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In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtonian–Navier–Stokes — with Boussinesq approximation — and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh–Bénard convection cells. Finally, it is discussed how our result may be related to nonlinear or chaotic dynamical regimes. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]