Normalization of Neutral Gas Equations

Here,

(4.70) |

Note that

(4.71) | ||

(4.72) |

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

where

(4.76) |

The only large or small quantities remaining in the previous equations are the parameters and .

Suppose that . In other words, suppose that the flow velocity is much greater than the thermal speed. Retaining only the largest terms in Equations (4.73)-(4.75), our system of fluid equations reduces to (in unnormalized form):

(4.77) | ||

(4.78) |

These are called the

Suppose that . In other words, suppose the flow velocity is of similar magnitude to the thermal speed. Again, retaining only the largest terms in Equations (4.73)-(4.75), our system of fluid equations reduces to (in unnormalized form):

(4.79) | ||

(4.80) | ||

(4.81) |

The previous equations can be rearranged to give:

(4.82) | ||

(4.83) | ||

(4.84) |

These are called the

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

(4.85) |

to lowest order. To next order, we obtain a set of equations describing the relatively slow viscous and thermal evolution of the gas:

(4.86) | ||

(4.87) | ||

(4.88) |

Clearly, this set of equations is only appropriate to relatively quiescent, quasi-equilibrium, gas dynamics. Virtually all of the terms in our original fluid equations, (4.55)-(4.57), must be retained in this limit.

The previous investigation reveals an important truth in gas dynamics, which also applies to plasma dynamics. Namely, the form of the fluid equations crucially depends 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 gets faster, and vice versa.