  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-B\`enard
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.
