User:Tohline/AxisymmetricConfigurations/Equilibria
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  =Axisymmetric Configurations (  +  =Axisymmetric Configurations (SteadyState Structures)= 
  +  Equilibrium, axisymmetric '''structures''' are obtained by searching for timeindependent, steadystate solutions to the [[User:Tohline/AxisymmetricConfigurations/PGE#Axisymmetric_Configurations_.28Part_I.29identified set of simplified governing equations]].  
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Given that our aim is to construct steadystate configurations, we should set the partial timederivative of all scalar quantities to zero; in addition, we will assume that both meridionalplane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the lefthandsides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:  Given that our aim is to construct steadystate configurations, we should set the partial timederivative of all scalar quantities to zero; in addition, we will assume that both meridionalplane velocity components, <math>\dot{r}</math> and <math>~\dot{\theta}</math>, to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the lefthandsides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:  
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<span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />  <span id="PGE:Poisson"><font color="#770000">'''Poisson Equation'''</font></span><br />  
Current revision as of 17:15, 3 August 2019
Contents 
Axisymmetric Configurations (SteadyState Structures)
Equilibrium, axisymmetric structures are obtained by searching for timeindependent, steadystate solutions to the identified set of simplified governing equations.
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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
We begin with an Eulerian formulation of the principle governing equations written in spherical coordinates for an axisymmetric configuration, namely,
Equation of Continuity



The Two Relevant Components of the
Euler Equation
: 

= 

: 

= 

Adiabatic Form of the
First Law of Thermodynamics



Poisson Equation



where the pair of "relevant" components of the Euler equation have been written in terms of the specific angular momentum,
,
which is a conserved quantity in axisymmetric systems.
Given that our aim is to construct steadystate configurations, we should set the partial timederivative of all scalar quantities to zero; in addition, we will assume that both meridionalplane velocity components, and , to zero — initially as well as for all time. As a result of these imposed conditions, both the equation of continuity and the first law of thermodynamics are automatically satisfied; the Poisson equation remains unchanged; and the lefthandsides of the pair of relevant components of the Euler equation go to zero. The governing relations then take the following, considerably simplified form:
Spherical Coordinate Base  

Poisson Equation
The Two Relevant Components of the

See Also
 Part I of Axisymmetric Configurations: Simplified Governing Equations
© 2014  2020 by Joel E. Tohline 