Motion of a Rough Disc in Newtonian Aerodynamics
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Dynamics of a rough disc in a rarefied medium is considered. We prove that any finite rectifiable curve can be approximated in the Hausdorff metric by trajectories of centers of rough discs provided that the parameters of the system are carefully chosen. To control the dynamics of the disc, we use the so-called inverse Magnus effect which causes deviation of the trajectory of a spinning body. We study the so-called response laws for scattering billiards e.g. relationship between the velocity of incidence of a particle and that of reflection. We construct a special family of such laws that is weakly dense in the set of symmetric Borel measures. Then we find a shape of cavities that provides selected law of reflections. We write down differential equations that describe motions of rough discs. We demonstrate how a given curve can be approximated by considered trajectories.
KeywordsBilliards Shape optimization Magnus effect Rarified medium Retroreflectors
This work was supported by Russian Foundation for Basic Researches under Grants 14-01-00202-a and 15-01-03797-, by Saint-Petersburg State University under Thematic Plans 22.214.171.1240 and 126.96.36.1994, by FEDER funds through COMPETE – Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) from the “Fundação para a Ciência e a Tecnologia” (FCT), cofinanced by the European Community Fund FEDER/POCTI under FCT research projects (PTDC/MAT/113470/2009 and PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690). The author is grateful to Prof. Alexandre Plakhov from University of Aveiro for his ideas, remarks and corrections.
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