Solving the Hydrodynamical System of Equations of Inhomogeneous Fluid Flows with Thermal Diffusion: A Review

Abstract

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown that the exact integration of the thermal diffusion equations is carried out in the Lin–Sidorov–Aristov class. This class of exact solutions is a generalization of the Ostroumov–Birikh family of exact solutions. The use of the class of exact solutions by Lin–Sidorov–Aristov makes it possible to take into account not only the inhomogeneity of the pressure field, the temperature field and the concentration field, but also the inhomogeneous velocity field. The present review shows that there is a class of exact solutions for describing the flows of incompressible fluids, taking into account the Soret and Dufour cross effects. Accurate solutions are important for modeling and simulating natural, technical and technological processes. They make it possible to find new physical mechanisms of momentum transfer for the design of new types of equipment.

Stability Estimates of Optimal Solutions for the Steady Magnetohydrodynamics-Boussinesq Equations

“…Let us turn to the inequalities (62), (63), which ensure the uniqueness of the solution of Problem 1. It is clear that for fixed elements f, b, f , k and α, the right-hand sides M u , M H , M T and M p included in the inequalities (62), (63) depend on the control u = (g, q, ψ, χ) and on the element j.…”

“…Theorem 6. Let the conditions of Theorem 5 and the inequalities (62), (63) be satisfied for all u ∈ K and j ∈ L 2 (Ω). Then, every nontrivial Lagrange multiplier (λ 0 , y * ) satisfying (93)-( 95) is regular, i.e., it has the form (1, y * ) and, moreover, it is defined in a unique way.…”

“…We also mention papers [59][60][61][62][63][64][65][66][67][68] which touch upon issues close to the subject of our paper from fields of electromagnetism, acoustics, nonlinear diffusion, engineering mechanics, viscoelasticity.…”

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

“…These works can be divided into several groups. The first group contains papers that develop methods for finding exact solutions to these equations (see, for example, [11][12][13][14][15] and reviews [16]). The second group contains works devoted to application of the Li-Ovsyannikov symmetry method to study qualitative properties of solutions of equations of HMT in viscous binary and/or heat-conducting liquids.…”

Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions

“…The classical Boussinesq approximation system [1,2] was proposed as a simplified model for heat and mass transfer in a linear viscous fluid and is widely used in studying non-isothermal flows (see, for example, [3][4][5][6][7][8][9][10] and the references cited therein). This system of partial differential equations (PDEs) includes the motion equations, the incompressibility condition, and the energy balance equation.…”