Curvature effect can be neglected

[werson tan]

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 ??

 

[haruspex]

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?

 

[werson tan]

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 ) ?

 

[SammyS]

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?

 

[Chestermiller]

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.

 

[werson tan]

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?

 

[werson tan]

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?

 

[Chestermiller]

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.

 

[Chestermiller]

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.

 

[werson tan]

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

 

[Chestermiller]

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.