6.10 The nature of viscosity
Before discussing turbulence further, it is useful to examine the origins of viscosity. Viscosity is introduced in Sec. 2.12 of this book as part of the Newtonian constitutive model. The model was originally phenomenological, but was later derived directly from kinetic theory which describes a gas as a number of submicroscopic particles, e.g. atoms or molecules, in random motion.
The kinetic view of viscosity imagines a ﬂuid in two dimensions, and , subjected to a shear force in the direction. Although the mean ﬂow is in the direction, particles move in the direction due to random ﬂuctuations with a mean speed .
Consider a plane at . A particle will pass through the plane if its path towards it is not interrupted by a collision which sends it moving away from the plane. Particles passing through the plane arrive from an average distance , where is some factor of the average distance travelled by a moving particle between successive collisions, the mean free path .
From kinetic theory, the mass ﬂow rate of particles passing through a surface of unit area . The mean velocity of particles crossing the plane from the direction is ; similarly from the direction, it is .
The net momentum of the particles, positive on the side of the plane is then

(6.16) 
The net momentum is equivalent to the shear stress on the side of the plane, , as described in Sec. 2.6 . By comparison with Eq. (6.16 ), the dynamic viscosity in terms of molecular properties. The kinematic viscosity is

(6.17) 
A more thorough analysis^{13} begins with the Boltzmann equation and applies the Chapman–Enskog expansion to ﬁrst order in Knudsen number

(6.18) 

(6.19) 