User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3
Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)
This chapter extends the accompanying chapters titled, Construction Challenges (Pt. 1) and [[. The focus here is on an SCF technique that will incorporate specification of a Lagrangian flow-flied.
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Motivation
Where Are We Headed?
In a separate discussion, we have shown that, as viewed from a frame that "tumbles" with the (purple) body of a Type 1 Riemann ellipsoid, each Lagrangian fluid element moves along an elliptical path in a plane that is tipped by an angle <math>~\theta</math> about the x-axis of the body. (See the yellow-dotted orbits in Figure panels 1a and 1b below). As viewed from the (primed) coordinates associated with this tipped plane, by definition, z' = 0 and dz'/dt = 0, and the planar orbit is defined by the expression for an,
| Off-Center Ellipse | ||
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<math>~1</math> |
<math>~=</math> |
<math>~\biggl(\frac{x'}{x_\mathrm{max}} \biggr)^2 + \biggl(\frac{y' - y_0}{y_\mathrm{max}} \biggr)^2 \, .</math> |
As a function of time, the x'-y' coordinates and associated velocity components of each Lagrangian fluid element are given by the expressions,
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<math>~x'</math> |
<math>~=</math> |
<math>~x_\mathrm{max}\cos(\dot\varphi t)</math> |
and, |
<math>~y' - y_0</math> |
<math>~=</math> |
<math>~y_\mathrm{max}\sin(\dot\varphi t) \, ,</math> |
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<math>~\dot{x}'</math> |
<math>~=</math> |
<math>~- x_\mathrm{max}~ \dot\varphi \cdot \sin(\dot\varphi t) = (y_0 - y') \biggl[ \frac{x_\mathrm{max}}{y_\mathrm{max}} \biggr] \dot\varphi </math> |
and, |
<math>~\dot{y}' </math> |
<math>~=</math> |
<math>~y_\mathrm{max}~\dot\varphi \cdot \cos(\dot\varphi t) = x' \biggl[ \frac{y_\mathrm{max}}{x_\mathrm{max}}\biggr] \dot\varphi \, .</math> |
See Also
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© 2014 - 2021 by Joel E. Tohline |