Rayleigh-Bènard convection in the generalized Oberbeck-Boussinesq system

•We investigated the more than 100 years old problem Rayleigh-Benard convection or the related Oberbeck-Boussinesq equations with a self-similar Ansatz.•We gave physical interpretation of the applied self-similar Ansatz.•We found analytic solutions which are new and has nothing to do with the chaotic solutions obtained from truncated Fourier series in the last 50 years.•A possible answer is given, from the analytic temperature distribution, how the Rayleigh-Benard convection cells could appear.

The original Oberbeck-Boussinesq (OB) equations which are the coupled two dimensional Navier-Stokes(NS) and heat conduction equations have been investigated by E.N. Lorenz half a century ago with Fourier series and opened the way to the paradigm of chaos. In our former study-Chaos, Solitons and Fractals 78, 249 (2015)-we presented fully analytic solutions for the velocity, pressure and temperature fields with the aim of the self-similar Ansatz and gave a possible explanation of the Rayleigh-Bènard convection cells. Now we generalize the Oberbeck-Boussinesq hydrodynamical system, going beyond the first order Boussinesq approximation and consider a non-linear temperature coupling. We investigate more general, power law dependent fluid viscosity or heat conduction material equations as well. Our analytic results obtained via the self-similar Ansatz may attract the interest of various fields like meteorology, oceanography or climate studies.