Question of paper for Dean vortex

FluidKo · June 28, 2022, 08:30

Hello

I have a question of paper at below.
https://doi.org/10.1080/14786440708564324

1.Velocity profile for straight pipe in third page.
$U=V=0, W=A(a^{2}-r^{2}), P/\rho =Cz$

I'm wondering how W is derived.
I've guessed it may be derived from continuity equation but I'm not sure.
Where it is derived?

2. Equation 5 in third page
For the slightly curved pipe we assume
$U=u, V=v, W=A(a^{2}-r^{2})+w, P/\rho =Cz+p/\rho$
where u, v, w and p are all small and of order a/R

A. Meaning of u, v, w and p
I don't understand meaning of 'u, v, w and p are small and of order a/R'
I guess meaning of small is just magnitude of u, v, w and p are small.
But I have no idea of 'of order a/R'

B. W is changed
W is changed like $W=A(a^{2}-r^{2})+w$ . But I don't understand why w is added for case of slightly curved pipe.
I think because I don't know w exactly, I can't guess why w is added to W for case of slightly curved pipe.

3. Equation 6 in third page
If terms of order $a^{2}/R^{2}$ are ignored, the equation of continuity is
$\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{1}{r}\frac{\partial v}{\partial \psi }$

I don't understand meaning of 'terms of order $a^{2}/R^{2}$ '. What is $a^{2}/R^{2}$ ? And why it can be neglected? I think the reason why we can ignore is because they are too small. But I can't understand exact meaning of 'terms of order $a^{2}/R^{2}$ '.

4. 'The terms of this equation that are not small must vanish ;' in fourth page.

What I know is we can neglect some terms only if they are very very small like higher order term. But why author says that we can neglect not small terms? I have no idea.

5. 'these are $-C-4\nu A$ ' in fourth page.

Why this term can be vanished? Why this term can be considered as zero? What is the reason?

Please help me.

Thank you for reading

LuckyTran · June 29, 2022, 00:22

1. It's a parabolic velocity profile, which is the analytical solution for fully developed laminar flow in a straight circular pipe. It's the classic Hagen-Poiseuille problem. You get this from integrating the momentum equation twice after applying all the symmetry conditions and fully developed condition.

2A. u,v,w are small velocities up to order a/R. By itself it can seem that this came out of nowhere and was arbitrarily decided. But in the very next sentence, the subsequent analysis attempts to drop terms of order a^2/R^2 (i.e. higher order terms). Hence, u,v,w are small terms up to order a/R and no larger, otherwise you wouldn't be able to drop terms of order a^2/R^2.

2B. W= A(a^2-rj^2) is valid only for a straight pipe. If the flow the pipe is going to flow in a big circular loop then the flow at the outside of the loop (larger R) must be faster than the flow in the inside of the loop (smaller R). w must be non-zero or the solution is automatically a straight pipe flow. w should therefore be a function of R (although this is not fully explained in the note, it is correctly stated that w must be small function up to order a/R, which is equivalent).

3. Eq. (4) is the continuity equation. If u,v,w are of order a/R then the first, second, and fourth terms are of order a/R/a (or 1/R), the remaining are of order 1/R^2. This is just my quick dimensional analysis. A formal perturbation analysis will expand the continuity equation by the perturbation parameter a/R and a^2/R^2 (because 1/R is not a perturbation parameter, but a/R is). Intuitively it still means that R>>a, the radius of the bend is much much larger than the radius of the pipe cross-section.

4&5. There are "not small" terms appearing in equations. Small terms can be neglected, "not small" terms can never be neglected. However, the equations of motions are valid for vanishing small terms. So if you make all the small terms vanish, the remaining "not small" terms must somehow still satisfy the equations of motions. That means, whatever is left, no matter how large must balance each other out such that they sum to the right-hand-side $-C-4\nu A =0$ . The fact that the terms are not small constraints their behavior. They are not arbitrarily large, they can be large only if $C=-4\nu A$ . Otherwise, they must vanish. If neither occurs, then the starting ansatz that u,v,w are small has been violated.

Do this mental exercise. If x+A+B=0 for all x and all non-trivial A and B, the I can put x=0 (or take the limit as x approaches 0), I have that A+B=0. Therefore A=-B or there is no solution for x=0, which violates my starting ansatz that x+A+B=0 has a solution for all x. I'm not saying that A and B are negligible or that A and B vanish by themselves. But A+B (the sum) must vanish. Otherwise, if x=0, then you have a non-zero number must be equal to zero.

FluidKo · June 29, 2022, 05:08

Quote:

Originally Posted by LuckyTran

1. It's a parabolic velocity profile, which is the analytical solution for fully developed laminar flow in a straight circular pipe. It's the classic Hagen-Poiseuille problem. You get this from integrating the momentum equation twice after applying all the symmetry conditions and fully developed condition.

2A. u,v,w are small velocities up to order a/R. By itself it can seem that this came out of nowhere and was arbitrarily decided. But in the very next sentence, the subsequent analysis attempts to drop terms of order a^2/R^2 (i.e. higher order terms). Hence, u,v,w are small terms up to order a/R and no larger, otherwise you wouldn't be able to drop terms of order a^2/R^2.

2B. W= A(a^2-rj^2) is valid only for a straight pipe. If the flow the pipe is going to flow in a big circular loop then the flow at the outside of the loop (larger R) must be faster than the flow in the inside of the loop (smaller R). w must be non-zero or the solution is automatically a straight pipe flow. w should therefore be a function of R (although this is not fully explained in the note, it is correctly stated that w must be small function up to order a/R, which is equivalent).

3. Eq. (4) is the continuity equation. If u,v,w are of order a/R then the first, second, and fourth terms are of order a/R/a (or 1/R), the remaining are of order 1/R^2. This is just my quick dimensional analysis. A formal perturbation analysis will expand the continuity equation by the perturbation parameter a/R and a^2/R^2 (because 1/R is not a perturbation parameter, but a/R is). Intuitively it still means that R>>a, the radius of the bend is much much larger than the radius of the pipe cross-section.

4&5. There are "not small" terms appearing in equations. Small terms can be neglected, "not small" terms can never be neglected. However, the equations of motions are valid for vanishing small terms. So if you make all the small terms vanish, the remaining "not small" terms must somehow still satisfy the equations of motions. That means, whatever is left, no matter how large must balance each other out such that they sum to the right-hand-side $-C-4\nu A =0$ . The fact that the terms are not small constraints their behavior. They are not arbitrarily large, they can be large only if $C=-4\nu A$ . Otherwise, they must vanish. If neither occurs, then the starting ansatz that u,v,w are small has been violated.

Do this mental exercise. If x+A+B=0 for all x and all non-trivial A and B, the I can put x=0 (or take the limit as x approaches 0), I have that A+B=0. Therefore A=-B or there is no solution for x=0, which violates my starting ansatz that x+A+B=0 has a solution for all x. I'm not saying that A and B are negligible or that A and B vanish by themselves. But A+B (the sum) must vanish. Otherwise, if x=0, then you have a non-zero number must be equal to zero.

Hello LuckyTran!

Sorry I can't understand answer of qeustion 4 and 5.
Let's get something straight first.
Does 'not small terms must vanish' means
'not small terms eliminate small terms'(Because not small terms are much bigger that small terms )
or 'not small terms can be eliminated by themselves'?
I think I'm confused to understand English sentence.

LuckyTran · June 29, 2022, 05:12

In x + A + B =0, x is small and A and B are not necessarily small. A and B do not necessarily vanish because they are not small. But A+B must vanish or x + A + B = 0 is the wrong equation. Because put x=0 (or take the lim as x approaches 0 if you want to do it more formally) into x + A + B = 0, you have 0 + A + B = 0 which means A + B = 0. You are allowed to put x=0 because x is small. You cannot take the limit as A or B approaches 0 because they're not necessarily small. This is the meaning of not small terms must vanish. If you cannot understand the meaning of vanish just know that, after all negligible terms are removed, the remaining not small terms must still satisfy the equation which is 0.

Not small terms can eliminate small terms, but that doesn't help you advance in any way. Not small terms eliminating small terms means that A + B = -x also satisfies the equation. But so what? That's the same as what you had when you started.

FluidKo · June 29, 2022, 07:13

Quote:

Originally Posted by LuckyTran

In x + A + B =0, x is small and A and B are not necessarily small. A and B do not necessarily vanish because they are not small. But A+B must vanish or x + A + B = 0 is the wrong equation. Because put x=0 (or take the lim as x approaches 0 if you want to do it more formally) into x + A + B = 0, you have 0 + A + B = 0 which means A + B = 0. You are allowed to put x=0 because x is small. You cannot take the limit as A or B approaches 0 because they're not necessarily small. This is the meaning of not small terms must vanish. If you cannot understand the meaning of vanish just know that, after all negligible terms are removed, the remaining not small terms must still satisfy the equation which is 0.

Not small terms can eliminate small terms, but that doesn't help you advance in any way. Not small terms eliminating small terms means that A + B = -x also satisfies the equation. But so what? That's the same as what you had when you started.

Yes you are right.
There is no diffenrence that I understand first time.
I'm just confused the meaning of sentence because I'm not good at English you know.

I've misunderstood that author says not small terms can be eliminated by themselves.
So on the spur of moment, I'm confused that why authors says like that even though he has said that only small terms are negligible.
But it was just my misunderstood because of my low English level.
Now it is solved.
Thank you for your answer.

June 28, 2022, 08:30	Question of paper for Dean vortex	#1
FluidKo Senior Member - Join Date: Oct 2021 Location: - Posts: 139 Rep Power: 5	Hello I have a question of paper at below. https://doi.org/10.1080/14786440708564324 1.Velocity profile for straight pipe in third page. $U=V=0, W=A(a^{2}-r^{2}), P/\rho =Cz$ I'm wondering how W is derived. I've guessed it may be derived from continuity equation but I'm not sure. Where it is derived? 2. Equation 5 in third page For the slightly curved pipe we assume $U=u, V=v, W=A(a^{2}-r^{2})+w, P/\rho =Cz+p/\rho$ where u, v, w and p are all small and of order a/R A. Meaning of u, v, w and p I don't understand meaning of 'u, v, w and p are small and of order a/R' I guess meaning of small is just magnitude of u, v, w and p are small. But I have no idea of 'of order a/R' B. W is changed W is changed like $W=A(a^{2}-r^{2})+w$ . But I don't understand why w is added for case of slightly curved pipe. I think because I don't know w exactly, I can't guess why w is added to W for case of slightly curved pipe. 3. Equation 6 in third page If terms of order $a^{2}/R^{2}$ are ignored, the equation of continuity is $\frac{\partial u}{\partial r}+\frac{u}{r}+\frac{1}{r}\frac{\partial v}{\partial \psi }$ I don't understand meaning of 'terms of order $a^{2}/R^{2}$ '. What is $a^{2}/R^{2}$ ? And why it can be neglected? I think the reason why we can ignore is because they are too small. But I can't understand exact meaning of 'terms of order $a^{2}/R^{2}$ '. 4. 'The terms of this equation that are not small must vanish ;' in fourth page. What I know is we can neglect some terms only if they are very very small like higher order term. But why author says that we can neglect not small terms? I have no idea. 5. 'these are $-C-4\nu A$ ' in fourth page. Why this term can be vanished? Why this term can be considered as zero? What is the reason? Please help me. Thank you for reading

June 29, 2022, 00:22		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,754 Rep Power: 66	1. It's a parabolic velocity profile, which is the analytical solution for fully developed laminar flow in a straight circular pipe. It's the classic Hagen-Poiseuille problem. You get this from integrating the momentum equation twice after applying all the symmetry conditions and fully developed condition. 2A. u,v,w are small velocities up to order a/R. By itself it can seem that this came out of nowhere and was arbitrarily decided. But in the very next sentence, the subsequent analysis attempts to drop terms of order a^2/R^2 (i.e. higher order terms). Hence, u,v,w are small terms up to order a/R and no larger, otherwise you wouldn't be able to drop terms of order a^2/R^2. 2B. W= A(a^2-rj^2) is valid only for a straight pipe. If the flow the pipe is going to flow in a big circular loop then the flow at the outside of the loop (larger R) must be faster than the flow in the inside of the loop (smaller R). w must be non-zero or the solution is automatically a straight pipe flow. w should therefore be a function of R (although this is not fully explained in the note, it is correctly stated that w must be small function up to order a/R, which is equivalent). 3. Eq. (4) is the continuity equation. If u,v,w are of order a/R then the first, second, and fourth terms are of order a/R/a (or 1/R), the remaining are of order 1/R^2. This is just my quick dimensional analysis. A formal perturbation analysis will expand the continuity equation by the perturbation parameter a/R and a^2/R^2 (because 1/R is not a perturbation parameter, but a/R is). Intuitively it still means that R>>a, the radius of the bend is much much larger than the radius of the pipe cross-section. 4&5. There are "not small" terms appearing in equations. Small terms can be neglected, "not small" terms can never be neglected. However, the equations of motions are valid for vanishing small terms. So if you make all the small terms vanish, the remaining "not small" terms must somehow still satisfy the equations of motions. That means, whatever is left, no matter how large must balance each other out such that they sum to the right-hand-side $-C-4\nu A =0$ . The fact that the terms are not small constraints their behavior. They are not arbitrarily large, they can be large only if $C=-4\nu A$ . Otherwise, they must vanish. If neither occurs, then the starting ansatz that u,v,w are small has been violated. Do this mental exercise. If x+A+B=0 for all x and all non-trivial A and B, the I can put x=0 (or take the limit as x approaches 0), I have that A+B=0. Therefore A=-B or there is no solution for x=0, which violates my starting ansatz that x+A+B=0 has a solution for all x. I'm not saying that A and B are negligible or that A and B vanish by themselves. But A+B (the sum) must vanish. Otherwise, if x=0, then you have a non-zero number must be equal to zero. thedal and FluidKo like this.

June 29, 2022, 05:12		#4
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,754 Rep Power: 66	In x + A + B =0, x is small and A and B are not necessarily small. A and B do not necessarily vanish because they are not small. But A+B must vanish or x + A + B = 0 is the wrong equation. Because put x=0 (or take the lim as x approaches 0 if you want to do it more formally) into x + A + B = 0, you have 0 + A + B = 0 which means A + B = 0. You are allowed to put x=0 because x is small. You cannot take the limit as A or B approaches 0 because they're not necessarily small. This is the meaning of not small terms must vanish. If you cannot understand the meaning of vanish just know that, after all negligible terms are removed, the remaining not small terms must still satisfy the equation which is 0. Not small terms can eliminate small terms, but that doesn't help you advance in any way. Not small terms eliminating small terms means that A + B = -x also satisfies the equation. But so what? That's the same as what you had when you started. FluidKo likes this.

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