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Supplemental Relations

Apart from the independent variables <math>~t</math> and <math>~\vec{x}</math>, our principal governing equations involve the vector velocity <math>~\vec{v}</math>, and the four scalar variables, <math>~\Phi</math>, <math>~P</math>, <math>~\rho</math>, and <math>~\epsilon</math>. Because the variables outnumber the equations by one, one (additional) supplemental relationship between the physical variables must be specified in order to close the set of equations.

Also, in order to complete the unique specification of a particular physical problem, either a steady-state flow field or initial conditions must be specified, depending on whether one is studying a time-independent (structure) or time-dependent (stability or dynamics) problem, respectively. Throughout this H_Book, the following strategy will be adopted in order to complete the physical specification of each examined system:

For time-independent problems, we will ...
- adopt a structural relationship between <math>~P</math> and <math>~\rho</math>, and
- specify a steady-state flow-field.
For time-dependent problems, we will ...
- adopt an equation of state, and
- specify initial conditions.

Time-Independent Problems

Barotropic Structure

For time-independent problems, a structural relationship between <math>~P</math> and <math>~\rho</math> is required to close the system of principal governing equations. [Tassoul (1978) refers to this as a "geometrical" rather than a "structural" relationship; see the discussion associated with his Chapter 4, Eq. 14.] Generally throughout this H_Book, we will assume that all time-independent configurations can be described as barotropic structures; that is, we will assume that <math>~P</math> is only a function of <math>~\rho</math> throughout such structures. More specifically, we generally will adopt one of the following two analytically prescribable <math>~P</math>(<math>~\rho</math>) relationships displayed in the following Table.

Barotropic Relations

Polytropic

Zero-temperature Fermi (degenerate electron) Gas

<math>~P = K_\mathrm{n} \rho^{1+1/n}</math>

	<math>~P_\mathrm{deg} = A_\mathrm{F} F(\chi) </math>
where: <math>F(\chi) \equiv \chi(2\chi^2 - 3)(\chi^2 + 1)^{1/2} + 3\sinh^{-1}\chi</math>
and:	<math>\chi \equiv (\rho/B_\mathrm{F})^{1/3}</math>

Reference (original): Chandraskehar, S. (1935)

In the polytropic relation, the "polytropic index" <math>~n</math> and the "polytropic constant" <math>~K_\mathrm{n}</math> are assumed to be independent of both <math>~\vec{x}</math> and <math>~t</math>. In the zero-temperature Fermi gas relation, the two constants <math>~A_\mathrm{F}</math> and <math>~B_\mathrm{F}</math> are expressible in terms of various fundamental physical constants, as detailed in the accompanying variables appendix.

Steady-state Flow-Field Specification

A steady-state velocity flow-field must be specified for time-independent problems. The specification can be as simple as stating that <math>~\vec{v}</math> = 0 everywhere in space. Other examples are uniform rotation; or, for axisymmetric configurations, uniform specific angular momentum. In this H_Book, the flow-field specification will generally vary from chapter to chapter.

Time-Dependent Problems

Equation of State

The equation of state that generally will be adopted for time-dependent problems is one that describes an ideal gas. As the accompanying discussion illustrates, the ideal gas equation of state can assume a variety of different forms. Throughout this H_Book, we frequently will use either "Form A" or "Form B" of the ideal gas equation of state, as displayed in the following Table, to supplement the principal governing equation.

Ideal Gas Equation of State

Form A

Form B

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

"Form A" provides a relationship between the gas temperature <math>~T</math> and the state variables <math>~P</math> and <math>~\rho</math>, where <math>~\Re</math> is the gas constant and <math>~\bar{\mu}</math> is the mean molecular weight of the gas. "Form B" relates the state variables <math>~P</math> and <math>~\rho</math> to <math>~\epsilon</math>, where the ratio of specific heats <math>~\gamma_\mathrm{g}</math> is assumed to be independent of both <math>~\vec{x}</math> and <math>~t</math>. [See Tassoul (1978) — specifically the discussion associated with Chapter 4, Eq. 13 — for a more general statement related to the proper specification of the supplemental, equation of state relationship.]

Initial Conditions

For time-dependent problems, the principal governing equations must be supplemented further through the specification of initial conditions. Frequently throughout this H_Book, we will select as initial conditions a specification of <math>~\rho</math>( <math>~\vec{x}</math>, <math>~t</math> = 0), <math>~P</math>( <math>~\vec{x}</math>, <math>~t</math> = 0), and <math>~\vec{v}</math>( <math>~\vec{x}</math>, <math>~t</math> = 0) that, as a group themselves, define a static or steady-state equilibrium structure. Perturbation or computational fluid dynamic (CFD) techniques can be used to test the stability of such structures.