Local Solvability of the Flow Problem for the Equations of Motion of Two Interpene Crating Fluids
УДК 532.511
Abstract
The paper is dedicated to the one-dimensional problem of the nonisothermal flow of a two-phase mixture of viscous incompressible fluids with inhomogeneous boundary conditions. The mathematical model describing the two viscous fluids mixture flow is based on the equations of mass conservation, momentum conservation for each phase, and on the energy conservation equation, in the large. Local in time solvability of the initial boundary value problem in S.L.Sobolev and Helder spaces is proved. Section 1 sets the problem set up and provides the short literature review on the topic close papers and the main result formulation. Section 2 explains the transformation of the original system of equations. Sections 3, 4 prove the existence of the strong and classic solutions on a small time interval with constant true density using the Bubnov-Galerkin method. Notionally, the proof of the theorem is based on the similar result proof for viscous heatconducting gas (Antonsev S.N., Kazhihov A.V., Monahov V.N. Boundary value problems of heterogeneous fluid mechanic). The particularity of the considered problem is the presence of inhomogeneous boundary conditions.
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Нигматулин Р.И. Динамика многофазных сред. М., 1987. Ч. 1.
Рахматулин Х.А. Основы газодинамики взаимопроникающих движений сжимаемых сред // Прикл. математика и механика. 1956. Т. 20. Вып. 2.
Файзуллаев Д.Ф., Умаров У.И., Шакиров У.У. Гидродинамика одно- и двухфазных сред и ее практическое приложение. Ташкент, 1980.
Антонцев С.Н., Кажихов У.В., Монахов В.Н. Краевые задачи механики неоднородных жидкостей. Новосибирск, 1983.
Peter Eshuis, Ko van der Weele, Meheboob Alam, Henk Jan van Gerner, Martin van der Hoef, Hans Kuipers, Stefan Luding, Devaraj van der Meer, Detlef Lohse Buoyancy driven convection in vertically shaken granular matter: experiment, numerics, and theory // Granular Matter. 2013. № 15. DOI 10.1007/s10035-013-0440-x.
Gard S.K., Pritchett J.W. Dynamics of gas -fluidized beds // Journal of Applied Phisics. 1975. Vol. 46. № 10.
Goz M. Existence and uniqueness of time-dependent spatially periodic solutions of fluidized bed equations // ZAMM.Z.angew. Math. Mech. 1991. Vol. 71. № 6.
Akhmerova I.G., Papin A.A. Solvability of the Boundary-Value Problem for Equations of One-Dimensional Motion of a Two-Phase Mixture // Mathematical Notes. 2014. Vol. 96. № 2.
Papin A.A., Akhmerova I.G. Solvability of the system of equations of one-dimensional motion of a heat-conducting two-phase mixture // Mathematical Notes. 2010. Vol. 87. № 2.
Папин А.А., Ахмерова И.Г. Задача протекания для уравнений движения двух взаимопроникающих вязких жидкостей // Тед. Сиб. мат. журн. Сиб. отд. УН ТФ. Новосибирск. 2004. Деп. ВИНИТИ № 37.
Рождественский Б.И., Яненко Н.Н. Системы квазилинейных уравнений и их приложения к газовой динамике. М., 1978.
Хартман Ф. Обыкновенные дифференциальные уравнения. М., 1970.
Папин А.А. Краевые задачи для уравнений двухфазной фильтрации. Барнаул. 2009.
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