Analytic self-similar solutions of the Oberbeck-Boussinesq equations

Barna, Imre Ferenc and Mátyás, László (2015) Analytic self-similar solutions of the Oberbeck-Boussinesq equations. CHAOS SOLITONS & FRACTALS, 78. pp. 249-255. ISSN 0960-0779

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Abstract

In this article we will present pure two-dimensional analytic solutions for the coupled noncompressible Newtoniain 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 has a strongly damped oscillating behavior which is an interesting feature.

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