Bernoulli's Principle and Energetics of Flowing Blood
Because flowing blood has mass and velocity it has kinetic energy (KE). This KE is proportionate to the mean velocity squared (V2; from KE = ½ mV2). Furthermore, as the blood flows inside a vessel, pressure is exerted laterally against the walls of the vessel; this pressure represents the potential or pressure energy (PE). The total energy (E) of the blood flowing within the vessel, therefore, is the sum of the kinetic and potential energies (assuming no gravitational effects) as shown below.
E = KE + PE (where KE ∝ V2) Therefore, E ∝ V2 + PE
There are two important concepts that follow from this relationship.
- Blood flow is driven by the difference in total energy between two points. Although pressure is normally considered as the driving force for blood flow, in reality it is the total energy that drives flow between two points (e.g., longitudinally along a blood vessel or across a heart valve). Throughout most of the cardiovascular system, KE is relatively low, so for practical purposes, it is stated that the pressure energy (PE) difference drives flow. When KE is high, however, adding KE to the PE significantly increases the total energy, E. To illustrate this, consider the flow across the aortic valve during cardiac ejection. Late during ejection, the intraventricular pressure (PE) falls slightly below the aortic pressure (PE), nevertheless, flow continues to be ejected into the aorta. The reason for this is that the KE of the blood as it moves across the valve at a very high velocity ensures that the total energy (E) in the blood crossing the valve is higher than the total energy of the blood more distal in the aorta.
- Kinetic energy and pressure energy can be interconverted so that total energy remains unchanged. This is the basis of Bernoulli's Principle. This principle can be illustrated by a blood vessel that is suddenly narrowed then returned to its normal diameter. In the narrowed region (stenosis), the velocity increases as the diameter decreases. Quantitatively, V ∝ 1/D2 because flow (F) is the product of mean velocity (V) and vessel cross-sectional area (A) (F = V ∙ A), and A is directly related to diameter (D) (or radius, r) squared (from A = π ∙ r2). If the diameter is reduced by one-half in the region of the stenosis, the velocity increases 4-fold. Because KE ∝ V2, the KE increases 16-fold. Assuming that the total energy is conserved within the stenosis (E actually decreases because of resistance), then the 16-fold increase in KE must result in a proportionate decrease in PE. Once past the narrowed segment, KE will revert back to its pre-stenosis value because the post-stenosis diameter is the same as the pre-stenosis diameter and flow is conserved. Because of the resistance of the stenosis, and the likelihood of turbulence, the post-stenosis PE and E will both fall. To summarize this concept, blood flowing at higher velocities has a higher ratio of kinetic energy to potential (pressure) energy.
An interesting, yet practical application of Bernoulli's Principle is found when blood pressure measurements are made from within the ascending aorta. As described above, during ventricular ejection, the velocity and hence kinetic energy of the flowing blood is very high.