Page 322 - Intro to Tensor Calculus

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This gives the stress relations due to random particle motion

σ xx = − ρU 2
r σ yx = − ρV r U r σ zx = − ρW r U r
σ yy = − ρV 2 (2.5.113)
σ xy = − ρU r V r r σ zy = − ρW r V r
2
r
σ xz = − ρU r W r σ yz = − ρV r W r σ zz = − ρW .
The Boltzmann equation is a basic macroscopic model used for the study of individual particle motion
where one takes into account the distribution of particles in both space, time and energy. The Boltzmann
equation for gases assumes only binary collisions as three-body or multi-body collisions are assumed to
rarely occur. Another assumption used in the development of the Boltzmann equation is that the actual
time of collision is thought to be small in comparison with the time between collisions. The basic problem
associated with the Boltzmann equation is to find a velocity distribution, subject to either boundary and/or
initial conditions, which describes a given gas flow.
The continuum equations involve trying to obtain the macroscopic variables of density, mean velocity,
stress, temperature and pressure which occur in the basic equations of continuum mechanics considered
earlier. Note that the moments of the Boltzmann equation, derived for gases, also produced these same
continuum equations and so they are valid for gases as well as liquids.
In certain situations one can assume that the gases approximate a Maxwellian distribution

m 3/2 m
~
~
~
f(~r, V, t) ≈ n(~r, t) exp − V · V (2.5.114)
2πkT 2kT
thereby enabling the calculation of the pressure tensor and temperature from statistical considerations.
In general, one can say that the Boltzmann integral-differential equation and the Maxwell transfer
equation are two important formulations in the kinetic theory of gases. The Maxwell transfer equation
depends upon some gas-particle property φ which is assumed to be a function of the gas-particle velocity.
The Boltzmann equation depends upon a gas-particle velocity distribution function f which depends upon
~
position ~,velocity V and time t. These formulations represent two distinct and important viewpoints
r
considered in the kinetic theory of gases.