divergence problems

[PG1995]

Hi

Could you please help me the problems below? Thank you.

Problem #1: I believe this problem is similar to the Problem #2 below. It seems comparatively simpler and more explanatory.

Problem #2: As I mentioned above, this problem is related to the problem #1 above but is more complex.


Reference(s):
1: https://www.electro-tech-online.com/attachments/vec_chap4_ref1-jpg.70407/


Regards
PG

 

[steveB]

Problem 1

Q1. The exercise asks to determine the fluid loss per unit volume from the box defined by Δx, Δy and Δz, not the fluid loss from the pond.

Q2. In green, they are calculating the fluid flow velocity at the center and on the faces. In yellow, they are determining the volumetric fluid flowing through each of the six faces of the box.

Q3. Division by ΔxΔyΔz provides the quantity per unit volume, because ΔxΔyΔz is the volume of the box.

 

[steveB]

Problem 2:

Q1: This means exactly what it says. They are restricting the analysis to a case where those functions do not change sign. Presumably, if there happened to be a sign change in a particular example, you could make the region smaller and smaller, or move the region infinitesimally with an offset, until this assumption is true. I expect you can still do the analysis if there is a sign change, but perhaps this is conceptually more confusing to a typical reader, and the sign change is not particularly crucial to the understanding.

Q2: They explain this. They are calculating the component of velocity in the direction of the outward normal. The dot product is a good tool for calculating components in a direction. Just dot the vector with the component direction (unit vector) and you get the answer.

Q3: You could define an inward normal, which is just the negative direction of the outward normal. However, the outward normal is typically used in vector analysis.

Q4: Look at the expression. Doesnt' the equation given look like the approximate derivative we used in basic calculus? Then, we take the limit at Δx goes to zero to get the exact derivative (no longer an approximation). The same thing is happening here. The rectangle is small and eventually we take the limit as the size of the rectangle goes to zero (to get the equality rather than the approximate form).

 

[MrAl]

Hi,


Just to add a little here...

For the first Q1, the point P is a single point inside the box, and what we want to know is how the fluid is behaving at THIS point. In fact, from this microscopic view we can't even see the whole pond, just what is passing that point alone, and we also can not influence the flow around us and that also means we can't impede the flow either. So we are only concerned with what is happening at this point alone. We are attempting to measure the dynamics of the fluid without changing anything about the way it behaves.

This gives rise to questions about why we might be concerned about a single point, what good would it do. What it says is how much the fluid is expanding away from that point or contracting toward that point. This sounds funny too, but if we had a hole in the bottom of the 'tank' and we only had a top view, we'd see fluid leaving that point. Also, if the fluid was compressible, we might see more material per unit volume enter or leave that node. IF the fluid was incompressible we would never see any change (div v=0).

In terms of writing that paper i have to say that i find another unusual wording which i'd like to revise...
I'd change that first paragraph that currently reads this:

"Show that the loss of fluid per unit volume per unit time in a small parallelpiped having center at P and edges parallel to the coordinate axes and having magnitude dx, dy, dz respectively, is given approximately by div v=D*v."

to this:

"Show that the approximate loss of fluid per unit volume per unit time in a small parallelpiped having center at P and edges parallel to the coordinate axes and having magnitude dx, dy, dz respectively, is given more exactly by div v=D*v."

or maybe more like this:
"Show that the loss of fluid per unit volume per unit time calculated approximately using a small parallelpiped having center at P and edges parallel to the coordinate axes and having sides dx, dy, dz respectively, is calculated exactly by div v=D*v"

Where 'D' here is the inverted triangle used in various texts to show the div and other.

It occurred to me that this paper may have been written by someone who used English as a second language, or perhaps was translated from another language.

 

[PG1995]

Thanks a lot, Steve, MrAl.

This is about Problem #2 from post #1 above. It says that loss in volume per unit time in x direction is "(2) - (1)". I was thinking that loss in volume per unit time should be rather (1)+(2). Why should the (1) be subtracted from the (2)? Please help me with it. Thanks.

Regards
PG

 

[steveB]

It says that loss in volume per unit time in x direction is "(2) - (1)". I was thinking that loss in volume per unit time should be rather (1)+(2). Why should the (1) be subtracted from the (2)?

The best thing to do here is to carefully think about the meaning of each expression, as defined.

(2) is the volume of fluid flowing through the face that is on the positive side of the x-axis. Hence, positive velocity implies that fluid is going in the positive x-direction, which then means that when the number is positive, fluid is being lost from the box.

(1) is the volume of fluid flowing through the face that is on the negative side of the x-axis. Hence, positive velocity implies that fluid is (again) going in the positive x-direction, which then means that when the number is positive, fluid is being added to the box.

Hence, if you want to know the total loss in volume from both faces, you must use (2)-(1) because the -(1) creates a positive loss when (1) is negative.

Think about it another way. Let's say velocity is constant in a stream of fluid that is incompressible. Any box will have 0 loss (and 0 gain) in volume per unit time. This is because the front face and the back face have equal velocity and the same direction of flow. Hence, whatever flows in one face, must flow out the other face. If you used (2)+(1), you would double the number and think that a large amount of volume is being lost per unit time, when clearly this is not happening.

 

[PG1995]

Thanks a lot, Steve.

I'm sorry for asking you again. My last query resulted from a confusion which I think needs to cleared. So, kindly help me. Here is the confusion. Thank you.

Regards
PG

 

[steveB]

Those two formulas are providing first order approximations of the velocity on the front face and on the back face. The value of v1 is the velocity at the center. The distance from the center to each face is Δx/2. The partial derivative of velocity with respect to x times the distance of travel Δx/2 is the change in velocity.

Lets say the derivative or slope in velocity is positive at the center of the box. Then, we expect the back face to have a smaller velocity and the front face to have a larger velocity. Hence we subtract the change to find the back-face velocity and we add the change to find the front face velocity.

 

[PG1995]

Thank you.

What I don't get here is that what it means by "components M and N do not change sign" (that is Q1). Could you please help me with it? Thanks.

Regards
PG

 

[steveB]

Thank you.

What I don't get here is that what it means by "components M and N do not change sign" (that is Q1). Could you please help me with it? Thanks.

Regards
PG

My guess is that you know what it means, but you are confused by why it is important and why they mention it.

Remember that these types of analyses are considering very small distances, areas and volumes. In this case, the assumption is automatically held true if you make the region small enough. We want the side lengths to be so small that the velocity changes look linear. If a linear change does create a sign change, then the side lengths can be made smaller and smaller until the zero crossing is not in the region.

Unless I'm missing a subtle point, this specification causes more confusion and is unnessessary. I suggest you ignore it if it causes confusion

 

[PG1995]

My guess is that you know what it means...

But seriously I don't get what it really means! If a molecule has M component 5i and N component 3j, then what does it mean to say the M and N components change (or, do not change) sign in the region?

Besides, don't you think the figure given is somewhat misleading? It shows the fluid (assume some gas because it helps me to think of divergence in terms of gas' expansion and contraction) flowing from left to right along y-axis without any change in velocity; both magnitude and direction of velocity are shown to be constant (look at those arrows in the figure). This would mean divergence of zero and even if there is any divergence then it would be only along y-axis. But I think the figure is only there to give you general sense of the question so it doesn't need to convey information fully and accurately.

Thank you.

Regards
PG

 

[steveB]

Ok, you are correct that a tiny piece of the fluid has one velocity, but what about the tiny piece of fluid next to that one? Remember, before we take the limit as area goes to zero, we have a tiny, but finite, area. However, once we take the limit, we are dealing with one point in the fluid. So, there is a velocity distribution over a larger area, then as the area is made small, the variations look linear, then if the area is made arbitrarily small, zero crossings can be eliminated.

Your comments about the drawing seem reasonable to me. It looks like the flow of an incompressible fluid, for which the divergence is zero, unless the is a source or a sink at that point.

 

[PG1995]

But seriously I don't get what it really means! If a molecule has M component 5i and N component 3j, then what does it mean to say the M and N components change (or, do not change) sign in the region?

Ok, you are correct that a tiny piece of the fluid has one velocity, but what about the tiny piece of fluid next to that one? Remember, before we take the limit as area goes to zero, we have a tiny, but finite, area. However, once we take the limit, we are dealing with one point in the fluid. So, there is a velocity distribution over a larger area, then as the area is made small, the variations look linear, then if the area is made arbitrarily small, zero crossings can be eliminated.

Thank you. So, speaking in terms of motion of molecules, then by saying that M and N components do not change sign, it really means that we are restricting the analysis to a region in which all the molecules are moving with the same pattern - i.e. if one molecule of that region has M component 5i then all others also move with this component. But if the M and N components do change sign then it might mean that one molecule would be having 5i component and some other in the same region might have M component of -5i. Do I have it correct? Kindly let me know. Thanks.

Best wishes
PG

 

[steveB]

Thank you. So, speaking in terms of motion of molecules, then by saying that M and N components do not change sign, it really means that we are restricting the analysis to a region in which all the molecules are moving with the same pattern - i.e. if one molecule of that region has M component 5i then all others also move with this component. But if the M and N components do change sign then it might mean that one molecule would be having 5i component and some other in the same region might have M component of -5i. Do I have it correct? Kindly let me know. Thanks.

Best wishes
PG

This doesn't sound right at all. As I mentioned, this specification of the sign not changing should be ignored if it causes confusion. However, whether due to this comment, or due to something else, it's clear that you are missing the concept here.

First, let's clarify the issue with "molecules". Fluid theory typically treats the fluid as a continuum, so it is best to not even think about molecules or atoms or any type of discrete structure. We are dealing with calculus concepts that require taking limits as volume elements go to zero size.

Also, the points in a fluid within a finite region, don't necessarily have to move with the same "pattern". The whole point is that there is a velocity distribution and there is likely to be a different velocity at various points in the volume of the fluid. The velocity change is specified to vary continuously, but, over a small distance, this change in velocity will vary almost linearly. The idea that the velocity might change from +5 to -5 is not the point of restricting the sign to not change. Rather, the idea is that the variation in velocity will not have a zero crossing over the region of interest. Again, I don't think this is a critical restriction because it will automatically be held if the volume element is made small enough.

I'm not sure what else to say to clarify, so I would ask you to carefully reread my previous comments and think about it for a while. I'm probably not mentioning a critical idea that might help you to visualize it, so feel free to ask more probing questions if it still gives you trouble after careful review. To help clarify my imperfect explanation, I've attached the first page of "Fluid Mechanics" by Landau and Lifshitz. I recommend that you use a book like this to help get the concepts down correctly. This is not an easy subject, so the carefully chosen words from the great physicists can be a great help to the student.