Microscopic Study of Fluid Flow

A new numerical technique for the study of fluid flow is developed which is particularly well adapted for the study of dilute gases. In this initial study, the systems studied were limited to two-dimensions. The system is divided into a grid of rectangular cells. Each cell contains hundreds or even thousands of molecules. The molecules are assumed to be randomly located in their cell. Each cell is assigned a velocity spectrum constructed by specifying the probabilities that a molecule will have a particular speed and direction of motion. The ranges of speeds and of angle are divided into sectors which cover the ranges completely. Each sector is characterized by a central (i.e., average) value, and the probability that a member of the cell will have a speed (or angle) spanned by the sector is assigned to each sector. Between them, these data constitute a "velocity spectrum". A basic time step is chosen such that there is a high probability that a substantial fraction of the population will suffer a collision in that time, but not so long that large changes in a cell's spectrum occur. The collisions are assumed to be hard -sphere interactions, although that is not a limitation of the method. After initial values in each cell have been set, the system is interaction through one time step as follows: in a given cell the consequences of all possible collisions are computed, and taking the probabilities into account, a new (intermediate) velocity spectrum is found. By using the apparatus of random walk theory, estimates are made of what fractions of the population of a cell will move to adjacent cells and what fraction will remain. The pertinent transfers of populations from each cell to each of its 8 neighbors are then carried out. Each transfer involves transferring an appropriate velocity spectrum with the transferred molecules. The system is now ready to be advanced another time step. Two different geometries of flow are studied: (1) straight-channel flow, and (2) channel flow with a constriction. The quality of the solution is monitored by following the conservation of energy, the continuity, the entropy behavior, and the general nature of the calculated flows.