Influence of the Delay and Dispersion in Mechanics

Abstract

The aim of this work is to clarify the new mathematical model describing the mechanics of continuous media and rarefied gas. The present study is associated with the formulation of conservation laws as conditions of equilibrium of angular momentums, while usually formulated in terms of balance of force. The equations for gas are found from the modified Boltzmann equation and the phenomenological theory. For a rigid body, the equations used the phenomenological theory, but changed their interpretation. We elucidate the contribution of cross-effects in the conservation laws of continuum mechanics, including the self-diffusion, thermal diffusion, etc., which indicated S. Wallander. The paradox of Hilbert in the solution of the Boltzmann equation by the Chapman-Enskog method was resolved. Refined model of the boundary conditions for rarefied gas flows and transient flow were near the moving surfaces. We establish conditions for the existence of the A. N. Kolmogorov inertial range on the basis of the proposed theory. Based on the theory, derivation of the Prandtl formula for boundary layer was received. Delay in mechanics plays an important role on commensurability of relaxation times and lateness. New accounting delay option is proposed to consider the difference between the time derivative as a limit and end values of the mean free path in a rarefied gas. The role of individual time delay for each particle velocity and the average time is debated. The Boltzmann equation is written with an additional term. This situation is typical for discrete medium. The transition from discrete to continuous environment is a key issue mechanics. Summary records of all effects lead to a cumbersome system of equations and therefore require the selection of main effects in a particular situation. The role of the time has similar problems in quantum mechanics. Some examples are suggested.

Dispersion Effects in the Falkner-Skan Problem and in the Kinetic Theory

“…In general case three positions are important for understanding the causes of modification the theory [1] [2] [3]. Another type of including angular momentum is contained in [6].…”

“…It was built with modified the Lagrangian function. The classical phenomenological theory was used for a solid body but interpretation was varied in this case [1] [2] [3]. Classical ideas are obtained from consideration the volume as exclusive.…”

“…Even classic computation of solution presents considerable difficulties [5]. In [1] [2] [3] influence of the angular momentum on the equations of continuum mechanics was obtained. The proposed theoretical method of consideration for the angular momentum without new empirical constants was suggested.…”

“…Being open system, the elementary volume exchanged components physical quantities in all directions. Very importance is the role of the inertia сentre [3]. In this paper we numerically investigate the influence of small perturbations of vertical velocity on the longitudinal velocity in the modified Falkner-Skan problem to reflect changes the influence of the angular momentum in the elementary volume.…”