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Unformatted text preview: Page 1 of 1 MEC-501/ Spring 1999, Instructor: V. Prasad Rayleigh-Bnard Convection Introduction The Rayleigh-Bnard (henceforth referred to as RB) system has been studied by researchers for a complete century by now and is still a topic of interest to them. It would be interesting to find out why? The R-B problem is relevant to applications ranging from astrophysics (where e.g., the validity of star models depends to a large degree on the validity of energy transport in the outer regions of stellar atmospheres), geophysics (where e.g., the current theories of continental drift depend on possible convective motion in the earth's mantle caused by internal heat generation due to radioactive decay), and atmospheric sciences (where e.g., theories and prediction of current weather phenomena as well as long-term effects like ice ages depend on the validity of theories of convective energy transport in the Earth's atmosphere). It's applications to various engineering systems, such as Solar Energy systems, material processing, energy storage, and nuclear systems are numerous. Apart from its relevance to the various branches of engineering and physical sciences, the R-B system is being investigated for purely theoretical and fundamental reasons as well. The classical "Standard" mathematical model of this problem consists of a set of non-linear coupled partial differential equations, the solution of which is degenerate and non-unique. It, therefore, serves as a paradigm of a nonlinear system, which if investigated properly, can provide insight to the researchers studying nonlinear phenomena. It is now, generally recognized that time dependence in the R-B system offers clues to the transition from laminar to turbulent flow. It is worthwhile to note here that the transition phenomena, in general, is independent of the geometrical systems and is more a flow property, e.g., a proper choice of length scale makes the transition Reynolds Number to be identical for both flat plate and pipe flow case. In another related issue, R-B system is the most carefully studied example of nonlinear systems exhibiting self-organization or pattern forming systems , of special interest to researchers from Synergetic. It demonstrates essential features typical not only of various hydrodynamic instabilities but also of many non-linear pattern-forming processes differing in their nature. Formation of patterns close to spatially periodic ones can be observed in crystal growth, propagation of solidification fronts, electrohydrodynamic instabilities of nematic liquid crystals, chemical reaction-diffusion processes, auto catalytic reactions, buckling of thin plates and shells, morphogenesis of plants and animals, etc. Such patterns are also seen in cloud streets, sand ripples on flat beaches, and desert Page 2 of 2 MEC-501/ Spring 1999, Instructor: V. Prasad dunes, in genealogical formations, in interacting laser beams, in the grid-scale instabilities of numerical algorithms, and in many other objects. R-B convection can be considered, as stated by numerical algorithms, and in many other objects....
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