Fluid Closure

Lack of closure is an endemic problem in fluid theory. Since each moment is
coupled to the next higher moment (*e.g.*, the density evolution depends on
the flow velocity, the flow velocity evolution depends on the viscosity
tensor, *etc*.), any finite set of exact moment equations is bound to
contain more unknowns than equations.

There are two basic types of fluid closure schemes. In *truncation* schemes,
higher order moments are arbitrarily
assumed to vanish, or simply prescribed in terms of lower
moments. Truncation schemes can often provide quick insight into fluid
systems, but always involve uncontrolled approximation. *Asymptotic* schemes
depend on the rigorous exploitation of some small parameter.
They have the advantage of being systematic, and providing some
estimate of the error involved in the closure. On the other hand,
the asymptotic approach to closure is mathematically very demanding, since it
inevitably involves working with the kinetic equation.

The classic example of an asymptotic closure scheme is the Chapman-Enskog
theory of a neutral gas dominated by collisions. In this case, the small
parameter is the ratio of the mean-free-path between collisions to the
macroscopic variation length-scale. It is instructive to briefly examine this theory,
which is very well described in a classic monograph by Chapman and
Cowling.^{}

Consider a neutral gas consisting of identical hard-sphere molecules of
mass and
diameter . Admittedly, this is not a particularly
physical model of a neutral gas,
but we are only considering it for illustrative purposes. The fluid
equations for such a gas are similar to Eqs. (220)-(222):

Here, is the (particle) density, the flow velocity, the scalar pressure, and the acceleration due to gravity. We have dropped the subscript because, in this case, there is only a single species. Note that there is no collisional friction or heating in a single species system. Of course, there are no electrical or magnetic forces in a neutral gas, so we have included gravitational forces instead. The purpose of the closure scheme is to express the viscosity tensor, , and the heat flux density, , in terms of , , or , and, thereby, complete the set of equations.

The mean-free-path for hard-sphere molecules is given by

(232) |

In the Chapman-Enskog scheme, the distribution function is expanded, order by order,
in the small parameter :

(233) |

(234) |

It is possible to *linearize* the kinetic equation, and then rearrange
it so as to obtain an *integral equation* for in terms of .
This rearrangement depends crucially on the *bilinearity* of the collision
operator.
Incidentally, the equation is integral because the collision operator is an integral
operator. The integral equation is solved by expanding in velocity space
using Laguerre polynomials (sometime called Sonine polynomials). It is
possible to reduce the integral equation to an infinite set of simultaneous
algebraic equations for the coefficients in this expansion. If the expansion
is truncated, after terms, say, then these algebraic equations can be solved for
the coefficients. It turns out that the Laguerre polynomial expansion
converges very rapidly. Thus, it is conventional to only keep the first *two*
terms in this expansion, which is usually sufficient to ensure an accuracy of
about in the final result. Finally, the appropriate moments
of are taken, so as to obtain expression for the heat flux density
and the viscosity
tensor. Strictly speaking, after evaluating , we should then go on to
evaluate , so as to ensure that really is negligible compared to .
In reality, this is never done because the mathematical difficulties involved
in such a calculation are prohibitive.

The Chapman-Enskog method outlined above can be applied to *any* assumed
force law between molecules, provided that the force is sufficiently short-range
(*i.e.*, provided that it falls off faster with increasing
separation than the Coulomb force). For all sensible force laws, the viscosity
tensor is given by

Here, is the

(237) | |||

(238) |

where is the

Here,

is the

Equations (239)-(240) have a simple physical interpretation: the viscous and thermal
diffusivities of a neutral gas can be accounted for in terms of the *random-walk diffusion* of molecules with excess momentum and energy, respectively.
Recall the standard result in stochastic theory that if particles
jump an average distance , in a random direction, times a second, then
the diffusivity associated with such motion is
.
Chapman-Enskog theory basically allows us to calculate the numerical constants
and ,
multiplying in the expressions for and ,
for a given force law between molecules.
Obviously, these coefficients are different for different force laws. The
expression for the
mean-free-path, , is also different for different force laws.

Let , , and be typical values of the
particle density, the thermal velocity, and the mean-free-path, respectively.
Suppose that the typical flow velocity is
,
and the typical variation length-scale is . Let us
define the following normalized quantities:
,
,
,
,
,
,
,
,
,
,
,
. Here,
.
Note that

(242) | |||

(243) |

All hatted quantities are designed to be . The normalized fluid equations are written:

where

(247) |

Suppose that . In other words, the flow velocity is much
greater than the thermal speed. Retaining only the largest terms in Eqs. (244)-(246),
our system of fluid equations reduces to (in unnormalized form):

(248) | |||

(249) |

These are called the

Suppose that
. In other words, the flow velocity is of order the
thermal speed. Again, retaining only the largest terms in Eqs. (244)-(246),
our system of fluid equations reduces to (in unnormalized form):

(250) | |||

(251) | |||

(252) |

The above equations can be rearranged to give:

These are called the

Suppose, finally, that
. In other words, the flow
velocity is of order the viscous and thermal diffusion velocities. Our system
of fluid equations now reduces to a force balance criterion,

(256) |

(257) | |||

(258) | |||

(259) |

Clearly, this set of equations is only appropriate to relatively quiescent, quasi-equilibrium, gas dynamics. Note that virtually all of the terms in our original fluid equations, (228)-(230), must be retained in this limit.

The above investigation reveals an important truth in gas dynamics, which also
applies to plasma dynamics. Namely, the form of the
fluid equations depends crucially on the typical fluid *velocity*
associated with the type of dynamics under investigation. As a general rule,
the equations get simpler as the typical velocity get faster, and *vice versa*.