User:Tohline/VE/RiemannEllipsoids
From VisTrailsWiki
(→The Three Diagonal Elements) 
(→The Six OffDiagonal Elements) 

Line 535:  Line 535:  
<td align="left">  <td align="left">  
<math>~  <math>~  
  2 \mathfrak{T}_{ij}  \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j  +  2 \mathfrak{T}_{ij}  \Omega_i\Omega_k I_{kj} + 2\epsilon_{ilm}\Omega_m \int_V \rho u_lx_j d^3x 
\, .  \, .  
</math>  </math>  
Line 554:  Line 554:  
<td align="left">  <td align="left">  
<math>~  <math>~  
  2 \mathfrak{T}_{23}  \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3  +  2 \mathfrak{T}_{23}  \Omega_2\Omega_3 I_{33} + 2\Omega_1 \cancelto{0}{\int_V \rho u_3x_3 d^3x} 
   2\Omega_3 \int_V \rho u_1x_3  +   2\Omega_3 \int_V \rho u_1x_3 d^3x \, , 
</math>  </math>  
</td>  </td>  
Line 573:  Line 573:  
<td align="left">  <td align="left">  
<math>~  <math>~  
  2 \mathfrak{T}_{32}  \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2  +  2 \mathfrak{T}_{32}  \Omega_3 \Omega_2 I_{22} + 2\Omega_2 \int_V \rho u_1x_2 d^3x 
   2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2  +   2\Omega_1 \cancelto{0}{\int_V \rho u_2 x_2 d^3x} 
\, .  \, .  
</math>  </math>  
Line 588:  Line 588:  
<tr>  <tr>  
<td align="right">  <td align="right">  
  <math>~\int_V \rho u_i x_j  +  <math>~\int_V \rho u_i x_j d^3x</math> 
</td>  </td>  
<td align="center">  <td align="center">  
Line 623:  Line 623:  
</td></tr></table>  </td></tr></table>  
+  Our first offdiagonal element is, then,  
+  <table border="0" cellpadding="5" align="center">  
+  <tr>  
+  <td align="right">  
+  <math>~0</math>  
+  </td>  
+  <td align="center">  
+  <math>~=</math>  
+  </td>  
+  <td align="left">  
+  <math>~  
+  2 \mathfrak{T}_{23}  \Omega_2\Omega_3 I_{33}  
+   2\Omega_3 \rho \int_V u_1 z d^3x  
+  </math>  
+  </td>  
+  </tr>  
+  </table>  
+  
+  
+  ===How Solution is Obtained ===  
Adding this pair of governing expressions we obtain,  Adding this pair of governing expressions we obtain,  
Revision as of 16:17, 6 August 2020
Contents

SteadyState 2^{nd}Order Tensor Virial Equations
By satisfying all six — not necessarily unique — components of the SecondOrder Tensor Virial Equation, the entire set of Riemann Ellipsoids can be determined.
 Tiled Menu  Tables of Content  Banner Video  Tohline Home Page  
Here we are only interested in determining the equilibrium conditions of uniformdensity ellipsoids that have semiaxes, .
General Coefficient Expressions
As has been detailed in an accompanying chapter, the gravitational potential anywhere inside or on the surface, , of an homogeneous ellipsoid may be given analytically in terms of the following three coefficient expressions:









where, and are incomplete elliptic integrals of the first and second kind, respectively, with arguments,

and 

[ EFE, Chapter 3, §17, Eq. (32) ] 
Adopted (Internal) Velocity Field
EFE (p. 130) states that the … kinematical requirement, that the motion , associated with , preserves the ellipsoidal boundary, leads to the following expressions for its components:









[ EFE, Chapter 7, §47, Eq. (1) ] 
Equilibrium Expressions
[EFE §11(b), p. 22] Under conditions of a stationary state, [the tensor virial equation] gives,



[This] provides six integral relations which must obtain whenever the conditions are stationary.
When viewing the (generally ellipsoidal) configuration from a rotating frame of reference, the 2^{nd}order TVE takes on the more general form:



[ EFE, Chapter 2, §12, Eq. (64) ] 
EFE (p. 57) also shows that … The potential energy tensor … for a homogeneous ellipsoid is given by



[ EFE, Chapter 3, §22, Eq. (128) ] 
where



[ EFE, Chapter 3, §22, Eq. (129) ] 
is the moment of inertia tensor. Expressions for all nine components of the kinetic energy tensor, are derived in Appendix E, below; and expressions for each of the six Coriolis components can be found in Appendices B, C, & D.
The Three Diagonal Elements
For , we have,


















Once we choose the values of the (semi) axis lengths of an ellipsoid — from which the value of can be immediately determined — along with a specification of , this equation has the following five unknowns: . Similarly, for ,















This gives us a second equation, but an additional pair of (for a total of seven) unknowns: . For the third diagonal element — that is, for — we have,















This gives us three equations vs. seven unknowns.
The Six OffDiagonal Elements
Notice that the offdiagonal components of both and are zero. Hence, the equilibrium expression that is dictated by each offdiagonal component of the 2^{nd}order TVE is,



For example — as is explicitly illustrated on p. 130 of EFE — for and ,




[ EFE, Chapter 7, §47, Eq. (3) ] 
whereas for and ,




[ EFE, Chapter 7, §47, Eq. (4) ] 
Given our adoption of a uniformdensity configuration whose surface has a precisely ellipsoidal shape and, along with it, our adoption of the above specific prescription for the internal velocity field, , we recognize that,
This has allowed us to set to zero one of the integrals in each of these last two expressions. In what follows, we will benefit from recognizing, as well, that,

Our first offdiagonal element is, then,



How Solution is Obtained
Adding this pair of governing expressions we obtain,






[ EFE, Chapter 7, §47, Eq. (6) ] 
and subtracting the pair gives,






[ EFE, Chapter 7, §47, Eq. (7) ] 
Various Degrees of Simplification
Riemann SType Ellipsoids
Describe …
Jacobi and Dedekind Ellipsoids
Describe …
Maclaurin Spheroids
Describe …
Appendices: Various Integrals Over Ellipsoid Volume
Throughout this set of appendices, we work with a uniformdensity ellipsoid whose surface is defined by the expression,



Appendix A: Volume
Here we seek to find the volume of the ellipsoid via the Cartesian integral expression,



Preliminaries
First, we will integrate over and specify the integration limits via the expression,



second, we will integrate over and specify the integration limits via the expression,



third, we will integrate over and set the limits of integration as .
Carry Out the Integration
Following thestrategy that has just been outlined, we have,


















Appendix B: Coriolis Component u_{1}x_{2}






























[ EFE, Chapter 7, §47, p. 130, Eq. (9a) ] 
Appendix C: Coriolis Component u_{1}x_{3}
Here we will additionally make use of the integration limits,



Integration over the relevant Coriolis component gives,



























[ EFE, Chapter 7, §47, p. 130, Eq. (9b) ] 
Appendix D: The Other Four Coriolis Components
It follows that,
























Appendix E: Kinetic Energy Components
Diagonal Elements















Similarly,


















OffDiagonal Elements


















[ EFE, Chapter 7, §47, p. 130, Eq. (8) ] 
Similarly,
























And, finally,






and, 



See Also
© 2014  2020 by Joel E. Tohline 