Equilibrium equations - spherical coordinates

[jhongg7]

TL;DR

Hello guys, I would like to know if someone has developed the general equations of equilibrium for a differential element.

 

[Chestermiller]

I don't understand your question. What do you mean by "different element?"

 

[jhongg7]

I'm sorry Chester, it is Differential element.

 

[Chestermiller]

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.

 

[jhongg7]

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.

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.

 

[Chestermiller]

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.

 

[jhongg7]

Ok, thank you Chester!