User:Tohline/AxisymmetricConfigurations/PGE
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Axisymmetric Configurations (Governing Equations)
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If the selfgravitating configuration that we wish to construct is axisymmetric, then the coupled set of multidimensional, partial differential equations that serve as our principal governing equations can be simplified to a coupled set of twodimensional PDEs.
Cylindrical Coordinate Base
Here we choose to …
 Express each of the multidimensional spatial operators in cylindrical coordinates () (see, for example, the Wikipedia discussion of vector calculus formulae in cylindrical coordinates) and set to zero all spatial derivatives that are taken with respect to the angular coordinate :
Spatial Operators in Cylindrical Coordinates
=
[BT87], p. 649, Eq. (1B37)
=
[BT87], p. 650, Eq. (1B50)
=
=
[BT87], p. 650, Eq. (1B45)
=
[BT87], p. 651, Eq. (1B54)
From this last expression — the socalled convective operator — we conclude as well that, for axisymmetric systems,
=
 Express all vector timederivatives in cylindrical coordinates:
Vector TimeDerivatives in Cylindrical Coordinates
=
=
=
[BT87], p. 647, Eq. (1B23)
Governing Equations (CYL.)
Introducing the above expressions into the principal governing equations gives,
Equation of Continuity
Euler Equation
Adiabatic Form of the
First Law of Thermodynamics
Poisson Equation
Conservation of Specific Angular Momentum (CYL.)
The component of the Euler equation leads to a statement of conservation of specific angular momentum, j, as follows.
So, for axisymmetric configurations, the and components of the Euler equation become, respectively,

Eulerian Formulation (CYL.)
Each of the above simplified governing equations has been written in terms of Lagrangian time derivatives. An Eulerian formulation of each equation can be obtained by replacing each Lagrangian time derivative by its Eulerian counterpart. Specifically, for any scalar function, f,
Making this substitution throughout the set of governing relations gives:
Equation of Continuity
The Two Relevant Components of the
Euler Equation
: 



: 



Adiabatic Form of the
First Law of Thermodynamics
Poisson Equation
Spherical Coordinate Base
Here we choose to …
 Express each of the multidimensional spatial operators in spherical coordinates () (see, for example, the Wikipedia discussion of vector calculus formulae in spherical coordinates) and set to zero all spatial derivatives that are taken with respect to the angular coordinate :
Spatial Operators in Spherical Coordinates
=
[BT87], p. 649, Eq. (1B38)
=
[BT87], p. 650, Eq. (1B51)
=
=
[BT87], p. 650, Eq. (1B46)
=
[BT87], p. 651, Eq. (1B55)
From this last expression — the socalled convective operator — we conclude as well that, for axisymmetric systems,
=
 Express all vector timederivatives in spherical coordinates:
Vector TimeDerivatives in Spherical Coordinates
=
=
=
=
[BT87], p. 648, Eq. (1B30)
Governing Equations (SPH.)
Introducing the above expressions into the principal governing equations gives,
Equation of Continuity



Euler Equation



Adiabatic Form of the
First Law of Thermodynamics
Poisson Equation



Conservation of Specific Angular Momentum (SPH.)
The component of the Euler equation leads to a statement of conservation of specific angular momentum, , as follows.









So, for axisymmetric configurations, the and components of the Euler equation become, respectively,

Eulerian Formulation (SPH.)
Each of the above simplified governing equations has been written in terms of Lagrangian time derivatives. An Eulerian formulation of each equation can be obtained by replacing each Lagrangian time derivative by its Eulerian counterpart. Specifically, for any scalar function, f,
Making this substitution throughout the set of governing relations gives:
Equation of Continuity



The Two Relevant Components of the
Euler Equation
: 

= 

: 

= 

Adiabatic Form of the
First Law of Thermodynamics



Poisson Equation



See Also
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