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

One of the simplest model to couple viscous flow to heat conduction is the Oberbeck–Boussinesq (OB) system which were also investigated by E N Lorenz. In our former studies—2015 Chaos Solitons Fractals 78 249, 2017 Chaos Solitons Fractals 103 336—we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh–Bénard 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 analysis show that the viscous heating term smooths out any kind of Bénard oscillations and stabilizes the flow. All the velocity, pressure and temperature distributions are free of oscillations. These results may attract interest in micro or nanofluidics. (paper)