Curvature effect can be neglected

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In summary, when the ratio of the length (ℓ) to the radius (R) is very small (ℓ/R ≪ 1), the effect of curvature can be neglected. This is because as the ratio approaches zero, the system behaves as if it is between two parallel plates, and the effect of curvature on the fluid velocity profile becomes negligible. This simplifying approximation can save time and effort in solving fluid mechanics problems.
  • #1
werson tan
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Homework Statement



why when ℓ / R ≪ 1 , the curvature effect can be neglected ??

Homework Equations

The Attempt at a Solution


no matter how small or how big is R , it is strill considered as curvature , right ??
 

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  • #2
It would take a bit of analysis to demonstrate clearly that the curvature can be neglected, but it is not unusual that this should be the case. When solving a question for the range of an arrow, do you take the curvature of the Earth into account?
 
  • #3
haruspex said:
It would take a bit of analysis to demonstrate clearly that the curvature can be neglected, but it is not unusual that this should be the case. When solving a question for the range of an arrow, do you take the curvature of the Earth into account?
can you explain in detail by giving analogy (is arrow represent ℓ ) ?
 
  • #4
werson tan said:
can you explain in detail by giving analogy (is arrow represent ℓ ) ?
When you solve a problem involving projectile motion, do you take the curvature of the Earth into account?
 
  • #5
If you solve the fluid mechanics problem you are talking about exactly for the velocity distribution, and then take the limit of the solution as l/R approaches zero, you will find that the solution approaches that for shear flow between to infinite parallel plates. So, as l/r approaches, the curvature of the system can be neglected.
 
  • #6
Chestermiller said:
If you solve the fluid mechanics problem you are talking about exactly for the velocity distribution, and then take the limit of the solution as l/R approaches zero, you will find that the solution approaches that for shear flow between to infinite parallel plates. So, as l/r approaches, the curvature of the system can be neglected.
what do you mean ? can you explain further?
 
  • #7
Chestermiller said:
If you solve the fluid mechanics problem you are talking about exactly for the velocity distribution, and then take the limit of the solution as l/R approaches zero, you will find that the solution approaches that for shear flow between to infinite parallel plates. So, as l/r approaches, the curvature of the system can be neglected.
when R is big relative to l , we can only say that the two circle 'combined together' , right ? how can the curvature be neglected?
 
  • #8
werson tan said:
what do you mean ? can you explain further?
What they are saying is that if l/R is very small, the velocity profile for that flow approaches that for shear flow between two parallel plates. Basically, the system behaves as if the curvature is negligible. It isn't R and it isn't l individually that determines the effect of curvature in this system. It is the dimensionless group l/R. Imagine that the inner diameter is 1 mile, and the outer diameter is 1 mile plus 0.1 inch. On the local scale of the gap between the cylinders, you couldn't tell whether the system is curved, or whether you are dealing with two parallel plates.
 
  • #9
werson tan said:
when R is big relative to l , we can only say that the two circle 'combined together' , right ? how can the curvature be neglected?
It isn't that the curvature is not there. It is that the effect of the curvature on the fluid velocity profile in the gap between the cylinders is negligible.
 
  • #10
Chestermiller said:
What they are saying is that if l/R is very small, the velocity profile for that flow approaches that for shear flow between two parallel plates. Basically, the system behaves as if the curvature is negligible. It isn't R and it isn't l individually that determines the effect of curvature in this system. It is the dimensionless group l/R. Imagine that the inner diameter is 1 mile, and the outer diameter is 1 mile plus 0.1 inch. On the local scale of the gap between the cylinders, you couldn't tell whether the system is curved, or whether you are dealing with two parallel plates.

why ? i still can't understand
 
  • #11
werson tan said:
why ? i still can't understand
I don't know how to explain it any better. As I said, if you solved for the velocity profile between the two cylinders exactly and took the limit as the gap between the cylinders became very small (compared to the radii). you would approach the same velocity profile as that for flow between infinite parallel plates. In fluid mechanics, you need to be able to recognize the kinds of approximations you can make for specific situations. Otherwise, you will be wasting your valuable time spending hours to solve a problem that you could have done in minutes. I guess you are asking why it is that you are not able to recognize the simplifying approximation that can be made in this problem. I don't know how to answer that because it seems so obvious to me.
 

1. What is the curvature effect and why is it important?

The curvature effect refers to the change in a physical property or behavior at the curved surface of an object. It is important because it can affect the accuracy of calculations and predictions in various fields of science and engineering.

2. How is the curvature effect typically neglected in scientific research?

The curvature effect can be neglected by assuming that the curved surface can be approximated by a flat surface. This simplification is often made in mathematical models and simulations to make calculations more manageable.

3. Are there any cases where the curvature effect cannot be neglected?

Yes, there are cases where the curvature effect cannot be ignored. For example, in certain fluid dynamics problems, the curvature of a surface can significantly alter the flow behavior and must be taken into account.

4. How can the curvature effect be measured or accounted for in experiments?

Measuring and accounting for the curvature effect in experiments can be challenging. One approach is to use specialized equipment, such as a profilometer, to directly measure the curvature of a surface. Another method is to use mathematical corrections or calibrations based on known curvature effects.

5. What are some potential consequences of neglecting the curvature effect in scientific research?

If the curvature effect is neglected in scientific research, it can lead to inaccurate results and predictions. This can have significant consequences, especially in fields like engineering where small errors can have major impacts on the performance and safety of structures and systems.

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