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

In summary: 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.
  • #1
tajmann
2
0
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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  • #2
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.
 
  • #3
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
 
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  • #4
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?
 
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  • #5
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.
 
  • #6
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?
 
  • #7
Chestermiller said:
Is it sufficient now?

A proof that it's 1-(r/R)2 would suffice.
 
  • #8
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.
 

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

1. What is shear stress in a vessel?

Shear stress in a vessel refers to the force exerted on the walls of a vessel due to the movement of the fluid inside. This force is parallel to the surface of the vessel and can cause deformation or damage to the vessel if it exceeds a certain threshold.

2. How is shear stress calculated?

Shear stress is calculated by dividing the force applied to the vessel walls by the cross-sectional area of the vessel. It is typically measured in units of pascals (Pa) or newtons per square meter (N/m²).

3. What factors affect shear stress in a vessel?

The magnitude of shear stress in a vessel is influenced by various factors such as the velocity of the fluid, the viscosity of the fluid, the shape and size of the vessel, and the roughness of the vessel walls.

4. Why is shear stress important in vessel design?

Shear stress is an important consideration in vessel design because it can cause structural damage if it exceeds the vessel's strength or the material's yield point. It is also important to consider in order to ensure the safe operation of the vessel.

5. How can shear stress be reduced in a vessel?

Shear stress can be reduced in a vessel by decreasing the velocity of the fluid, using a less viscous fluid, or by altering the shape or roughness of the vessel walls. Additionally, the use of reinforcing materials or thickening the walls of the vessel can also help to reduce shear stress.

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