A New Class of Exact Solutions to the Navier–Stokes Equations with Allowance for Internal Heat Release

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

“…Aristov. He not only repeated (independently) their results for solving problems of magnetohydrodynamics and convection, but also generalized them to an exact solution for pressure, temperature and concentration fields in the form of quadratic forms with respect to two coordinates and showed the validity of these solutions for fluid flows, taking into account internal heat release (Rayleigh's dissipative function) [37,46].…”

“…By now, at least three families can be distinguished in the context of our research: the Gromeka-Beltrami-Trkal class [28][29][30][31][32][33][34], the Lin-Sidorov-Aristov class [7,[35][36][37] and other solutions that, with an appropriate change in coordinates, can be reduced to these two classes. It is useful to trace the history of obtaining exact solutions from reviews and books [7,[38][39][40][41][42][43][44][45][46][47][48][49][50].…”

“…In addition, in reviews [5,7,14,16,[38][39][40][41][42][43][44][45][46][47][48][49][50] it was shown that most of the known exact solutions belong to the Lin-Sidorov-Aristov class [35][36][37]. Probably, the first described approach to integrating the Navier-Stokes equations was implemented in the work of C.C.…”

“…First and mainly, in part of the basic solution, the velocity components U, V, W depend only on terms added in the first two components that are linear in two variables. Thus, the Lin-Sidorov-Aristov velocity field is written as [7,[35][36][37][38][39][40][41][42][43][44][45][46][47]:…”

Generalized Boussinesq System with Energy Dissipation: Existence of Stationary Solutions

“…The main motivation for this simplification lies in the fact that ignoring the effect of viscous dissipation greatly facilitates mathematical analysis and finding of solutions to heat and mass transfer models that are based on the Boussinesq system. However, from a physical point of view as well as for certain practical applications, it is interesting to consider the "full equations", that is, the equations that include all the nonlinear terms [11][12][13][14][15][16]. Studying the Boussinesq system with energy dissipation is important as it offers insights into complex dynamics of fluid motion and energy transfer, contributing to advancements in environmental science, engineering, and climate modelling.…”