How can I make a “further interpretation” of the 2nd term of system kinetic energy when it comes to “fluids”? (Possibly deformable body, too)

Linked pdf below is not lecture note of mine, however, my professor allude to the same topic; you may see either of formula in the middle or yellow box of the page 5. Well... writer commented (on page 1) further explanation would go on about fluid later but I can’t find anything relevant.


Thank you!

------edited below------

Thanks for all helping me adapt to this kind world. Since I was accustomed to be on QUORA, where the shorter the better, I forgot to abide by basic format required here.

I wrote the above in almost new fashion so that you do not need to read through it again. If something is still in wrong way, please let me know. :) (I'm not sure it would be fine to requestion like this?)

Here's the situation:

For system of particles, we usually come up with mass center to simplify their behavior.

From $∑_i\frac{1}{2}m_iv_i^2 = \frac{1}{2}MVcm^2 + ∑_i\frac{1}{2}m_iv_i'^2$ , where notation "i" represents each particles and "'" represents 'measured with respect to center of mass', the statement below follows.

Kinetic energy of a system K = $\frac{1}{2}MVcm^2 +K'(measured\ from\ center\ of\ mass)$

Here's the question:

Regardless of the state of the particles, it is evident that the term $K'$ represent how much kinetic energy do they have when they are measured from mass center.

However, many asserts that there are further menifestation in that term indicating specific behaviors.

Then, how can I differentiate the behaviors of particles governed by $K$ (especially with regard to the term $K'$) from the fluid case to the rigid body case.

As one of such further interpretation, I noticed this; as relative distance between each of components of rigid body and the origin(mass center) remains the same, $K'$ would mean how much rotating energy do they have.

But for the fluid case, I cannot even imagine what's going on for the mass center of the fluid(or gas) particles and what's the meaningful way of describing their seemingly random behavior. Only some kind of feelings are drifting like 'deformation potential energy' or statistical approach, etc.

  • 1
    $\begingroup$ Welcome to Physics! When asking about context of a paper, please copy the relevant passage and give specific details what you don't understand (see this Meta post for more). Otherwise we can only guess at what you don't understand. $\endgroup$ – Kyle Kanos Oct 20 at 16:23
  • $\begingroup$ When you say "further interpretation" it implies you already have some sort of preexisting understanding of it. You should post that as well, because otherwise there will be answers that tell you something you already know, and that is not very useful. $\endgroup$ – S V Oct 20 at 16:33
  • $\begingroup$ Welcome to Physics! Stack Exchange posts are version controlled, so please do not make your post look like a revision table. Instead, just seamlessly integrate the new material into the post. There is an edit history button at the bottom of the post for those interested in seeing what changed. $\endgroup$ – Kyle Kanos Oct 22 at 3:08

If I'm not mistaken, you are referring to the second term on the right side of the following equation from page 5 of the paper

K = $\frac{1}{2}M_{CM}v^{2}$ + K(rel. to CM)

Well... writer commented (on page 1) further explanation would go on about fluid later but I can’t find anything relevant.

I believe the further explanation alluded to by the writer is probably related to application of the equation on page 5 to perhaps fluid dynamics or thermodynamics. As I am more familiar with the thermodynamics perspective, the following is from that viewpoint.

To understand the second term you need to understand that the total kinetic energy of a system consists of its external kinetic energy (the first term) and its internal kinetic energy (the second term). In order to help differentiate the two, let's start with the first term.

External kinetic energy:

Consider a system consisting of a fluid (could be a gas, or a liquid) that is confined in some sort of container in a lab. The mass of the fluid is $M$. Let the center of mass of the fluid in the container be moving at constant velocity $v$ with respect to the lab frame. The fluid therefore has a kinetic energy of $\frac{1}{2}M_{CM}v^2$ with respect to the external (to the container) reference frame of the lab. This is the external kinetic energy of the fluid..

Internal Kinetic Energy:

Now let's consider the kinetic energy at the microscopic level, that is, the internal kinetic energy of the molecules in the container. Imagine a microscopic version of yourself located at the center of mass of the fluid. You would observe that the molecules of the fluid are randomly zipping around in the container with various velocities relative to your location at the center of mass of the fluid in the container. Each particle therefore possesses kinetic energy due to its random motion about the center of mass. The "temperature" of the fluid measured by some macroscopic observer outside the container would be a measure of the average kinetic energy of all the molecules relative to the center of mass of the fluid in the container. The average kinetic energy of the fluid molecules is the $K$ in the second term of the equation. This is the internal kinetic energy of the fluid. It is important to note that this internal kinetic energy is independent of the external kinetic energy of the fluid as a whole (the first term in the equation) because it depends only on the velocity of the molecules relative to the center of mass of the container, and not on the velocity of the center of mass of the fluid. Because of this independence, the temperature of the fluid will be the same for the container of gas at rest in the lab, or moving at constant velocity with respect to the lab (at least for non-relativistic velocities)

Hope this helps.

  • $\begingroup$ Sorry for appreciating late, but, fortunately, now I fully understand what you tried to convey to me. As you indicated, the very thing my professor wanted to point out was the notion of (existence of) internal energy. Thinking back, I think I couldn't see the subtle difference of understanding the system 'in different scale from understading the system 'in different reference frame'. Thanks a lot! $\endgroup$ – user245305 Oct 30 at 15:34
  • $\begingroup$ @user245305 You're welcome. Glad it helped. $\endgroup$ – Bob D Oct 30 at 16:35

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