Blood flow, Bernoulli's equation and Poiseuille's equation

[gkangelexa]

Blood flow...
Relating the Continuity of flow equation (A₁v₁ = A₂v₂) with Bernoulli's equation, with Poiseuille's equation.

Continuity of flow equation tells us this: when the area decreases, the velocity increases in order to maintain a constant flow rate.
Bernoulli's equation tells us that when velocity increases, the pressure (that the fluid exerts on its walls) decreases.

Poiseuille's equation says that the flow rate Q is directly proportional to the pressure gradient (P1 - P2).

So, knowing all this, where am I thinking wrong in the following situation involving blood? (I'm assuming blood has laminar flow like my physics book does).

When you have cholesterol buildup and arterosclerosis, then the arteries decrease in area since the radius is smaller. From the continuity of flow equation, the velocity of the blood must increase to maintain the same flow rate Q. This increase in velocity results in a lower pressure at that area.

However, based on Poiseuille's equation, the arterosclerosis would cause a decrease in R in the equation, and consequently cause an increase in the pressure gradient in order to maintain the same flow rate. This means that the heart should increase the pressure (high blood pressure as is observed)
How can this be though? If P₂ is decreased (as was established in the previous paragraph), then P₁ should decrease, not increase. or It shouldn't have to increase since P₂ decreased, and this already created a greater pressure gradient.

 

[Studiot]

https://www.physicsforums.com/showthread.php?t=506575&highlight=blood

 

[Andy Resnick]

Blood flow...
Relating the Continuity of flow equation (A₁v₁ = A₂v₂) with Bernoulli's equation, with Poiseuille's equation.

Continuity of flow equation tells us this: when the area decreases, the velocity increases in order to maintain a constant flow rate.
Bernoulli's equation tells us that when velocity increases, the pressure (that the fluid exerts on its walls) decreases.

Poiseuille's equation says that the flow rate Q is directly proportional to the pressure gradient (P1 - P2).

So, knowing all this, where am I thinking wrong in the following situation involving blood? (I'm assuming blood has laminar flow like my physics book does).

When you have cholesterol buildup and arterosclerosis, then the arteries decrease in area since the radius is smaller. From the continuity of flow equation, the velocity of the blood must increase to maintain the same flow rate Q. This increase in velocity results in a lower pressure at that area.

However, based on Poiseuille's equation, the arterosclerosis would cause a decrease in R in the equation, and consequently cause an increase in the pressure gradient in order to maintain the same flow rate. This means that the heart should increase the pressure (high blood pressure as is observed)
How can this be though? If P₂ is decreased (as was established in the previous paragraph), then P₁ should decrease, not increase. or It shouldn't have to increase since P₂ decreased, and this already created a greater pressure gradient.

This is a good example of 'biology without biology'. In terms of fluid dynamics, the thread Studiot linked to is a reasonable discussion about the relevant mechanics, including the difference between pressure and pressure drop.

In terms of cardiovascular disease, however, the question is completely irrelevant. Plaques do not form a uniform constriction of an artery, for example. Blood flow is not steady, but highly variable in time. Blood is a nonNewtonian fluid. Blood pressure (systole and diastole) are systemic physiological features and are not driven by a single plaque- in fact, the main loss of driving pressure occurs in the arterioles. I could go on...

To be sure, there is good work being done:
http://www.cism.it/courses/c0204/
http://www.ncbi.nlm.nih.gov/pubmed/8302047
http://www.sciencedirect.com/science/article/pii/0021929096845441
http://www.google.com/url?sa=t&sour...sg=AFQjCNEMQOAbrdkMP81kgLBrqDJnXEB2DA&cad=rja

 

[russ_watters]

In addition, when you add an obstruction, you're no longer talking about the same system, so continuity doesn't apply!

 

[PJC01]

I can understand your confusion. However, there are a few things to consider in this situation. Firstly, the continuity of flow equation and Bernoulli's equation are idealized equations and do not take into account all the complexities of blood flow in the body. In reality, there are many factors that affect blood flow, such as the elasticity of blood vessels, the viscosity of blood, and the presence of other obstructions or constrictions in the arteries. These factors can all impact the flow rate and pressure in a more complicated manner than what is described by these equations.

Additionally, arterosclerosis not only decreases the radius of the arteries, but it also causes the walls of the arteries to become stiffer and less elastic. This can affect the flow of blood and create turbulence, which is not accounted for in the equations mentioned. This turbulence can also lead to an increase in pressure, as the blood is forced to flow through a narrower and more rigid space.

Furthermore, the body has regulatory mechanisms in place to maintain a constant blood pressure. When there is a decrease in blood flow due to arterosclerosis, the body may activate these mechanisms to increase the pressure in order to maintain adequate blood flow to the tissues and organs. This can also contribute to the observed increase in blood pressure.

In summary, while the equations you mentioned can provide a basic understanding of blood flow, they do not encompass all the complexities and factors involved in the human body. Arterosclerosis and other conditions can affect blood flow in ways that cannot be fully explained by these equations, and the body has mechanisms in place to compensate for these changes.