Hierarchial structure theory for basic equations of fluid mechanics (BEFM) and the simplified Navier-Stokes equations (SNSE)

Abstract

A hierarchial structure for the basic equations of fluid mechanics (BEFM) is found through the analysis of scales of length and time that proves a measure of the rate of change of the quantities describing the motion of the fluid as well as an estimation of the order of magnitude of various terms included in BEFM. The hierarchial structure theory shows that if (1) the characteristic Reynolds numbersRe is larger than unity and (2) the length scale in one coordinate direction is larger than that in other coordinate directions. BEFM can be classified into some levels according to the estimation of the order of magnitude of various terms included in BEFM. The hierarchial structure of BEFM has two branches: one is from BLE- to BEFM inner hierarchy, the other is from EE- to BEFM outer hierarchy, where BLE and EE are abbreviations of the boundary-layer equations and of Euler equations, respectively. The relationship between the two branches of the hierarchial structure, the characteristics, subcharacteristics and mathematical properties of the hierarchial equations are studied. A comparison between the present hierarchial equations and the Simplified Navier-Stokes equations (SNSE) appeared in literatures is also made. BLE-, EE-and Inner-outer-matched (IOM) equations hierarchies are the most important and useful three levels for solving viscous flow-fields approximately.