Anastasia A. Domnich

This paper is concerned with a boundary value problem describing the stationary non-isothermal flow of a viscous incompressible fluid through a porous medium. The model consists of the Stokes equations with the Darcy and Forchheimer resistance terms, the incompressibility condition, and the convection–diffusion equation for heat, subject to the no-slip and Robin-type heat exchange conditions on the solid parts of the boundary, and to flow-through boundary conditions (with prescribed pressure and temperature) on the inlet–outlet parts. The problem is nonlinear due to convective heat transfer, the quadratic Forchheimer resistance term, and the temperature dependence of the external forces and heat source intensity. The main result establishes the existence of a weak solution under natural assumptions on the data (measurability, continuity, quadratic growth of the external forces, and boundedness of the heat source intensity). The proof is based on the Leray–Schauder principle and on a priori estimates for solutions. In the case where the external forces and the heat source intensity are independent of temperature, the weak solution is shown to be unique. Finally, for any weak solution, we derive the energy equalities representing the balance of kinetic and thermal energy.

This article is devoted to the mathematical analysis of a heat and mass transfer model for the pressure-induced flow of a viscous fluid through a plane channel subject to Navier’s slip conditions on the channel walls. The important feature of our work is that the used model takes into account the effects of variable viscosity, thermal conductivity, and slip length, under the assumption that these quantities depend on temperature. Therefore, we arrive at a boundary value problem for strongly nonlinear ordinary differential equations. The existence and uniqueness of a solution to this problem is analyzed. Namely, using the Galerkin scheme, the generalized Borsuk theorem, and the compactness method, we proved the existence theorem for both weak and strong solutions in Sobolev spaces and derive some of their properties. Under extra conditions on the model data, the uniqueness of a solution is established. Moreover, we considered our model subject to some explicit formulae for temperature dependence of viscosity, which are applied in practice, and constructed corresponding exact solutions. Using these solutions, we successfully performed an extra verification of the algorithm for finding solutions that was applied by us to prove the existence theorem.

A mathematical model describing the flow of a nonuniformly heated viscous fluid through a bounded domain under slip boundary conditions is studied. Sufficient conditions are derived for the solvability of the problem in the class of weak solutions. Energy equalities for the solutions are obtained. It is shown that the set of weak solutions is sequentially weakly closed.

Under study is some stationary model describing non-Newtonian fluid flows with the viscosity dependent on the strain rate and the heat transfer in a bounded 3D domain. The model is a strongly nonlinear system of coupled partial differential equations for the velocity field, temperature, and pressure. The system is supplemented with a no-slip condition and a linear Robin-type boundary condition for the temperature on the boundary of the flow domain. We propose an operator formulation of this boundary-value problem. Using the properties of d-monotone operators and the Leray-Schauder Fixed Point Theorem, we prove the existence of weak solutions under natural conditions on the data of the model. The solution set is shown to be bounded and closed.

Рассматривается математическая модель, описывающая стационарное ползущее течение неравномерно нагретой несжимаемой жидкости через ограниченную трехмерную область с локально-липшицевой границей. Выбранная модель представляет собой систему нелинейных дифференциальных уравнений в частных производных второго порядка со смешанными краевыми условиями: на участке протекания заданы давление, температура и касательная составляющая поля скоростей, а на твердых стенках сосуда используется условие прилипания и краевое условие типа Робена для температуры. Для данной краевой задачи вводится понятие слабого решения (пара «скорость-температура»), которое определяется как решение некоторой системы интегральных тождеств. Основной результат работы - теорема о существовании слабых решений в подпространстве декартова произведения двух соболевских пространств. Для доказательства этой теоремы дается операторная трактовка краевой задачи, выводятся априорные оценки решений и применяется теорема Лерэ-Шаудера о неподвижной точке вполне непрерывного отображения. Установлены энергетические равенства, которым удовлетворяют слабые решения.

We study an optimal control problem for the mathematical model that describes steady non-isothermal creeping flows of an incompressible fluid through a locally Lipschitz bounded domain. The control parameters are the pressure and the temperature on the in-flow and out-flow parts of the boundary of the flow domain. We propose the weak formulation of the problem and prove the existence of weak solutions that minimize a given cost functional. It is also shown that the marginal function of this control system is lower semi-continuous.