Fluid Dynamics

Newton’s problem of constructing an axisymmetric forebody part of minimum drag is considered. The solution to this problem, although without any explanation, was given by Newton himself in the main work of his life, Philosophiae Naturalis Principia Mathematica. However, Newton’s prefabricated solution was not understood by aerodynamicists who turned to solving Newton’s problem and some of its generalizations in the middle of the twentieth century. A.N. Krylov translated Newton’s Principia into Russian, giving detailed explanations of many of Newton’s statements, including the discussed problem. Moreover, having explained one of these statements, Krylov formulated the necessary condition for the drag minimum, missed by all Newton readers, not just aerodynamicists, but also such an authority on the variational calculus as Legendre. However, even Krylov’s explanations did not help to understand Newton’s solution to the only who had access to them—Soviet aerodynamicists. The main goal of this article is to describe the history of the Newton problem solution, in which the author happened to participate.

We study the motion of an incompressible viscous conducting fluid, which initially rotates as a solid at a constant angular velocity together with parallel bounding walls under the action of longitudinal vibrations of one of the walls beginning suddenly and a magnetic field suddenly applied to one of them. The walls make an arbitrary angle with the axis of rotation. The magnetic field is applied along the wall normal. In the general case, the solution is presented in the form of a series. The vectors of tangential stresses that act on the gap walls from the fluid are presented. Some particular cases of the wall motion are discussed. The results are used to study individual structures of the boundary layers at the walls. This study generalizes studies [1-3].

The paper reviews the application of the formalism of a characteristic functional for statistical description of a random velocity field obeying the Navier-Stokes equation for incompressible fluids in the presence of regular and random external forces. The equation in functional derivatives for the characteristic functional is obtained using a representation of the characteristic functional in the form of a functional integral over two fields. From this equation one can obtain equations for various statistical characteristics of the velocity field such as the variance of velocity pulsations (the pair correlation function) or the mean response of velocity field to external forces (Green’s function). The method of skeleton Feynman diagrams is used in the analysis of the equations and of the solution structures. This fact follows directly from the functional formulation of the theory without referring to the commonly used methods of perturbation theory. The vertices of three types arising in the theory formulation appear to be linked. This enables considering the vertex of only one type and simplify the diagrammatic representations of various quantities.