Newton's aerodynamic problem in media of chaotically moving particles

Abstract

We study the problem of minimal resistance for a body moving with constant velocity in a rarefied medium of chaotically moving point particles, in Euclidean space R^d. The particles distribution over velocities is radially symmetric. Under some additional assumptions on the distribution function, the complete classification of bodies of least resistance is made. In the case of three and more dimensions there are two kinds of solutions: a body similar to the solution of classical Newton's problem and a union of two such bodies ``glued together'' by rear parts of their surfaces. In the two-dimensional case there are solutions of five different types: (a) a trapezium; (b) an isosceles triangle; (c) the union of a triangle and a trapezium with common base; (d) the union of two isosceles triangles with common base; (e) the union of two triangles and a trapezium. The cases (a)--(d) are realized for any distribution of particles over velocities, and the case (e) is only realized for some distributions. Two limit cases are considered, where the average velocity of particles is big and where it is small as compared to the body's velocity. Finally, using the obtained analytical results, we study numerically a particular case: the problem of body's motion in a rarefied homogeneous monatomic ideal gas of positive temperature in R^2 and in R^3.