Momentum density redistributions

Such an approach is conceptually different from the continuum description of momentum transport in a fluid in terms of the NS equations. It can be demonstrated, however, that, with a proper choice of the lattice (viz. its symmetry properties), with the collision rules, and with the proper redistribution of particle mass over the (discrete) velocity directions, the NS equations are obeyed at least in the incompressible limit. It is all about translating the above characteristic LB features into the physical concepts momentum, density, and viscosity. The collision rules can be translated into the common variable viscosity, since colliding particles lead to viscous behavior indeed. The reader interested in more details is referred to Succi (2001). 

In the following discussion, we shall review the momentum-space aspect of diatomic interactions, focusing upon the redistribution of the momentum density and its contribution to the interaction potential. Since all the readers may not be very familiar with the momentum-space treatment, we start with a brief introduction to the concepts of the momentum wave function and the momentum density. 

In Fig. 17, these redistributions of the momentum densities are summarized for the three typical cases of diatomic interactions (Koga, 1981). In the case of attractive interactions, all the density reorganizations are initially contractions and succeedingly change into expansions in the order Ap, Ap, and Ap (Fig. 17a). For repulsive interactions with no stable molecules, the reorganizations may be expansive throughout the interactions (Fig. 17b). When there is a potential barrier (Fig. 17c), the initial... 

Duration of Velocity Equalization If the upper plate is not constantly pushed forward giving it a constant supply of momentum px, the fluid flow between the plates will soon come to a stop. This also happens when the lower plate is not held so that momentum cannot flow off there. In this situation, the momentum behaves like a diffusing substance in a closed container. For the sake of simplicity, we will imagine the plates to have no mass so that they can adapt without inertia to the velocity of the adjacent layer of liquid. All that remains is the redistribution of momentum px in the liquid where the excess in the upper half is to be moved to the lower half. If p is the density of the liquid, mass (first term), average velocity (second term), and momentum (third term) are ... 

The close-to-equilibrium theory, LIT, is applicable when the system is in local equilibrium. In the case of local equilibrium - in contrast to complete or global equilibrium - small but nevertheless macroscopic parts of the system are in equilibrium. These small parts, the so-called fluid elements, are identified by their macroscopic position, r, at time, t. The densities of the extensive variables in the whole system - such as mass, momentum, and energy - are conserved. Thus, their redistribution among the fluid elements (in a one-component atomic fluid) can be expressed in exact mathematical relationships as follows ... 

The Navier-Stokes equations describe the above-mentioned redistribution of mass, momentum, and energy density taking into account the rates of the physical processes in a fluid. Given the necessary boundary conditions they can be solved numerically. 

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