- #1
jhongg7
- 4
- 1
- TL;DR
- Hello guys, I would like to know if someone has developed the general equations of equilibrium for a differential element.
Equilibrium equations - spherical coordinates
- Thread starter jhongg7
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Discussion Overview
The discussion revolves around the derivation of equilibrium equations in spherical coordinates, specifically focusing on how to arrive at certain equations that are typically presented without detailed development in various texts. The scope includes theoretical aspects of vector force balance and the use of unit vectors in spherical coordinates.
Discussion Character
- Technical explanation
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant seeks clarification on the term "different element," which is later corrected to "differential element."
- Another participant suggests that the original question may pertain to the development or presentation of equations, referencing a specific textbook for further reading.
- A participant expresses frustration that while many texts present the equations, they do not explain how to derive them, particularly in spherical coordinates.
- One participant proposes a method for deriving the equations by expressing vector force balance in terms of unit vectors and their spatial derivatives, suggesting that these derivatives can be expressed using trigonometric functions of latitude and longitude.
Areas of Agreement / Disagreement
Participants do not reach a consensus on the best approach to derive the equations, and multiple viewpoints on the clarity of existing resources are evident.
Contextual Notes
Limitations include the lack of detailed derivation steps in existing literature and the potential dependence on definitions of unit vectors and spatial derivatives in spherical coordinates.
Who May Find This Useful
This discussion may be useful for students or professionals interested in the mathematical foundations of equilibrium equations in spherical coordinates, particularly those seeking deeper insights into derivation methods.
- #2
- 23,735
- 5,938
I don't understand your question. What do you mean by "different element?"
- #3
jhongg7
- 4
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I'm sorry Chester, it is Differential element.
- #4
- 23,735
- 5,938
So isn't. that what your diagram says? Or are you asking for a development or a presentation of the equations? If the latter, see Transport Phenomena by Bird, Stewart, and Lightfoot.
- #5
jhongg7
- 4
- 1
Chestermiller said:
Hi Chester, I read the book, they have the equations but they don't develop the results. What I want to know is how to get to the last three equations. I have read many books, yet they present it, they don't say how to do it. The explain cylindrical but spherical they just present it.
- #6
- 23,735
- 5,938
I would do it by expressing the vector force balance in terms of the unit vectors in the radial, latitudinal, and longitudinal directions. The force balances are going to involve spatial derivatives of these unit vectors. Each derivative of each of the unit vectors can be expressed, in turn, in terms of the three unit vectors themselves (and trig functions of the latitude and longitude angles). I would derive these derivative relationships (or look them up in BSL). I would then substitute them into the appropriate places in the vector force balance. Then, the rest is easy, since all that is then needed is to dot the vector force balance, in turn, with each of the three unit vectors.
- #7
jhongg7
- 4
- 1
Ok, thank you Chester!
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