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1502.05039.pdf - Analytic self-similar solutions of the...

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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
I. INTRODUCTIONThe investigation of the dynamics of viscous fluids has a long past. Enormous scientificliterature is available from the last two centuries for fluid motion even without any kindof heat exchange. Thanks to new exotic materials like nanotubes, heat conduction in solidbulk phase (without any kind of material transport) is an other quickly growing independentresearch area as well.The combination of both processes are even more complex whichlacks general existence theorems for unique solutions. The most simple way to couple thesetwo phenomena together is the Boussinesq [1] approximation which is used in the field ofbuoyancy-driven flow (also known as natural convection). It states that density differencesare sufficiently small to be neglected, except where they appear in terms multiplied by g,the acceleration due to gravity. The main idea of the Boussinesq approximation is that thedifference in inertia is negligible but gravity is sufficiently strong to make the specific weightappreciably different between the two fluids. When the Boussinesq approximation is usedthan no sound wave can be described in the fluid, because sound waves move via densityvariation.Boussinesq flows are quite common in nature (such as oceanic circulations, atmosphericfronts or katabatic winds), industry (fume cupboard ventilation or dense gas dispersion),and the built environment (like central heating, natural ventilation). The approximation isextremely accurate for such flows, and makes the mathematics and physics much simplerand transparent.

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Term
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Fluid Dynamics, Chaos Theory, Partial differential equation, Joseph Fourier, M J Boussinesq

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