Consequences of the Ostrogradsky-Gauss Theorem for Numerical Simulation in Aeromechanics

Abstract

Using the Ostrogradsky-Gauss theorem to construct the laws of conservation and replacement of the integral over the surface by the integral over the volume, we neglect the integral term outside, i.e. neglect the circulation on the sides of the elementary volume (in the two-dimensional case, this is clearly visible). Circulation means the presence of rotation, which in turn means the presence of a moment of force (angular momentum). As a result, we have a symmetric stress tensor, a symmetric velocity tensor, etc. Static pressure, as follows from kinetic theory, there is a zero-order quantity; the terms associated with dissipative effects are first-order quantities. It does not follow from the Boltzmann equation and from the phenomenological theory that the pressure in the Euler equation is equal to one third of the sum of the pressures on the corresponding coordinate axes. The inaccuracy of determining the velocities in the stress tensor in the stress tensor does not strongly affect the results at low speeds. All these issues are discussed in the work. As example in this paper suggests task of flowing liquid at little distance of two parallel plates.

The Law of Conservation of Momentum and the Contribution of No Potential Forces to the Equations for Continuum Mechanics and Kinetics

“…If we consider equal pressures in different directions, we lose the moment of force but the pressure gradient is a force. Analyzing the results of solving the Euler equations [3] [4] and calculating potential flows, we obtain a vortex sheet, which indicates the existence of a moment. The numerical results were processed by the authors without averaging the values over the sides of the elementary volume.…”