Newton's aerodynamic problem in media of chaotically moving particles

A. Yu. Plakhov ; D. Torres

Sbornik. Mathematics, Tome 196 (2005) no. 6, pp. 885-933

Résumé

The problem of minimum resistance is studied for a body moving with constant velocity in a rarefied medium of chaotically moving point particles in the Euclidean space $\mathbb R^d$. The distribution of the velocities of the particles is assumed to be radially symmetric. Under additional assumptions on the distribution function a complete classification of the bodies of least resistance is carried out. In the case of dimension three or more there exist two kinds of solution: a body similar to the solution of the classical Newton problem and a union of two such bodies ‘glued together’ along the rear parts of their surfaces. In the two-dimensional case there exist solutions of five distinct types: (a) a trapezium; (b) an isosceles triangle; (c) the union of an isosceles triangle and a trapezium with a common base; (d) the union of two isosceles triangles with a common base; (e) the union of two triangles and a trapezium. Cases (a)–(d) are realized for an arbitrary velocity distribution of the particles, while case (e) is realized only for some distributions. Two limit cases are considered: when the average velocity of the particles is large and when it is small in comparison with the velocity of the body. Finally, the analytic results so obtained are used for the numerical study of a particular case: the problem of the motion of a body in a rarefied homogeneous monatomic ideal gas of positive temperature in $\mathbb R^2$ and in $\mathbb R^3$.