Why is shear stress highest at the wall of a vessel?

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Discussion Overview

The discussion revolves around the concept of shear stress in a fluid flowing within a vessel, specifically addressing why shear stress is highest at the wall and lowest at the center. Participants explore the relationship between fluid velocity and shear stress, examining the implications of different velocity profiles in the context of fluid mechanics.

Discussion Character

  • Conceptual clarification
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant seeks to understand shear stress in a vessel, noting that it relates to the force needed to overcome viscosity for blood flow, and questions why shear stress differs between the center and the wall.
  • Another participant states that shear stress is defined as viscosity times the derivative of velocity with respect to radial position, indicating that the derivative is zero at the center where velocity is maximum.
  • A participant challenges the explanation, questioning why the velocity profile could not be represented as 1-r/R and suggests that it may not be differentiable at the center.
  • Further contributions clarify that the solution to fluid mechanics equations follows a profile of (1 - (r/R)²) rather than 1 - r/R, prompting discussions on the sufficiency of the arguments presented.
  • Some participants express dissatisfaction with the explanations given, indicating a need for a proof of the velocity profile to fully address the original question.

Areas of Agreement / Disagreement

Participants express differing views on the sufficiency of the explanations provided regarding shear stress and velocity profiles. There is no consensus on the resolution of the original question, and multiple competing interpretations of the velocity profile exist.

Contextual Notes

Participants reference the need for a detailed derivation of equations related to laminar flow, suggesting that existing literature may contain the necessary proofs and explanations, but do not provide them in the discussion.

tajmann
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Hey guys,

I am trying to conceptualize as to why shear stress in a vessel is highest at the wall of the vessel and why it is at a minimum at the center.

First let me see if I actually understand shear stress - In a vessel with blood flow, it is the force required to overcome the viscosity that causes blood to flow at different rates.

Now, velocity of the blood (fluid) is highest at the center and lowest (~0) at the wall. I just don't understand why there is a difference b/w the center and the periphery. The way I understood it, there was a more or less gradual decrease in the velocity of blood as it reached the wall. As such, the shear stress would be more or less be equal at the wall and the center. I know I'm not understanding something. Thanks for the help. Much appreciated.
 
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The shear stress is equal to the viscosity times the derivative of the velocity with respect to radial position. The derivative of the velocity is equal to zero at the center of the tube, and highest at the surface. We know that the derivative of velocity is zero at the center of the tube, because this is the location at which the velocity is maximum.
 
Chestermiller said:
The shear stress is equal to the viscosity times the derivative of the velocity with respect to radial position. The derivative of the velocity is equal to zero at the center of the tube, and highest at the surface. We know that the derivative of velocity is zero at the center of the tube, because this is the location at which the velocity is maximum.
I don't think that really answers the question. Why could the velocity profile not look like 1-r/R? Msybe it's not differentiable at the centre.
Tajmann, try reading http://www.cs.cdu.edu.au/homepages/jmitroy/eng243/sect09.pdf
 
Last edited by a moderator:
haruspex said:
I don't think that really answers the question. Why could the velocity profile not look like 1-r/R? Msybe it's not differentiable at the centre.
Tajmann, try reading http://www.cs.cdu.edu.au/homepages/jmitroy/eng243/sect09.pdf

The solution to the fluid mechanics equations goes as (1- (r/R)2), not 1 - r/R. Are you asking how they get the solution to the fluid mechanics equations?
 
Last edited by a moderator:
Chestermiller said:
The solution to the fluid mechanics equations goes as (1- (r/R)2), not 1 - r/R.
Yes, I know. I was just pointing out that the argument you gave was not sufficient to resolve the question posed.
 
haruspex said:
Yes, I know. I was just pointing out that the argument you gave was not sufficient to resolve the question posed.

Is it sufficient now?
 
Chestermiller said:
Is it sufficient now?

A proof that it's 1-(r/R)2 would suffice.
 
haruspex said:
A proof that it's 1-(r/R)2 would suffice.

I don't feel like providing that here. The detailed derivation of the equations for laminar flow in a tube can be found in any book on transport phenomena. I suggest Transport Phenomena by Bird, Stewart, and Lightfoot.
 

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