Blood flow and pressure ( and velocity)

[kay]

so i was reading on some applications of bernoulli's principle and i encountered a paragraph in which it was stated that 'the speed of the flow of blood in this region ( the region inside the artery) is raised which lower the pressure inside the artery and it may collapse due to external pressure. '
Now my doubt is that why does pressure decrease? When the velocity of blood flow increases inside the artery?

 

[bigfooted]

Bernoulli's principle is basically conservation of energy, and says that the total pressure remains constant on a streamline. The total pressure is the static pressure plus the dynamic pressure. Dynamic pressure is 'velocity pressure': P=p_s + \frac{1}{2}\rho V^2=\textrm{const}.
So if the velocity goes up (because for instance the flow-through area of the artery is decreasing), the dynamic pressure goes up and therefore the static pressure goes down.

There is a derivation based on conservation of energy here:
http://en.wikipedia.org/wiki/Bernoulli's_principle#Derivations_of_Bernoulli_equation

 

[kay]

Can you give me the theoretical reason for it?

 

[FactChecker]

Imagine billions of particles with velocity vectors. Each velocity vector is the sum of a velocity component toward the artery wall and a velocity component parallel to the wall. The component that is pushing into the wall contributes to the pressure against the artery wall. The component parallel to the wall contributes to the flow of the fluid. The Pythagorean theorem tells how to divide the vector (hypotenuse) between velocity toward the wall and velocity in the direction of flow. As the fluid flow increases, the velocity vectors are tilting more in the direction of flow and less toward the artery wall. So more flow velocity => less pressure against the artery wall.

 

[Chestermiller]

Imagine billions of particles with velocity vectors. Each velocity vector is the sum of a velocity component toward the artery wall and a velocity component parallel to the wall. The component that is pushing into the wall contributes to the pressure against the artery wall. The component parallel to the wall contributes to the flow of the fluid. The Pythagorean theorem tells how to divide the vector (hypotenuse) between velocity toward the wall and velocity in the direction of flow. As the fluid flow increases, the velocity vectors are tilting more in the direction of flow and less toward the artery wall. So more flow velocity => less pressure against the artery wall.

If you have a literature reference for this explanation, please provide it.

 

[FactChecker]

If you have a literature reference for this explanation, please provide it.

I don't know a reference. I just wanted to give some intuition as to why there was a trade off between pressure and flow velocity squared.
constant energy =~ V_total² = V_{toward wall}² + V_{parallel to wall}² =~ Pressure + 1/2 ρ V_{parallel to wall}²
The references I have seen seemed more rigorous, but less intuitive to me.

 

[Chestermiller]

I don't know a reference. I just wanted to give some intuition as to why there was a trade off between pressure and flow velocity squared.
constant energy =~ V_total² = V_{toward wall}² + V_{parallel to wall}² =~ Pressure + 1/2 ρ V_{parallel to wall}²
The references I have seen seemed more rigorous, but less intuitive to me.

The reason I asked for a reference is that this doesn't seem correct (to me). The Bernoulli equation, which is what Bigfoot was alluding to in post #2 and you were alluding to in this quote, is based on Newton's 2nd law. If the velocity is higher downstream than upstream, the pressure must be higher upstream than downstream in order to accelerate the fluid. (This, of course, neglects viscous drag).

Chet

 

[FactChecker]

If the velocity is higher downstream than upstream, the pressure must be higher upstream than downstream in order to accelerate the fluid.

I agree. I must have said something confusing in my earlier post if I implied the opposite. And your way of looking at it may be the most intuitive of all.
To continue your logic: If the velocity slows down farther downstream, there must be some higher pressure downstream that slowed it down. So the trade-off between flow velocity and pressure is intuitive in all cases.