Solution of the Synthesis Problem of Boundary Optimal Control of a Rod Cooling Process with a Heat Conductive Viscosity

The problem of synthesis of the boundary optimal control of the cooling process of media with heat conductive viscosity is investigated. In addition to the distributed parameters, the concentrated parameters act on the system. This is due to the fact that the temperature of the external environment is unknown and varies according to a given law. As a result, the process is described by a system of partial differential equations and ordinary differential equations. In this case, heat transfer occurs at the right end of the rod. This complicates the obtaining of a solution of this boundary-value problem in an explicit form. But it is possible to establish the existence and uniqueness of the solution of the corresponding boundary-value problem for concrete admissible controls.The criterion of quality is a quadratic functional and it is required to build control in the form of feedback. First by the Fourier method, the problem under consideration is formulated in an infinite-dimensional phase space. As a result, the problem of synthesis of optimal control in a functional space is obtained. To solve this problem, the dynamic programming method is used. To do this, let's introduce the Bellman functional and obtain the Bellman equation, which this functional satisfies. The solution of this equation allows to find the control parameter in the form of a functional defined on the set of the state function. Further, by introducing the corresponding functions, feedback control is constructed for the original problem. Unlike program control, this allows to influence the behavior of the system at any time, that is, to ensure the self-regulation of the process. However, let's note that the difficulties in solving this problem are connected with the justification of the proposed method. This is established by the investigation of a closed system

Investigation of the Influence of Gravitational Forces on the Process of Displacement of Viscoplastic Fluids

The object of research is a numerical simulation of the process of two-dimensional two-phase filtration of viscoplastic oil and water, taking into account the gravitational forces, some properties of liquids, as well as relative phase permeabilities and capillary forces.As is known, the problems of multiphase filtration have specific features. Therefore, there is a need to develop difference schemes in adaptive grids that reduce the artificial viscosity and oscillation of the numerical solution. They also make it possible to obtain acceptable results with a small number of nodes in the computational grid.To take into account the singularities of the solution, a difference-iteration method is used in moving grids. Based on the computational experiment, the influence of the initial pressure gradient and gravity on the displacement process is investigated.Economical difference schemes that combine the advantages of explicit and implicit schemes are constructed and make it possible to reduce the two-dimensional problem to a chain of one-dimensional problems. A difference-iterative method is also proposed in moving grids for solving two-dimensional (axisymmetric) non-stationary filtration problems of anomalous liquids, by means of which an iterative process is constructed to find the distribution of water saturation.The carried out calculations to determine the influence of gravity on the displacement process have shown that at z=0, even at low productive-bed thicknesses, gravitational forces influence the displacement process. And over time this influence increases: if at the time t=0.08 on the circuit the difference of water saturation was 0.0077; at t=0.24–0.0122, then at t=1.04 it becomes equal to 0.0292.It is shown that when modeling the process without taking gravity into account it is expedient to simplify the geometry of the filtration region, i. e., to consider a plane-radial flow in view of the considerable simplicity of the calculations.The developed algorithms can be used for hydro-gas dynamic calculations related to the development and operation of oil fields containing anomalous oil