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- Problems of maximal mean resistance on the planePublication . Plakhov, Alexander; Gouveia, Paulo D.F.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. There are presented numerical and analytical results concerning this problem. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1.5 times resistance of K. The maximum is attained on a sequence of bodies with 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 to the analytical solutions.
- Spinning rough disc moving in a rarefied mediumPublication . Plakhov, Alexander; Tchemisova, Tatiana; Gouveia, Paulo D.F.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 nonelastic 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
- Two-dimensional body of maximum mean resistancePublication . Gouveia, Paulo D.F.; Plakhov, Alexander; Torres, Delfim F.M.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.
- Bodies of maximal aerodynamic resistance on the planePublication . Plakhov, Alexander; Gouveia, Paulo D.F.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. There are presented numerical and analytical results concerning this problem. In particular, the maximum resistance in the class of bodies contained in a convex body K is proved to be 1.5 times resistance of K. The maximum is attained on a sequence of bodies with 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 to the analytical solutions.