User:Tohline/AxisymmetricConfigurations/Equilibria
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Axisymmetric Configurations (Structure — Part II)
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Equilibrium, axisymmetric structures are obtained by searching for timeindependent, steadystate solutions to the identified set of simplified governing equations.
Cylindrical Coordinate Base
We begin by writing each governing equation in Eulerian form and setting all partial timederivatives to zero:
Equation of Continuity
The Two Relevant Components of the
Euler Equation






Adiabatic Form of the
First Law of Thermodynamics
Poisson Equation
The steadystate flow field that will be adopted to satisfy both an axisymmetric geometry and the timeindependent constraint is, . That is, but, in general, is not zero and can be an arbitrary function of and , that is, . We will seek solutions to the above set of coupled equations for various chosen spatial distributions of the angular velocity , or of the specific angular momentum, .
After setting the radial and vertical velocities to zero, we see that the 1^{st} (continuity) and 4^{th} (first law of thermodynamics) equations are trivially satisfied while the 2^{nd} & 3^{rd} (Euler) and 5^{th} (Poisson) give, respectively,









As has been outlined in our discussion of supplemental relations for timeindependent problems, in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between and .
Spherical Coordinate Base
See Also
 Part I of Axisymmetric Configurations: Simplified Governing Equations
© 2014  2020 by Joel E. Tohline 