Пределы применимости модели непрерывной среды и регуляризация уравнений динамики жидкости и газа

The continuous media is a fruitful mathematical notion for the descriptions of the variety of mechanical processes. In continuous mechanics basic equations of the balance of mass, momentum, and energy are derived first for the finite volumes and next transformed to the differential form. Together with physically reasonable boundary and initial conditions they are expected to give full description required processes. However the transition from a finite volume description to a differential one involves the mathematical assumption of the infinitesimal volume.
Notice that the governing equations of hydro and gas dynamics are nonlinear and a variety of crucial mathematical features are to be solved.

But taking into account the molecular structure of matter the infinitesimal volume necessarily requires dozens of molecules inside. It means that physically there is the lower limit on its size.
For instance the air of normal density is containing 1019 particles per cm3. It means that physically infinitesimal volume has the minimum size of the order d~10–6 cm3. It is much less than the mean free path, which is in the range of 10–4–10–5 cm, and is much greater than the characteristic size of gas molecules which are ~10–8 cm. Thus on the d scale values of gas-dynamic parameters must be constant.
Note that for the computer calculations the boundary-initial value problem must be discretized. That in turn requires a discrete description of the continuous media. That is why in the computer modelling differential equations serves for the transition from one discrete description to another one. That is why a regularization of equations of hydro and gas dynamics should be physically reasonable and additional terms ensuring regularization of solutions at distance d must be included in the differential equations. The absence of such terms in the classical equations of gas dynamics can be explained by the fact that the distance d = 10–6 cm, over which an essential change of the gas dynamic parameters is ensured, is significantly smaller than the discretization error. These resolution limits are determined by both the physical necessity of the detailed problem description, the capabilities of the numerical method, and the performance characteristics of the computer facilities employed in the calculations. The main aim of the lectures is to give an introduction to quasi-gas dynamical systems of equations and theory of lattice Boltzmann and kinetically consistent difference schemes.