Equations of Motion of an Ideal Incompressible Fluid; Kelvin’s Circulation Theorem

Abstract

Fluid motion as a physical process is associated with the Euler or Navier–Stokes equations in hydrodynamics. These equations describe an immense set of qualitatively different phenomena—from the simplest small oscillations of a continuum, such as the propagation of sound in a homogeneous fluid (gas), to the mysterious phenomenon of turbulence observed in a vast majority of natural and technological flows. This “comprehensive nature” of the equations means that it is impossible (at least, as of today) to construct their general solutions. Consequently, it also means that there is a need for an appropriate reduction of these equations based on both observations and on the physical nature of the class of motions under study. For this reason, by now individual areas of hydrodynamics, such as the theory of sound, vortex dynamics, hydrodynamical stability theory, magnetohydrodynamics, convection theory, aerodynamics and many others, have all taken on the status of independent domains of science with their own physical features, applications, and often with a specially developed mathematical toolbox, as is the case in nonlinear wave theory.