Influence of the Delay and Dispersion in Mechanics

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

The conservation laws of continuum mechanics and of the kinetic theory with the influence of the angular momentum and associated with its rotation of the elementary volume are considered, the variant of accounting lag is investigated for discrete environment. The analysis of the recording of the Lagrangian function for the collective interaction of the particles with the change of the center of inertia of the moving particles and the effect influence of the angular momentum were used. The equations for gas are calculated from the modified Boltzmann equation and the phenomenological theory. For a rigid body the equations were used of the phenomenological theory, but their interpretation was changed. The nonsymmetric stress tensor was obtained. The Boltzmann equation is written with an additional summand. This situation is typical for discrete environment as the transition from discrete to continuous environment is a key to the issue of 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 Hilbert paradox was being solved. The simplest problem of the boundary layer continuum (the Falkner-Skan task) and the kinetic theory are discussed. A draw attention at the delay process would be suggested for the description of discrete environment. Results are received for some special cases.

Effect of Mathematical Models on Experimental Data for the Gas and Liquids

The new analysis of the influence the method of calculating macro-parameters to experimental data is made for the mixture a rarefied gas and gas with internal degrees of freedom. The delay process is counted, which is important in describing of the discrete space and in describing the relaxation of the complicated molecules. The analysis of the recording the Lagrangian function for the collective interaction of the particles is made with counting of changing position of the inertia center. This equation should have a modified Liouville equation and the Boltzmann equation. General consideration of all effects gives us cumbrous system of equations. New another definition of temperature is obtained for molecules with vibration and rotation degree of freedom and for mixture. This is making another value for pressure of the mixture a rarefied gas and gas with internal degrees of freedom. Probably, exactly, these values are measured in all experiments. The simplest interaction of two homogeneous flows is studied which move in the same direction at different speeds.