Difference between revisions of "User:Tohline/ThreeDimensionalConfigurations/ChallengesPt3"
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=Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)= | =Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)= | ||
This chapter extends the accompanying chapters titled, [[User:Tohline/ThreeDimensionalConfigurations/Challenges|''Construction Challenges (Pt. 1)'']] and [[. The focus here is on | This chapter extends the accompanying chapters titled, [[User:Tohline/ThreeDimensionalConfigurations/Challenges|''Construction Challenges (Pt. 1)'']] and [[User:Tohline/ThreeDimensionalConfigurations/ChallengesPt2|''(Pt. 2)'']]. The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames. | ||
{{LSU_HBook_header}} | {{LSU_HBook_header}} | ||
==Various Coordinate Frames== | |||
==Motivation== | ==Motivation== | ||
Revision as of 20:14, 25 April 2021
Challenges Constructing Ellipsoidal-Like Configurations (Pt. 2)
This chapter extends the accompanying chapters titled, Construction Challenges (Pt. 1) and (Pt. 2). The focus here is on firming up our understanding of the relationships between various "tilted" Cartesian coordinate frames.
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Various Coordinate Frames
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 |