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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 vaiables 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.
Euler's Equation
(Momentum Conservation)
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<math>\frac{d\vec{v}}{dt} = - \frac{1}{\rho} \nabla P - \nabla \Phi</math> |
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© 2014 - 2021 by Joel E. Tohline |