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 ...

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