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==Principal Governing Equations==

According to the eloquent discussion of the broad subject of ''Fluid Mechanics'' presented by Landau and Lifshitz (1975), <FONT COLOR="#007700">the state of a moving fluid is determined by five quantities:  the three components of the velocity</FONT> {{User:Tohline/Math/VAR_VelocityVector01}} <FONT COLOR="#007700">and, for example, the pressure</FONT> {{User:Tohline/Math/VAR_Pressure01}} <FONT COLOR="#007700">and the density</FONT> {{User:Tohline/Math/VAR_Density01}} <FONT COLOR="#007700">.</FONT> For our discussions of astrophysical fluid systems throughout this Hypertext Book [H_Book], we will add to this the gravitational potential {{User:Tohline/Math/VAR_NewtonianPotential01}}. <FONT COLOR="#007700">Accordingly, a complete system of equations of fluid dynamics should be</FONT> six <FONT COLOR="#007700">in number. For an ideal fluid these are:</FONT>

<div align="center">
<span id="PGE:Euler"><font color="#770000">'''Euler's Equation'''</font></span><br />
('''Momentum Conservation''')

{{User:Tohline/Math/EQ_Euler01}}


<span id="PGE:Continuity"><font color="#770000">'''Equation of Continuity'''</font></span><br />
('''Mass Conservation''')

{{User:Tohline/Math/EQ_Continuity01}}


<span id="PGE:FirstLaw">'''Adiabatic Form of the'''<br>
<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />
('''Specific Entropy Conservation''')

{{User:Tohline/Math/EQ_FirstLaw01}}


<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />

{{User:Tohline/Math/EQ_Poisson01}}
</div>


In the next-to-last equation, {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} is the specific internal energy of the gas, and in the last expression, {{User:Tohline/Math/C_GravitationalConstant}} is the Newtonian gravitational constant. These differential equations relate the spatial and temporal variations of the principal variables to one another in a physically consistent fashion.  Temporal variations are accounted for through first-order derivatives with respect to time, {{User:Tohline/Math/VAR_Time01}}, and variations in space, {{User:Tohline/Math/VAR_PositionVector01}}, are accounted for through the differential operators: gradient (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>).


By restricting our discussions to physical systems that are governed by this set of equations, for the most part '''we will be considering the structure, stability, and dynamical behavior of compressible, inviscid fluid systems that are self-gravitating.'''  We will assume that no electromagnetic forces act on the fluid (''e.g.,'' the effects of magnetic fields on an ionized plasma fluid will not be considered) and, in the absence of dynamically generated shocks, we will assume that all compressions and rarefactions occur adiabatically.


To complete the description of any specific astrophysical system, this set of differential equations must be supplemented by additional relations &#150; for example a relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_SpecificInternalEnergy01}} &#150; which (at least in the context of this H_Book) usually will be algebraic expressions motivated by the specific physics that is relevant to the chosen system.


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