Newton’s Problem in Media with Positive Temperature

Abstract

In the original setting of Newton’s aerodynamic problem, it is supposed that there is no thermal motion of medium particles, that is, in an appropriate coordinate system (and prior to collisions with the body), all particles are resting. However, it is much more realistic to assume that thermal motion is present. In this chapter, we address the generalization of Newton’s problem for bodies moving in media with positive temperature. We shall see that the method of solution is quite conventional as compared with the original problem. On the other hand, a larger variety of optimal form is revealed here. In the three-dimensional case, an optimal body can (a) have a shape resembling the optimal Newtonian shape and (b) be the union of two Newton-like bodies “glued together” along their rear parts. Cases (a) and (b) are realized when the velocity of the body in the medium exceeds a critical value and when it is smaller than this value, respectively. In the two-dimensional case, there exist five different classes of solutions, while in the 2D analog of the original Newton problem, there are only two classes: an isosceles triangle and a trapezium.