Scattering in billiards and problems of Newtonian aerodynamics

Plakhov, Alexander

doi:10.1070/rm2009v064n05abeh004642

Cited by 21 publications

(23 citation statements)

References 27 publications

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“…In the 2D case the greatest value of decrease is nearly 1%, and of increase, 50% [8]. In the 3D case the greatest value of decrease is nearly 3%, and of increase, 100% [9].…”

Section: Introductionmentioning

confidence: 92%

“…Denote by π v,n and π v + ,n the projections π v,n : (v, v + , n) → (v, n) and The proof of the theorem in the two-dimensional case is given in [8], and the full proof is given in [9].…”

Section: Characterization Theorem and Optimization Of Resistancementioning

confidence: 99%

“…This part of the proof is much more difficult than the former one and is proved by making a direct construction. For this construction we refer to the papers [8] and [9].…”

Section: Characterization Theorem and Optimization Of Resistancementioning

confidence: 99%

“…The optimal transportation can be described as follows (for a detailed exposition see [7,9]). The mass on both the lower and the upper quarter of the circle is divided into two parts, as shown in Fig.…”

Section: Characterization Theorem and Optimization Of Resistancementioning

confidence: 99%

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Billiards, scattering by rough obstacles, and optimal mass transportation

Plakhov

2012

J Math Sci

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This article presents a brief exposition of recent results of the author on billiard scattering by rough obstacles. We define the notion of a rough body and give a characterization of scattering by rough bodies. Then we define the resistance of a rough body; it can be interpreted as the aerodynamic resistance of the somersaulting body moving through a rarefied medium. We solve the problems of maximum and minimum resistance for rough bodies (more precisely, for bodies obtained by roughening a prescribed convex set) in arbitrary dimension. Surprisingly, these problems are reduced to special problems of optimal mass transportation on the sphere.Mathematics subject classifications: 37D50, 49Q10

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“…In the 2D case the greatest value of decrease is nearly 1%, and of increase, 50% [8]. In the 3D case the greatest value of decrease is nearly 3%, and of increase, 100% [9].…”

Section: Introductionmentioning

confidence: 92%

Section: Characterization Theorem and Optimization Of Resistancementioning

confidence: 99%

“…This part of the proof is much more difficult than the former one and is proved by making a direct construction. For this construction we refer to the papers [8] and [9].…”

Section: Characterization Theorem and Optimization Of Resistancementioning

confidence: 99%

Section: Characterization Theorem and Optimization Of Resistancementioning

confidence: 99%

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Billiards, scattering by rough obstacles, and optimal mass transportation

Plakhov

2012

J Math Sci

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“…These results were announced, with a brief outline of the proof, in [57], and their detailed exposition is given in [60]. Remark 2.3.…”

Section: Bodies Modified In a Neighborhood Of Their Boundarymentioning

confidence: 99%

Exterior Billiards

Plakhov¹

2013

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The book contains an account of results obtained by the author and his collaborators on billiards in the complement of bounded domains and their applications in aerodynamics and geometrical optics.We consider several problems related to aerodynamics of bodies in highly rarefied media. It is assumed that the medium particles do not interact with each other and are elastically reflected when colliding with the body boundary; these assumptions drastically simplify the aerodynamics and allow to reduce it to a number of purely mathematical problems.First we examine problems of minimal resistance in the case of translational motion of bodies. These problems generalize the Newton problem of least resistance; the difference is that the bodies are generally nonconvex in our case and therefore the particles can make multiple reflections from the body surface. It is proved that typically the infimum of resistance equals zero; thus, there exist 'almost perfectly streamlined' bodies.Next we consider the generalization of Newton's problem on minimal resistance of convex axisymmetric bodies to the case of media with thermal motion of particles. Two kinds of solutions are found: first, Newton-like bodies and second, shapes obtained by gluing together two Newton-like bodies along their rear ends.Further, we state results on characterization of billiard scattering by nonconvex and rough bodies; next we solve some special problems of optimal mass transportation. These two groups of results are applied to problems of minimal and maximal resistance for bodies that move forward and at the same time slowly rotate. It is found, in particular, that the resistance of a three-dimensional convex body can be increased at most twice and decreased at most by 3.05% by roughening its surface.Next, we consider a rapidly rotating rough disc moving in a rarefied medium on the plane. It is shown that the force acting on the disc is not generally parallel to the direction of the disc motion, that is, has a nonzero transversal component. This phenomenon is called Magnus effect (proper or inverse, depending on the direction of the transversal component). We show that the kind of Magnus effect depends on the kind of disc roughness, and study this dependence. The problem of finding all admissible values of the force acting on the disc is formulated in terms of a vector-valued problem of optimal mass transportation.Finally, we describe bodies that have zero resistance when translating through a medium, and state results on existence or non-existence of bodies with mirror surface invisible in one or several directions. We also consider the problem of constructing retroreflectors: bodies with specular surface that reverse the direction of any incident beam of light.

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Problem of Minimum Resistance to Translational Motion of Bodies

Plakhov

2012

Exterior Billiards

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Scattering in billiards and problems of Newtonian aerodynamics

Cited by 21 publications

References 27 publications

Billiards, scattering by rough obstacles, and optimal mass transportation

Billiards, scattering by rough obstacles, and optimal mass transportation

Exterior Billiards

Problem of Minimum Resistance to Translational Motion of Bodies

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