are considered. The Herschel–Bulkley model is adopted to describe the rheological properties of moving fluids. The effect of pressure drop fluctuations on velocity profiles, as well as on the mean flow rates, friction on the pipe walls, and the thickness of the “quasi-solid” core, is studied numerically, depending on the amplitude and frequency of pressure drop fluctuations, the generalized Bingham number, and the fluid power index. It is shown that in flows of viscoplastic fluids with a power index greater than one, the effect of pressure drop fluctuations is qualitatively different in different ranges of shear rates. Additionally, flows are investigated in which the relative amplitudes of pressure drop oscillations are large. At large amplitudes and low frequencies of pressure drop oscillations, counter-flows periodically arise in the flow and two (rather than one) zones of “quasi-solid” flow are formed; moreover, finite time intervals periodically appear in which the flow rate is zero. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040181
G. G. Tsypkin
Abstract
A mathematical model is proposed for the replacement of methane with carbon dioxide in a hydrate contained in a reservoir in thermodynamic equilibrium with water and free methane. The substitution reaction region is assumed to be narrow enough to be approximated by the conversion front. A self-similar solution is found that reduces the problem to the numerical analysis of a system of transcendental equations. Numerical experiments show that there exist three characteristic regimes of injection, whose implementation depends on the amount of water in the free state in the reservoir.
{"title":"Mathematical Model of Replacing Methane in Hydrate with Carbon Dioxide When It Is Injected into a Reservoir Saturated with a Mixture of Hydrate, Methane, and Water","authors":"G. G. Tsypkin","doi":"10.1134/s0081543823040181","DOIUrl":"https://doi.org/10.1134/s0081543823040181","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A mathematical model is proposed for the replacement of methane with carbon dioxide in a hydrate contained in a reservoir in thermodynamic equilibrium with water and free methane. The substitution reaction region is assumed to be narrow enough to be approximated by the conversion front. A self-similar solution is found that reduces the problem to the numerical analysis of a system of transcendental equations. Numerical experiments show that there exist three characteristic regimes of injection, whose implementation depends on the amount of water in the free state in the reservoir. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821798","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040077
A. N. Golubyatnikov, D. V. Ukrainskii
Abstract
We discuss the well-known problem of energy concentration, which is the inverse of the strong explosion or the expanding piston problem. Using a number of physical examples, we show that under certain conditions and with certain forces involved in the focusing process, one can achieve the concentration of any amount of energy. First of all, this applies to the gravity force and its manifestation in problems of relativity theory with viscosity and heat conduction taken into account.
{"title":"On the Problem of Energy Concentration","authors":"A. N. Golubyatnikov, D. V. Ukrainskii","doi":"10.1134/s0081543823040077","DOIUrl":"https://doi.org/10.1134/s0081543823040077","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We discuss the well-known problem of energy concentration, which is the inverse of the strong explosion or the expanding piston problem. Using a number of physical examples, we show that under certain conditions and with certain forces involved in the focusing process, one can achieve the concentration of any amount of energy. First of all, this applies to the gravity force and its manifestation in problems of relativity theory with viscosity and heat conduction taken into account. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"42 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040211
V. A. Shargatov, A. P. Chugainova, A. M. Tomasheva
Abstract
We study the structure of the set of traveling wave solutions for the generalized Korteweg–de Vries–Burgers equation with the flux function having four inflection points. In this case there arise two monotone structures of stable special discontinuities propagating at different velocities (such a situation has not been described earlier in the literature). Both structures of special discontinuities are linearly stable. To analyze the linear stability of the structures of classical and special discontinuities, we apply a method based on the use of the Evans function. We also propose a conjecture that establishes the admissibility of classical discontinuities in the case when there are two stable special discontinuities.
{"title":"Structures of Classical and Special Discontinuities for the Generalized Korteweg–de Vries–Burgers Equation in the Case of a Flux Function with Four Inflection Points","authors":"V. A. Shargatov, A. P. Chugainova, A. M. Tomasheva","doi":"10.1134/s0081543823040211","DOIUrl":"https://doi.org/10.1134/s0081543823040211","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the structure of the set of traveling wave solutions for the generalized Korteweg–de Vries–Burgers equation with the flux function having four inflection points. In this case there arise two monotone structures of stable special discontinuities propagating at different velocities (such a situation has not been described earlier in the literature). Both structures of special discontinuities are linearly stable. To analyze the linear stability of the structures of classical and special discontinuities, we apply a method based on the use of the Evans function. We also propose a conjecture that establishes the admissibility of classical discontinuities in the case when there are two stable special discontinuities. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"136 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817119","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040168
Ivan Yu. Polekhin
Abstract
We present a topological–analytical method for proving some results of the N. N. Bogolyubov averaging method for the case of an infinite time interval. The essence of the method is to combine topological methods of proving the existence of a periodic solution applied to the averaged system with Bogolyubov’s theorem on the averaging on a finite time interval. The proposed approach allows us to dispense with the nondegeneracy condition for the Jacobi matrix from the classical theorems of the averaging method.
摘要 我们提出了一种拓扑分析方法，用于证明 N. N. Bogolyubov 平均法在无限时间间隔情况下的一些结果。该方法的实质是将证明周期解存在的拓扑方法与博格留波夫关于有限时间间隔上的平均定理结合起来。所提出的方法使我们可以省去平均法经典定理中雅各比矩阵的非退化条件。
{"title":"A Topological–Analytical Method for Proving Averaging Theorems on an Infinite Time Interval in a Degenerate Case","authors":"Ivan Yu. Polekhin","doi":"10.1134/s0081543823040168","DOIUrl":"https://doi.org/10.1134/s0081543823040168","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We present a topological–analytical method for proving some results of the N. N. Bogolyubov averaging method for the case of an infinite time interval. The essence of the method is to combine topological methods of proving the existence of a periodic solution applied to the averaged system with Bogolyubov’s theorem on the averaging on a finite time interval. The proposed approach allows us to dispense with the nondegeneracy condition for the Jacobi matrix from the classical theorems of the averaging method. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"70 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040041
Sergey V. Bolotin
Abstract
We obtain explicit formulas for the separatrix map of a multidimensional slow–fast Hamiltonian system. This map is used to partly extend Neishtadt’s results on the jumps of adiabatic invariants at the separatrix to the multidimensional case.
{"title":"Separatrix Maps in Slow–Fast Hamiltonian Systems","authors":"Sergey V. Bolotin","doi":"10.1134/s0081543823040041","DOIUrl":"https://doi.org/10.1134/s0081543823040041","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We obtain explicit formulas for the separatrix map of a multidimensional slow–fast Hamiltonian system. This map is used to partly extend Neishtadt’s results on the jumps of adiabatic invariants at the separatrix to the multidimensional case. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"70 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040090
S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova
Abstract
By coastal waves we mean time-periodic or nearly time-periodic gravity waves on water in a basin of depth (D(x)), (x=(x_1,x_2)), that are localized in the vicinity of the coastline (Gamma^0={D(x)=0}). In this paper, for the system of nonlinear shallow water equations, we construct asymptotic solutions corresponding to coastal waves in two specific examples. The solutions are presented in the form of parametrically defined functions corresponding to asymptotic solutions of the linearized system, which, in turn, are related to the asymptotic eigenfunctions of the operator (-nablacdot (g D(x)nabla)) that are generated by billiards with semi-rigid walls.
Abstract by coastal waves we mean time-periodic or nearly time-periodic gravity waves on water in a basin of depth (D(x)), (x=(x_1,x_2)), that are localized in the vicinity of the coastline (Gamma^0={D(x)=0})。本文针对非线性浅水方程系统，在两个具体例子中构建了与海岸波相对应的渐近解。这些解以参数定义函数的形式呈现，这些函数与线性化系统的渐近解相对应，而线性化系统的渐近解又与具有半刚性壁的台球产生的算子 (-nablacdot (g D(x)nabla)) 的渐近特征函数相关。
{"title":"Nonlinear Effects and Run-up of Coastal Waves Generated by Billiards with Semi-rigid Walls in the Framework of Shallow Water Theory","authors":"S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova","doi":"10.1134/s0081543823040090","DOIUrl":"https://doi.org/10.1134/s0081543823040090","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> By coastal waves we mean time-periodic or nearly time-periodic gravity waves on water in a basin of depth <span>(D(x))</span>, <span>(x=(x_1,x_2))</span>, that are localized in the vicinity of the coastline <span>(Gamma^0={D(x)=0})</span>. In this paper, for the system of nonlinear shallow water equations, we construct asymptotic solutions corresponding to coastal waves in two specific examples. The solutions are presented in the form of parametrically defined functions corresponding to asymptotic solutions of the linearized system, which, in turn, are related to the asymptotic eigenfunctions of the operator <span>(-nablacdot (g D(x)nabla))</span> that are generated by billiards with semi-rigid walls. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"143 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138821535","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040120
A. G. Kulikovskii, J. S. Zayko
Abstract
We consider the evolution of linear waves of small perturbations of an unstable flow of a viscous fluid layer over a curved surface. The source of perturbations is assumed to be given by initial conditions defined in a small domain (in the limit, in the form of a (delta)-function) or by an instantaneous localized external impact. The behavior of perturbations is described by hydrodynamic equations averaged over the thickness of the layer, with the gravity force and bottom friction taken into account (Saint-Venant equations). We study the asymptotic behavior of one-dimensional perturbations for large times. The inclination of the surface to the horizon is defined by a slowly varying function of the spatial variable. We focus on the perturbation amplitude as a function of time and the spatial variable. To study the asymptotics of perturbations, we use a simple generalization of the well-known method, based on the saddle-point technique, for finding the asymptotics of perturbations developing against a uniform background. We show that this method is equivalent to the one based on the application of the approximate WKB method for constructing solutions of differential equations. When constructing the asymptotics, it is convenient to assume that (x) is a real variable and to allow time (t) to take complex values.
{"title":"On Waves on the Surface of an Unstable Layer of a Viscous Fluid Flowing Down a Curved Surface","authors":"A. G. Kulikovskii, J. S. Zayko","doi":"10.1134/s0081543823040120","DOIUrl":"https://doi.org/10.1134/s0081543823040120","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider the evolution of linear waves of small perturbations of an unstable flow of a viscous fluid layer over a curved surface. The source of perturbations is assumed to be given by initial conditions defined in a small domain (in the limit, in the form of a <span>(delta)</span>-function) or by an instantaneous localized external impact. The behavior of perturbations is described by hydrodynamic equations averaged over the thickness of the layer, with the gravity force and bottom friction taken into account (Saint-Venant equations). We study the asymptotic behavior of one-dimensional perturbations for large times. The inclination of the surface to the horizon is defined by a slowly varying function of the spatial variable. We focus on the perturbation amplitude as a function of time and the spatial variable. To study the asymptotics of perturbations, we use a simple generalization of the well-known method, based on the saddle-point technique, for finding the asymptotics of perturbations developing against a uniform background. We show that this method is equivalent to the one based on the application of the approximate WKB method for constructing solutions of differential equations. When constructing the asymptotics, it is convenient to assume that <span>(x)</span> is a real variable and to allow time <span>(t)</span> to take complex values. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"70 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040144
V. Yu. Liapidevskii
Abstract
The flow of a stratified fluid over a slope is considered. In the one-layer shallow water approximation, a mathematical model is constructed for a turbulent flow of a denser fluid over a uniform slope, with the entrainment of the ambient fluid at rest and the sediment entrainment at the wave front taken into account. The main focus is on analyzing the structure of a self-sustaining wave (underwater avalanche) and on estimating its propagation velocity. The mathematical model arises from the equilibrium conditions in a more complete three-parameter model and contains only one numerical parameter that represents a combination of the parameters of the original model characterizing the slope, vortex energy dissipation rate, and entrainment rate. The structure of traveling waves is studied, exact self-similar solutions are constructed, and transition of the flow to a self-similar regime is analyzed numerically. It is shown that depending on the thickness and initial density of the sediment layer, self-similar solutions have different structures and front propagation velocities.
{"title":"Equilibrium Model of Density Flow","authors":"V. Yu. Liapidevskii","doi":"10.1134/s0081543823040144","DOIUrl":"https://doi.org/10.1134/s0081543823040144","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The flow of a stratified fluid over a slope is considered. In the one-layer shallow water approximation, a mathematical model is constructed for a turbulent flow of a denser fluid over a uniform slope, with the entrainment of the ambient fluid at rest and the sediment entrainment at the wave front taken into account. The main focus is on analyzing the structure of a self-sustaining wave (underwater avalanche) and on estimating its propagation velocity. The mathematical model arises from the equilibrium conditions in a more complete three-parameter model and contains only one numerical parameter that represents a combination of the parameters of the original model characterizing the slope, vortex energy dissipation rate, and entrainment rate. The structure of traveling waves is studied, exact self-similar solutions are constructed, and transition of the flow to a self-similar regime is analyzed numerically. It is shown that depending on the thickness and initial density of the sediment layer, self-similar solutions have different structures and front propagation velocities. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"73 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-20DOI: 10.1134/s0081543823040065
V. V. Bulatov
Abstract
Issues related to the statement of problems of describing the dynamics of linear internal gravity waves in stratified media with horizontal shear flows in critical modes of wave generation are considered. Model physical statements of problems in which critical levels may arise are discussed in the two-dimensional case. Analytic properties of the solutions near critical levels are studied. A system describing a flow of a stratified medium incident on an obstacle behind which outgoing waves may arise is discussed, in which case a singularity at the critical level is formed far away from the obstacle. Asymptotics of the solutions near the critical level are constructed and expressed in terms of the incomplete gamma function.
{"title":"Analytic Properties of Solutions to the Equation of Internal Gravity Waves with Flows for Critical Modes of Wave Generation","authors":"V. V. Bulatov","doi":"10.1134/s0081543823040065","DOIUrl":"https://doi.org/10.1134/s0081543823040065","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> Issues related to the statement of problems of describing the dynamics of linear internal gravity waves in stratified media with horizontal shear flows in critical modes of wave generation are considered. Model physical statements of problems in which critical levels may arise are discussed in the two-dimensional case. Analytic properties of the solutions near critical levels are studied. A system describing a flow of a stratified medium incident on an obstacle behind which outgoing waves may arise is discussed, in which case a singularity at the critical level is formed far away from the obstacle. Asymptotics of the solutions near the critical level are constructed and expressed in terms of the incomplete gamma function. </p>","PeriodicalId":54557,"journal":{"name":"Proceedings of the Steklov Institute of Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138817379","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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