arXiv:1502.05039v1[physics.flu-dyn]17 Feb 2015Analytic self-similar solutions of the Oberbeck-BoussinesqequationsI.F. Barna1,2and L. M´aty´as31Wigner Research Center of the Hungarian Academy of SciencesKonkoly-Thege ´ut 29 - 33, 1121 Budapest, Hungary2ELI-HU Nonprofit Kft., Dugonics T´er 13, 6720 Szeged, Hungary3Sapientia University, Faculty of Science,Libert˘atii sq.1, 530104 Miercurea Ciuc, Romania(Dated: February 19, 2015)AbstractIn this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtoniain Navier-Stokes — with Boussinesq approximation — and the heat con-duction equation. The system was investigated from E.N. Lorenz half a century ago with Fourierseries and pioneered the way to the paradigm of chaos. We present a novel analysis of the samesystem where the key idea is the two-dimensional generalization of the well-known self-similarAnsatz 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 errorfunctions. The temperature field has a strongly damped oscillating behavior which is an interestingfeature.PACS numbers: 47.10.ad,02.30.Jr1