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, eqn and eqn, subjected to a shear force in the eqn-direction. Although the mean flow is in the eqn-direction, particles move in the eqn-direction due to random fluctuations with a mean speed eqn.

PICT\relax \special {t4ht=

Consider a plane at eqn. 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 eqn, where eqn is some factor of the average distance travelled by a moving particle between successive collisions, the mean free path eqn.

From kinetic theory, the mass flow rate of particles passing through a surface of unit area eqn. The mean eqn-velocity of particles crossing the plane from the eqn-direction is eqn ; similarly from the eqn-direction, it is eqn .

The net eqn-momentum of the particles, positive on the eqn side of the plane is then

1 - - @ux 1 - - @ux 1 -- @ux 4- v ux + a @y-- 4 -v ux a -@y- = 2 -va -@y-: \relax \special {t4ht=

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

= av -: 2 \relax \special {t4ht=
The original analysis of Maxwell12 uses eqn in Eq. (6.17 ). It was later recognised the average distance described by eqn was larger due to persistence of velocities, i.e. a particle will sometimes maintain a path towards the plane after a collision.

A more thorough analysis13 begins with the Boltzmann equation and applies the Chapman–Enskog expansion to first order in Knudsen number

 -- Kn = L : \relax \special {t4ht=
The analysis derives the conservation laws with Newtonian and Fourier constitutive models, i.e. Eq. (2.19 ), Eq. (2.51 ), Eq. (2.41 ) and Eq. (2.54 ). Treating particles as rigid spheres and collisions as elastic, yields a value eqn, leaving a simple expression for viscosity which is
|--------| | 1-- | | ' 2v | ---------- \relax \special {t4ht=

12James Clerk Maxwell, Illustrations of the dynamical theory of gases. Part I. On the motions and collisions of perfectly elastic spheres, 1860.
13see Sydney Chapman and Thomas Cowling, The mathematical theory of non-uniform gases (3rd ed.), 1970.

Notes on CFD: General Principles - 6.10 The nature of viscosity