# Chapter 4

# Fundamental Equations of Fluid Mechanics

## 4.1. Continuous regime and molecular regime

A gas such as air consists of molecules moving in a random way. At the atomic level, the gas is a discontinuous medium. The mean distance separating the molecules is characterized by the (molecular) mean free path *λ*, which in general we know to calculate from thermodynamic quantities. In the case where a body with the characteristic dimension *L* (diameter of the cylinder, chord of the airfoil, length of the vehicle) is placed in a flow where the mean free path is *λ*, the following two extreme situations can be met.

– The mean free path is very small compared to the size of the body *λ*/*L* << 1: then the molecules are so close to each other that their action on the body is seen as a continuous action, the impact of each molecule cannot be distinguished. The fluid is considered as a continuous medium of which local properties can be defined by going to the limit *δu* → 0 of a small volume *δu*. If the volume *δu* contains a mass *δm*:

when *δu* → 0, which defines the density *ρ*. The fluid motion is amenable to the Navier–Stokes equations, which are the theoretical basis of conventional aerodynamics.

– The mean free path is of the same order as the size of the body *λ*/*L* = *O*(1) then every impact of a molecule is felt isolated by the body, therefore, the definition of local quantities from a passage to the ...