[Text] : статья / V. K. Andreev, I. V. Stepanova // Journal of Applied and Industrial Mathematics. - 2011. - Vol. 5, Iss. 4. - p. 491-499DOI 10.1134/S199047891104003X
. -
Аннотация: Under study is an invariant solution of the equations of thermal diffusive convection which describes a stationary process of a binary mixture flow in a vertical layer under the action of the pressure gradient and the buoyancy force that depends nonlinearly on temperature and concentration. Some general properties of this solution are established and an existence theorem is proved. Analysis of the numerical solution of the problem is carried out in the cases of a power-law and exponential dependence of the buoyancy force on its argument.
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Доп.точки доступа:
Stepanova, I.V.; Степанова, Ирина Владимировна; Андреев, Виктор Константинович
Труды сотрудников ИВМ СО РАН
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Найдено документов в текущей БД: 13
Analytical assessments of electron temperature and kinetics of gas ionization in the resonance field
/ A. P. Gavrilyuk // Teplofizika Vysokikh Temperatur. - 1995. - Vol. 33, Is. 1. - P144-146
. - ISSN 0040-3644
Кл.слова (ненормированные):
Electrons -- Equations of state of gases -- Laser beams -- Plasma heating -- Resonance -- Temperature measurement -- Electron temperature -- Resonance field -- Ionization of gases
Аннотация: The paper assesses coefficients and rates of ionization caused by an electron shock of atomic gas. It is meant that this gas is excited by resonance radiation and that ionization rate depends on the tune out of radiation frequency from the center of absorption line. It has been shown that in the central part of the line, where saturation is implemented, the coefficient depends only on characteristics of the resonance transition and ionization potential of the atom, and that the growth of electron concentration in time is exponential. When the far end of the line is tuned out, saturation disappears and dynamics of concentration growth becomes nonlinear.
Scopus
Держатели документа:
VTs SO RAN, Krasnoyarsk, Russian Federation
ИВМ СО РАН
Доп.точки доступа:
Gavrilyuk, A.P.; Гаврилюк, Анатолий Петрович
Кл.слова (ненормированные):
Electrons -- Equations of state of gases -- Laser beams -- Plasma heating -- Resonance -- Temperature measurement -- Electron temperature -- Resonance field -- Ionization of gases
Аннотация: The paper assesses coefficients and rates of ionization caused by an electron shock of atomic gas. It is meant that this gas is excited by resonance radiation and that ionization rate depends on the tune out of radiation frequency from the center of absorption line. It has been shown that in the central part of the line, where saturation is implemented, the coefficient depends only on characteristics of the resonance transition and ionization potential of the atom, and that the growth of electron concentration in time is exponential. When the far end of the line is tuned out, saturation disappears and dynamics of concentration growth becomes nonlinear.
Scopus
Держатели документа:
VTs SO RAN, Krasnoyarsk, Russian Federation
ИВМ СО РАН
Доп.точки доступа:
Gavrilyuk, A.P.; Гаврилюк, Анатолий Петрович
The evolution of the thermocapillary motion of three fluids in a plane layer
[Text] : статья / V. K. Andreev, Ye. N. Lemeshkhova // PMM JOURNAL OF APPLIED MATHEMATICS AND MECHANICS. - 2014. - Vol. 78, Iss. 4. - p. 341-347DOI 10.1016/j.jappmathmech.2014.12.004
. -
Аннотация: The unidirectional motion of three immiscible incompressible viscous heat-conducting liquids in a plane layer is considered. It is assumed that the motion occurs only under the action of thermocapillary forces from a state of rest. The analysis of the motion is reduced to solving linear conjugate initial boundary value problems for a system of parabolic equations. A non-stationary solution is sought by the Laplace transformation method and is obtained in the form of finite analytical expressions in transforms. It is proved that, as the time increases, the solution always reaches the steady state obtained earlier and an exponential estimate of the rate of convergence is given with an indicator which depends on the physical properties of the media and the layer thicknesses. The evolution of the velocity and temperature perturbation fields to a steady state for specific liquid media is obtained by numerical inversion of the Laplace transformation. (C) 2015 Elsevier Ltd. All rights reserved.
Доп.точки доступа:
Lemeshkhova, Ye.N.; Лемешкова, Елена Николаевна; Андреев, Виктор Константинович
Аннотация: The unidirectional motion of three immiscible incompressible viscous heat-conducting liquids in a plane layer is considered. It is assumed that the motion occurs only under the action of thermocapillary forces from a state of rest. The analysis of the motion is reduced to solving linear conjugate initial boundary value problems for a system of parabolic equations. A non-stationary solution is sought by the Laplace transformation method and is obtained in the form of finite analytical expressions in transforms. It is proved that, as the time increases, the solution always reaches the steady state obtained earlier and an exponential estimate of the rate of convergence is given with an indicator which depends on the physical properties of the media and the layer thicknesses. The evolution of the velocity and temperature perturbation fields to a steady state for specific liquid media is obtained by numerical inversion of the Laplace transformation. (C) 2015 Elsevier Ltd. All rights reserved.
Доп.точки доступа:
Lemeshkhova, Ye.N.; Лемешкова, Елена Николаевна; Андреев, Виктор Константинович
Symmetry analysis of nonlinear heat and mass transfer equations under Soret effect
/ I. V. Stepanova // Comm. Nonlinear Sci. Numer. Simul. - 2015. - Vol. 20, Is. 3. - P684-691, DOI 10.1016/j.cnsns.2014.06.043
. - ISSN 1007-5704
Кл.слова (ненормированные):
Exact solutions -- Group classification problem -- Lie symmetry analysis -- Thermodiffusion equations -- Transport coefficients -- Exact solution -- Group classification -- Lie symmetry analysis -- Thermodiffusion -- Transport coefficient
Аннотация: Three-dimensional equations describing heat and mass transfer in fluid mixtures with variable transport coefficients are studied. Using Lie group theory the forms of unknown thermal diffusivity, diffusion and thermal diffusion coefficients are found. The symmetries of the governing equations are calculated. It is shown that cases of Lie symmetry extension arise when arbitrary elements have the power-law, logarithmic and exponential dependencies on temperature and concentration. An exact solution is constructed for the case of linear dependence of diffusion and thermodiffusion coefficients on temperature. The solution demonstrates differences in concentration distribution in comparison with the same distribution under constant transport coefficients in the governing equations. © 2014 Elsevier B.V.
Scopus
Держатели документа:
Institute of Computational Modeling SB RAS, 50/44, Krasnoyarsk, Akademgorodok, Russian Federation
Доп.точки доступа:
Stepanova, I. V.
Кл.слова (ненормированные):
Exact solutions -- Group classification problem -- Lie symmetry analysis -- Thermodiffusion equations -- Transport coefficients -- Exact solution -- Group classification -- Lie symmetry analysis -- Thermodiffusion -- Transport coefficient
Аннотация: Three-dimensional equations describing heat and mass transfer in fluid mixtures with variable transport coefficients are studied. Using Lie group theory the forms of unknown thermal diffusivity, diffusion and thermal diffusion coefficients are found. The symmetries of the governing equations are calculated. It is shown that cases of Lie symmetry extension arise when arbitrary elements have the power-law, logarithmic and exponential dependencies on temperature and concentration. An exact solution is constructed for the case of linear dependence of diffusion and thermodiffusion coefficients on temperature. The solution demonstrates differences in concentration distribution in comparison with the same distribution under constant transport coefficients in the governing equations. © 2014 Elsevier B.V.
Scopus
Держатели документа:
Institute of Computational Modeling SB RAS, 50/44, Krasnoyarsk, Akademgorodok, Russian Federation
Доп.точки доступа:
Stepanova, I. V.
Моделирование нестационарного контакта в подшипнике качения
[Текст] : статья / В. А. Иванов, Н. В. Еркаев // Вестник Сибирского государственного аэрокосмического университета им. академика М.Ф. Решетнева. - 2015. - Т. 16, № 3. - С. 580-586
. - ISSN 1816-9724
Перевод заглавия: Simulation of non-steady contact in rolling bearings
Кл.слова (ненормированные):
Transient mode -- contact interaction -- lubricant layer -- нестационарный режим -- контактное взаимодействие -- смазочный слой
Аннотация: Рассмотрена задача нестационарного гидродинамического контакта ролика с упругим слоем с учетом прогиба поверхности, а также влияния давления на коэффициент вязкости. Зависимость вязкости от давления задана экспоненциальной функцией. В работе использовался итерационный метод численного решения уравнения Рейнольдса совместно с интегральным уравнением связи прогиба поверхности с давлением в смазочном слое. Показано, что вертикальное перемещение ролика вызывает дополнительное существенное возрастание давления в смазочном слое, пропорциональное вертикальной скорости. Коэффициент линейной зависимости несущей способности от вертикальной скорости назван коэффициентом демпфирования. В результате расчетов получены зависимости несущей способности и коэффициента демпфирования смазочного слоя от величины минимального зазора между роликом и пластиной. С использованием найденных функций изучен переходный процесс установления стационарного режима при резком изменении внешней нагрузки. Найдено характерное время установления и определены временные вариации пиковых значений давления. Исследовано влияние пьезокоэффициента вязкости на максимальные значения давления, достигаемые в процессе установления. Найдено критическое значение пьезокоэффициента, при котором эффект возрастания давления, обусловленный увеличением вязкости, компенсируется влиянием деформации упругой поверхности.
This article deals with the problem of non-steady hydrodynamic contact of a roller with finite size elastic plate. The lubricant viscosity coefficient is assumed to be exponential function of the pressure. For this problem, an iterative numerical method was elaborated to solve the 2-D Reynolds’ equation consistently with the integral equation of relationship between the surface deflection and pressure distribution in the lubrication layer. A normal motion of the roller causes additional pressure enhancement in the lubrication layer, which is proportional to the normal velocity. Coefficient of proportionality is called as damping coefficient. Carrying capacity and damping coefficient are determined from numerical solution as functions of minimal distance between the roller and plate. The obtained functions were used for modeling of the roller oscillations due to sudden variations of the external loading. Characteristic relaxation time and temporal variations of the pressure maximum are determined. Dependence of the pressure maximum on a special piezo-coefficient was investigated, which is a parameter of the exponential function approximating relationship between viscosity and pressure. Higher values of the piezo-coefficient yield larger values of the pressure maximum in the lubrication layer during the relaxation period. However, deflection of the body surfaces makes an opposite effect on the pressure. Therefore behavior of the pressure maximum is determined by two opposite factors related to the viscosity piezo-effect and surface deformations. From numerical simulations, a critical value of the piezo-coefficient is found when the influence of the piezo-coefficient is compensated by that of deformation of the elastic plate.
РИНЦ,
Полный текст сборника
Держатели документа:
Институт вычислительного моделирования CО РАН
Сибирский федеральный университет, Политехнический институт
Доп.точки доступа:
Еркаев, Николай Васильевич; Erkaev N.V.; Ivanov V.A.
Перевод заглавия: Simulation of non-steady contact in rolling bearings
| УДК |
Кл.слова (ненормированные):
Transient mode -- contact interaction -- lubricant layer -- нестационарный режим -- контактное взаимодействие -- смазочный слой
Аннотация: Рассмотрена задача нестационарного гидродинамического контакта ролика с упругим слоем с учетом прогиба поверхности, а также влияния давления на коэффициент вязкости. Зависимость вязкости от давления задана экспоненциальной функцией. В работе использовался итерационный метод численного решения уравнения Рейнольдса совместно с интегральным уравнением связи прогиба поверхности с давлением в смазочном слое. Показано, что вертикальное перемещение ролика вызывает дополнительное существенное возрастание давления в смазочном слое, пропорциональное вертикальной скорости. Коэффициент линейной зависимости несущей способности от вертикальной скорости назван коэффициентом демпфирования. В результате расчетов получены зависимости несущей способности и коэффициента демпфирования смазочного слоя от величины минимального зазора между роликом и пластиной. С использованием найденных функций изучен переходный процесс установления стационарного режима при резком изменении внешней нагрузки. Найдено характерное время установления и определены временные вариации пиковых значений давления. Исследовано влияние пьезокоэффициента вязкости на максимальные значения давления, достигаемые в процессе установления. Найдено критическое значение пьезокоэффициента, при котором эффект возрастания давления, обусловленный увеличением вязкости, компенсируется влиянием деформации упругой поверхности.
This article deals with the problem of non-steady hydrodynamic contact of a roller with finite size elastic plate. The lubricant viscosity coefficient is assumed to be exponential function of the pressure. For this problem, an iterative numerical method was elaborated to solve the 2-D Reynolds’ equation consistently with the integral equation of relationship between the surface deflection and pressure distribution in the lubrication layer. A normal motion of the roller causes additional pressure enhancement in the lubrication layer, which is proportional to the normal velocity. Coefficient of proportionality is called as damping coefficient. Carrying capacity and damping coefficient are determined from numerical solution as functions of minimal distance between the roller and plate. The obtained functions were used for modeling of the roller oscillations due to sudden variations of the external loading. Characteristic relaxation time and temporal variations of the pressure maximum are determined. Dependence of the pressure maximum on a special piezo-coefficient was investigated, which is a parameter of the exponential function approximating relationship between viscosity and pressure. Higher values of the piezo-coefficient yield larger values of the pressure maximum in the lubrication layer during the relaxation period. However, deflection of the body surfaces makes an opposite effect on the pressure. Therefore behavior of the pressure maximum is determined by two opposite factors related to the viscosity piezo-effect and surface deformations. From numerical simulations, a critical value of the piezo-coefficient is found when the influence of the piezo-coefficient is compensated by that of deformation of the elastic plate.
РИНЦ,
Полный текст сборника
Держатели документа:
Институт вычислительного моделирования CО РАН
Сибирский федеральный университет, Политехнический институт
Доп.точки доступа:
Еркаев, Николай Васильевич; Erkaev N.V.; Ivanov V.A.
On the solution of an inverse problem simulating two-dimensional motion of a viscous fluid
/ V. K. Andreev // Bull. South Ural State Univ. Ser. Math. Model. Program. Comput. Softw. - 2016. - Vol. 9, Is. 4. - С. 5-16, DOI 10.14529/mmpl60401
. - ISSN 2071-0216
Кл.слова (ненормированные):
A priori assessment -- Exponential stability -- Inverse problem -- Laplace transform
Аннотация: An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
Scopus,
Смотреть статью
Держатели документа:
Institute Computational Modelling, SB RAS, Krasnoyarsk, Russian Federation
Доп.точки доступа:
Andreev, V. K.
Кл.слова (ненормированные):
A priori assessment -- Exponential stability -- Inverse problem -- Laplace transform
Аннотация: An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
Scopus,
Смотреть статью
Держатели документа:
Institute Computational Modelling, SB RAS, Krasnoyarsk, Russian Federation
Доп.точки доступа:
Andreev, V. K.
Stationary and transient Soret separation in a binary mixture with a consolute critical point
/ I. I. Ryzhkov, S. V. Kozlova // Eur. Phys. J. E. - 2016. - Vol. 39, Is. 12, DOI 10.1140/epje/i2016-16130-6
. - ISSN 1292-8941
Кл.слова (ненормированные):
Topical Issue: Non-isothermal transport in complex fluids -- Bins -- Diffusion -- Mixtures -- Polymer blends -- Separation -- Complex fluids -- Concentration gradients -- Concentration variation -- Critical temperatures -- Initial concentration -- Local relaxation time -- Soret coefficients -- Temperature differences -- Binary mixtures
Аннотация: Abstract.: The stationary and transient Soret separation in a binary mixture with a consolute critical point is studied theoretically. The mixture is placed between two parallel plates kept at different temperatures. A polymer blend is used as a model system. Analytical solutions are constructed to describe the stationary separation in a binary mixture with variable Soret coefficient. The latter strongly depends on temperature and concentration and enhances near a consolute critical point due to reduced diffusion. As a result, a large concentration gradient is observed locally, while much smaller concentration variations are found in the rest of the layer. It is shown that complete separation can be obtained by applying a small temperature difference first, waiting for the establishment of stationary state, and then increasing this difference again. In this case, the critical temperature lies between hot and cold wall temperatures, while the mixture still remains in the one-phase region. When the initial (mean) temperature or concentration are shifted away from the near-critical values, the separation decreases. The analysis of transient behavior shows that the Soret separation occurs much faster than diffusion to the homogeneous state when the initial concentration is close to the critical one. It happens due to the decrease (increase) of the local relaxation time during the Soret (Diffusion) steps. The transient times of these steps become comparable for small temperature differences or off-critical initial concentrations. An unusual (non-exponential) separation dynamics is observed when the separation starts in the off-critical domain, and then enhances greatly when the system enters into the near-critical region. It is also found that the transient time decreases with increasing the applied temperature difference. Graphical abstract: [Figure not available: see fulltext.] © 2016, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.
Scopus,
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WOS
Держатели документа:
Institute of Computational Modelling SB RAS, Akademgorodok, Krasnoyarsk, Russian Federation
Siberian Federal University, Svobodny 79, Krasnoyarsk, Russian Federation
Доп.точки доступа:
Kozlova, S.V.; Kozlova S.V.; Рыжков, Илья Игоревич
Кл.слова (ненормированные):
Topical Issue: Non-isothermal transport in complex fluids -- Bins -- Diffusion -- Mixtures -- Polymer blends -- Separation -- Complex fluids -- Concentration gradients -- Concentration variation -- Critical temperatures -- Initial concentration -- Local relaxation time -- Soret coefficients -- Temperature differences -- Binary mixtures
Аннотация: Abstract.: The stationary and transient Soret separation in a binary mixture with a consolute critical point is studied theoretically. The mixture is placed between two parallel plates kept at different temperatures. A polymer blend is used as a model system. Analytical solutions are constructed to describe the stationary separation in a binary mixture with variable Soret coefficient. The latter strongly depends on temperature and concentration and enhances near a consolute critical point due to reduced diffusion. As a result, a large concentration gradient is observed locally, while much smaller concentration variations are found in the rest of the layer. It is shown that complete separation can be obtained by applying a small temperature difference first, waiting for the establishment of stationary state, and then increasing this difference again. In this case, the critical temperature lies between hot and cold wall temperatures, while the mixture still remains in the one-phase region. When the initial (mean) temperature or concentration are shifted away from the near-critical values, the separation decreases. The analysis of transient behavior shows that the Soret separation occurs much faster than diffusion to the homogeneous state when the initial concentration is close to the critical one. It happens due to the decrease (increase) of the local relaxation time during the Soret (Diffusion) steps. The transient times of these steps become comparable for small temperature differences or off-critical initial concentrations. An unusual (non-exponential) separation dynamics is observed when the separation starts in the off-critical domain, and then enhances greatly when the system enters into the near-critical region. It is also found that the transient time decreases with increasing the applied temperature difference. Graphical abstract: [Figure not available: see fulltext.] © 2016, EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg.
Scopus,
Смотреть статью,
WOS
Держатели документа:
Institute of Computational Modelling SB RAS, Akademgorodok, Krasnoyarsk, Russian Federation
Siberian Federal University, Svobodny 79, Krasnoyarsk, Russian Federation
Доп.точки доступа:
Kozlova, S.V.; Kozlova S.V.; Рыжков, Илья Игоревич
ON THE SOLUTION OF AN INVERSE PROBLEM SIMULATING TWO-DIMENSIONAL MOTION OF A VISCOUS FLUID
/ V. K. Andreev // Bull. South Ural State U. Ser.-Math Model Program Comput. - 2016. - Vol. 9, Is. 4. - С. 5-16, DOI 10.14529/mmp160401. - Cited References:16
. - ISSN 2071-0216. - ISSN 2308-0256
РУБ Mathematics, Applied
Аннотация: An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
WOS,
Смотреть статью
Держатели документа:
Inst Computat Modelling SB RAS, Krasnoyarsk, Russia.
Доп.точки доступа:
Andreev, V. K.
Рубрики:
EQUATIONS
Кл.слова (ненормированные):
inverse problem -- a priori assessment -- Laplace transform -- exponential -- stability
EQUATIONS
Кл.слова (ненормированные):
inverse problem -- a priori assessment -- Laplace transform -- exponential -- stability
Аннотация: An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
WOS,
Смотреть статью
Держатели документа:
Inst Computat Modelling SB RAS, Krasnoyarsk, Russia.
Доп.точки доступа:
Andreev, V. K.
517.956.27
О-110
О-110
О решении одной обратной задачи, моделирующей двумерное движение вязкой жидкости
[Текст] : статья / Виктор Константинович Андреев // Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование. - 2016. - Т. 9, № 4. - С. 5-16, DOI 10.14529/mmp160401
. - ISSN 2071-0216
Перевод заглавия: On the Solution of an Inverse Problem Simulating Two-Dimensional Motion of a Viscous Fluid
Кл.слова (ненормированные):
inverse problem -- a priori assessment -- Laplace transform -- exponential stability -- обратная задача -- априорные оценки -- преобразование Лапласа -- экспоненциальная устойчивость
Аннотация: Рассматривается обратная начально-краевая задача для линейного параболического уравнения, которая возникает при математическом моделировании двумерных ползущих движений вязкой жидкости в плоском канале. Неизвестная функция времени входит в правую часть уравнения аддитивно и находится из дополнительного условия интегрального переопределения. Поставленная задача имеет два разных интегральных тождества, которые позволяют получить априорные оценки решения в равномерной метрике и доказать теорему единственности. При некоторых ограничениях на входные данные решение построено в виде ряда по специальному базису. Для этого задача путем дифференцирования по пространственной переменной сводится к прямой неклассической задаче с двумя интегральными условиями вместо обычных краевых. Новая задача решается методом разделения переменных, позволяющим найти неизвестные функции в виде быстро сходящихся рядов. Другой, стандартный, метод решения исходной задачи состоит в сведении ее к нагруженному уравнению и первой начально-краевой задаче для него. В свою очередь, эта задача сведена к одномерному по времени операторному уравнению Вольтерры со специальным ядром. Доказано, что оно имеет решение в виде ряда. Установлены некоторые вспомогательные формулы, полезные при численном решении этого уравнения методом преобразования Лапласа. Установлены достаточные условия, при которых решение с ростом времени выходит на стационарный режим по экспоненциальному закону.
An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
РИНЦ
Держатели документа:
Институт вычислительного моделирования СО РАН
Доп.точки доступа:
Андреев, Виктор Константинович; Andreev V.K.
Перевод заглавия: On the Solution of an Inverse Problem Simulating Two-Dimensional Motion of a Viscous Fluid
| УДК |
Кл.слова (ненормированные):
inverse problem -- a priori assessment -- Laplace transform -- exponential stability -- обратная задача -- априорные оценки -- преобразование Лапласа -- экспоненциальная устойчивость
Аннотация: Рассматривается обратная начально-краевая задача для линейного параболического уравнения, которая возникает при математическом моделировании двумерных ползущих движений вязкой жидкости в плоском канале. Неизвестная функция времени входит в правую часть уравнения аддитивно и находится из дополнительного условия интегрального переопределения. Поставленная задача имеет два разных интегральных тождества, которые позволяют получить априорные оценки решения в равномерной метрике и доказать теорему единственности. При некоторых ограничениях на входные данные решение построено в виде ряда по специальному базису. Для этого задача путем дифференцирования по пространственной переменной сводится к прямой неклассической задаче с двумя интегральными условиями вместо обычных краевых. Новая задача решается методом разделения переменных, позволяющим найти неизвестные функции в виде быстро сходящихся рядов. Другой, стандартный, метод решения исходной задачи состоит в сведении ее к нагруженному уравнению и первой начально-краевой задаче для него. В свою очередь, эта задача сведена к одномерному по времени операторному уравнению Вольтерры со специальным ядром. Доказано, что оно имеет решение в виде ряда. Установлены некоторые вспомогательные формулы, полезные при численном решении этого уравнения методом преобразования Лапласа. Установлены достаточные условия, при которых решение с ростом времени выходит на стационарный режим по экспоненциальному закону.
An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
РИНЦ
Держатели документа:
Институт вычислительного моделирования СО РАН
Доп.точки доступа:
Андреев, Виктор Константинович; Andreev V.K.
Решение некоторых обратных задач для специальных движений вязкой жидкости
[Текст] : доклад, тезисы доклада / Виктор Константинович Андреев // Некоторые актуальные проблемы современной математики и математического образования : материалы научной конференции Герценовские чтения LXIX. - Санкт-Петербург : Российский государственный педагогический университет им. А.И. Герцена, 2016. - С. 260
. - ISBN 978-5-8064-2234-8
Перевод заглавия: Solution of some special inverse problem for the motion of viscous fluids
Кл.слова (ненормированные):
conjugate problem -- inverse problem -- apriori estimates -- exponential stability -- сопряжённая задача -- обратная задача -- априорные оценки -- экспоненциальная устойчивость
Аннотация: Исследуется линейная сопряжённая обратная задача для параболических уравнений, возникающая при описании ползущих двумерных движений вязких теплопроводных жидкостей или бинарных смесей в плоских слоях. Получены априорные оценки решения и найдены условия на входные данные, когда с ростом времени решение выходит на стационарный режим. В образах по Лапласу построено точное решение в виде квадратур и приведены численные результаты для конкретных жидких сред.
A linear conjugate inverse problem for parabolic equations is investigated. This problem arises under describing of creeping 2D motions of viscous heat conducting liquids or binary mixtures in plane layers. Some a priori estimates are obtained and input data conditions when solution tends to stationary one are found. In Laplace transforms the exact solution is obtained as quadratures and some numerical results for real liquids are calculated.
РИНЦ,
Источник статьи
Держатели документа:
Федеральный исследовательский центр "Красноярский научный центр Сибирского отделения Российской академии наук, Институт вычислительного моделирования СО РАН
Доп.точки доступа:
Андреев, Виктор Константинович; Andreev V. K.; Герценовские чтения LXIX "Некоторые актуальные проблемы современной математики и математического образования"(2016 ; 11.04 - 15.04 ; Санкт-Петербург)
Нет сведений об экземплярах (Источник в БД не найден)
Перевод заглавия: Solution of some special inverse problem for the motion of viscous fluids
Кл.слова (ненормированные):
conjugate problem -- inverse problem -- apriori estimates -- exponential stability -- сопряжённая задача -- обратная задача -- априорные оценки -- экспоненциальная устойчивость
Аннотация: Исследуется линейная сопряжённая обратная задача для параболических уравнений, возникающая при описании ползущих двумерных движений вязких теплопроводных жидкостей или бинарных смесей в плоских слоях. Получены априорные оценки решения и найдены условия на входные данные, когда с ростом времени решение выходит на стационарный режим. В образах по Лапласу построено точное решение в виде квадратур и приведены численные результаты для конкретных жидких сред.
A linear conjugate inverse problem for parabolic equations is investigated. This problem arises under describing of creeping 2D motions of viscous heat conducting liquids or binary mixtures in plane layers. Some a priori estimates are obtained and input data conditions when solution tends to stationary one are found. In Laplace transforms the exact solution is obtained as quadratures and some numerical results for real liquids are calculated.
РИНЦ,
Источник статьи
Держатели документа:
Федеральный исследовательский центр "Красноярский научный центр Сибирского отделения Российской академии наук, Институт вычислительного моделирования СО РАН
Доп.точки доступа:
Андреев, Виктор Константинович; Andreev V. K.; Герценовские чтения LXIX "Некоторые актуальные проблемы современной математики и математического образования"(2016 ; 11.04 - 15.04 ; Санкт-Петербург)
Нет сведений об экземплярах (Источник в БД не найден)
On some group properties of heat and mass transfer equations
/ I. V. Stepanova // J. Phys. Conf. Ser. - 2017. - Vol. 894, Is. 1, DOI 10.1088/1742-6596/894/1/012090
. - ISSN 1742-6588
Кл.слова (ненормированные):
Continuum mechanics -- Fluid dynamics -- Group theory -- Heat transfer -- Hydrodynamics -- Lie groups -- Thermal conductivity -- Group properties -- Heat and mass transfer -- Power-law -- Transport coefficient -- Mass transfer
Аннотация: Heat and mass transfer equations with variable transport coefficients are under study. The forms of unknown thermal conductivity, diffusion and Dufour coefficients are found by means of Lie group theory. It is shown that arbitrary elements have the power-law, logarithmic and exponential dependencies on temperature and concentration.
Scopus,
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Держатели документа:
Institute of Computational Modeling SB RAS, Krasnoyarsk, Russian Federation
Доп.точки доступа:
Stepanova, I. V.
Кл.слова (ненормированные):
Continuum mechanics -- Fluid dynamics -- Group theory -- Heat transfer -- Hydrodynamics -- Lie groups -- Thermal conductivity -- Group properties -- Heat and mass transfer -- Power-law -- Transport coefficient -- Mass transfer
Аннотация: Heat and mass transfer equations with variable transport coefficients are under study. The forms of unknown thermal conductivity, diffusion and Dufour coefficients are found by means of Lie group theory. It is shown that arbitrary elements have the power-law, logarithmic and exponential dependencies on temperature and concentration.
Scopus,
Смотреть статью
Держатели документа:
Institute of Computational Modeling SB RAS, Krasnoyarsk, Russian Federation
Доп.точки доступа:
Stepanova, I. V.
517.9
A10
A10
A Priori Estimates of the Adjoint Problem Describing the Slow Flow of a Binary Mixture and a Fluid in a Channel
[Text] : статья / Victor K. Andreev, Marina V. Efimova // Журнал Сибирского федерального университета. Серия: Математика и физика. - 2018. - Т. 11, № 4. - P482-493, DOI 10.17516/1997-1397-2018-11-4-482-493
. - ISSN 1997-1397
Перевод заглавия: Априорные оценки сопряженной задачи, описывающей совместное движение жидкости и бинарной смеси в канале
Кл.слова (ненормированные):
conjugate problem -- inverse problem -- a priori estimates -- asymptotic behavior -- сопряженная задача -- обратная задача -- априорные оценки -- асимтотическое поведение
Аннотация: We obtain a priori estimates of the solution in the uniform metric for a linear conjugate initial-boundary inverse problem describing the joint motion of a binary mixture and a viscous heat-conducting liquid in a plane channel. With their help, it is established that the solution of the non-stationary problem with time growth tends to a stationary solution according to the exponential law when the temperature on the channel walls stabilizes with time.
Для линейной сопряженной начально-краевой обратной задачи, описывающей совместное движение бинарной смеси и вязкой теплопроводной жидкости в плоском канале, получены априорные оценки решения в равномерной метрике. С их помощью установлено, что решение нестационарной задачи с ростом времени стремится к стационарному решению по экспоненциальному закону, если температура на стенках канала стабилизируется со временем.
РИНЦ
Держатели документа:
Institute of Computational Modeling SB RAS
Siberian Federal University
Доп.точки доступа:
Andreev, Victor K.; Андреев Виктор К.; Efimova, Marina V.; Ефимова Марина В.
Перевод заглавия: Априорные оценки сопряженной задачи, описывающей совместное движение жидкости и бинарной смеси в канале
| УДК |
Кл.слова (ненормированные):
conjugate problem -- inverse problem -- a priori estimates -- asymptotic behavior -- сопряженная задача -- обратная задача -- априорные оценки -- асимтотическое поведение
Аннотация: We obtain a priori estimates of the solution in the uniform metric for a linear conjugate initial-boundary inverse problem describing the joint motion of a binary mixture and a viscous heat-conducting liquid in a plane channel. With their help, it is established that the solution of the non-stationary problem with time growth tends to a stationary solution according to the exponential law when the temperature on the channel walls stabilizes with time.
Для линейной сопряженной начально-краевой обратной задачи, описывающей совместное движение бинарной смеси и вязкой теплопроводной жидкости в плоском канале, получены априорные оценки решения в равномерной метрике. С их помощью установлено, что решение нестационарной задачи с ростом времени стремится к стационарному решению по экспоненциальному закону, если температура на стенках канала стабилизируется со временем.
РИНЦ
Держатели документа:
Institute of Computational Modeling SB RAS
Siberian Federal University
Доп.точки доступа:
Andreev, Victor K.; Андреев Виктор К.; Efimova, Marina V.; Ефимова Марина В.
Properties of Solutions for the Problem of a Joint Slow Motion of a Liquid and a Binary Mixture in a Two-Dimensional Channel
/ V. K. Andreev, M. V. Efimova // J. Appl. Ind. Math. - 2018. - Vol. 12, Is. 3. - P395-408, DOI 10.1134/S1990478918030018
. - ISSN 1990-4789
Кл.слова (ненормированные):
a priori estimates -- asymptotic behavior -- conjugate problem -- inverse problem -- surface tension -- thermocapillarity -- Binary mixtures -- Channel flow -- Heat conduction -- Nonlinear equations -- Surface tension -- A-priori estimates -- Asymptotic behaviors -- Conjugate problems -- Horizontal components -- Non-stationary problems -- Properties of solutions -- Thermocapillarity -- Two dimensional channels -- Inverse problems
Аннотация: Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law. © 2018, Pleiades Publishing, Ltd.
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Держатели документа:
Institute of Computational Modeling, Akademgorodok 50/44, Krasnoyarsk, 660036, Russian Federation
Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660036, Russian Federation
Доп.точки доступа:
Andreev, V. K.; Efimova, M. V.
Кл.слова (ненормированные):
a priori estimates -- asymptotic behavior -- conjugate problem -- inverse problem -- surface tension -- thermocapillarity -- Binary mixtures -- Channel flow -- Heat conduction -- Nonlinear equations -- Surface tension -- A-priori estimates -- Asymptotic behaviors -- Conjugate problems -- Horizontal components -- Non-stationary problems -- Properties of solutions -- Thermocapillarity -- Two dimensional channels -- Inverse problems
Аннотация: Under study is a conjugate boundary value problemdescribing a joint motion of a binary mixture and a viscous heat-conducting liquid in a two-dimensional channel, where the horizontal component of the velocity vector depends linearly on one of the coordinates. The problemis nonlinear and inverse because the systems of equations contain the unknown time functions—the pressure gradients in the layers. In the case of small Marangoni numbers (the so-called creeping flow) the problem becomes linear. For its solutions the two different integral identities are valid which allow us to obtain a priori estimates of the solution in the uniform metric. It is proved that if the temperature on the channel walls stabilizes with time then, as time increases, the solution of the nonstationary problem tends to a stationary solution by an exponential law. © 2018, Pleiades Publishing, Ltd.
Scopus,
Смотреть статью
Держатели документа:
Institute of Computational Modeling, Akademgorodok 50/44, Krasnoyarsk, 660036, Russian Federation
Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660036, Russian Federation
Доп.точки доступа:
Andreev, V. K.; Efimova, M. V.