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

1. Jun 22, 2011

### 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.

2. Jun 23, 2011

### Studiot

3. Jun 23, 2011

### Andy Resnick

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