Alexander Plakhov

Invisibility in billiards

The problem of invisibility for bodies with a mirror surface is studied in the framework of geometrical optics. A closely related problem concerning the existence of bodies that have zero aerodynamical resistance is also studied here. We construct bodies that are invisible/have zero resistance in two directions, and prove that bodies which are invisible/have zero resistance do not exist in all possible directions of incidence.

Billiard Scattering on Rough Sets: Two-Dimensional Case

The notion of a rough two-dimensional (convex) body is introduced, and to each rough body there is assigned a measure on

Problems of maximal mean resistance on the plane

\mathbb{T}^3

Optimal Roughening of Convex Bodies

describing billiard scattering on the body. The main result is characterization of the set of measures generated by rough bodies. This result can be used to solve various problems of least aerodynamical resistance.

Spinning rough disc moving in a rarefied medium

A two-dimensional body moves through a rarefied medium; the collisions of the medium particles with the body are absolutely elastic. The body performs both translational and slow rotational motion. It is required to select the body, from a given class of bodies, such that the average force of resistance of the medium to its motion is maximal.Numerical and analytical results concerning this problem are presented. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1.5 times the resistance of K. The maximum is attained on a sequence of bodies with a very complicated boundary. The numerical study was made for somewhat more restricted classes of bodies. The obtained values of resistance are slightly lower, but the boundary of obtained bodies is much simpler, as compared with the analytical solutions.

Two-dimensional body of maximum mean resistance

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface, and do not interact with each other. We consider a generalization of Newton’s minimal resistance problem: given two bounded convex bodies C1 and C2 such that C1 ⊂ C2 ⊂ R and ∂C1 ∩∂C2 = ∅, minimize the resistance in the class of connected bodies B such that C1 ⊂ B ⊂ C2. We prove that the infimum of resistance is zero; that is, there exist ”almost perfectly streamlined” bodies. Mathematics subject classifications: 37D50, 49Q10

Comment on “Functions and Domains Having Minimal Resistance Under a Single-Impact Assumption”[SIAM J. Math. Anal., 34 (2002), pp. 101–120]

We study the Magnus effect: deflection of the trajectory of a spinning body moving in a gas. It is well known that in rarefied gases, the inverse Magnus effect takes place, which means that the transversal component of the force acting on the body has opposite signs in sparse and relatively dense gases. The existing works derive the inverse effect from non-elastic interaction of gas particles with the body. We propose another (complementary) mechanism of creating the transversal force owing to multiple collisions of particles in cavities of the body surface. We limit ourselves to the two-dimensional case of a rough disc moving through a zero-temperature medium on the plane, where reflections of the particles from the body are elastic and mutual interaction of the particles is neglected. We represent the force acting on the disc and the moment of this force as functionals depending on ‘shape of the roughness’, and determine the set of all admissible forces. The disc trajectory is determined for several simple cases. The study is made by means of billiard theory, Monge–Kantorovich optimal mass transport and by numerical methods.

Newton’s problem of minimal resistance under the single-impact assumption

A two-dimensional body, exhibiting a slight rotational movement, moves in a rarefied medium of particles which collide with it in a perfectly elastic way. In previously realized investigations by the first two authors, [Alexander Yu. Plakhov, Paulo D.F. Gouveia, Problems of maximal mean resistance on the plane, Nonlinearity, 20 (2007), 2271-2287], shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement. Giving continuity to this study, new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier. This result, now presented, consists of a two-dimensional shape which confers on the body a resistance which is very close to its theoretical supremum value. But its interest does not lie solely in the maximization of Newtonian resistance; on regarding its characteristics, other areas of application are seen to begin to appear which are thought to be capable of having great utility. The optimal shape which has been encountered resulted from numerical studies, thus it is the object of additional study of an analytical nature, where it proves some important properties which explain in great part its effectiveness.

THE PROBLEM OF MINIMAL RESISTANCE FOR FUNCTIONS AND DOMAINS

Recently Comte and Lachand-Robert [SIAM J. Math. Anal., 34 (2002), pp.101–120] stated a very interesting and actual problem of minimizing mean specific resistance of infinite surfaces in a parallel flow of noninteracting point particles. They also constructed surfaces having resistance 0.593 and proved that they are minimizers. Unfortunately, their proof is incorrect. In this comment we provide a counterexample showing that the least value of resistance is not attained and is less than 0.581 (but greater than or equal to 0.5). Therefore, the problem remains open.

Bodies with mirror surface invisible from two points

A parallel flow of non-interacting point particles is incident on a body at rest. When hitting the bodys surface, the particles are reflected elastically. Assume that each particle hits the body at most once (SIC condition); then the force of resistance of the body along the flow direction can be written down in a simple analytical form. The problem of minimal resistance within this model was first considered by Newton (1687) in the class of bodies with a fixed length M along the flow direction and with a fixed maximum orthogonal cross section, under the additional conditions that the body is convex and rotationally symmetric. Here we solve the problem (first stated by Ferone, Buttazzo, and Kawohl in 1995) for the wider class of bodies satisfying SIC and with the additional conditions removed. The scheme of solution is inspired by Besicovitchs method of solving the Kakeya problem. If the maximum cross section is a disc, the decrease of resistance as compared with the original Newton problem is more than twofold; the ratio tends to 2 as M goes to 0 and to 81/4 as M goes to infinity. We also prove that the infimum of resistance is 0 for a wider class of bodies with both single and double impacts allowed.