Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system

Abstract

The simplest model to couple the heat conduction and Navier-Stokes equations together is the Oberbeck-Boussinesq(OB) system which were investigated by E.N. Lorenz and opened the paradigm of chaos. In our former studies -Chaos Solitons and Fractals 78, 249 (2015), ibid, 103, 336 (2017) -we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh-Bènard convection cells with the help of the self-similar Ansatz. Now we generalize the OB hydrodynamical system, including a viscous source term in the heat conduction equation. Our results may attract the interest of various fields like micro or nanofluidics or climate studies.

Analytic Solutions of a Two-Fluid Hydrodynamic Model

“…The shape functions f, g, h could be any continuous functions with existing first and second continuous derivatives and will be evaluated later on. The logic, the physical and geometrical interpretation of the Ansatz were exhaustively analyzed in all our former publications [3,4,9,10,12] therefore we neglect it.…”

“…This study is organically linked to our long-term program in which we systematically goes over fundamental hydrodynamic systems and analyze physically relevant self-similar and traveling wave solutions. Till now we published about half a dozen papers [3,4,5,8,9,10] and a book chapter [12] in this field. Due to our knowledge there is no self-similar solution known and analyzed for time-dependent two-fluid models.…”

Self-similar analysis of the time-dependent compressible and incompressible boundary layers including heat conduction

“…The shape functions f, g, h and i could be any continuous functions with existing first and second continuous derivatives and will be evaluated later on. The logic, the physical and geometrical interpretation of the Ansatz were exhaustively analyzed in all our former publications [19][20][21] therefore we skip it here.…”

“…In our former studies we investigated three different kind of Rayleigh-Bénard heat conduction problems [19][20][21] which are full two dimensional viscous flows coupled to the heat conduction equation. In the first study [19] we gave a reasonable geometrical explanation of the possible birth of the Bénard cells which is a remarkable property of the analytic results.…”

“…These last five publications of us [17] - [21] can be considered as precursors of the present study.…”