Blood flow through a cylindrical blood vessel can be mathematically described using the principles of laminar flow, a regime in which fluid moves smoothly in parallel layers. In this model, the velocity of the blood is not uniform across the cross-section of the vessel; rather, it varies with the radial distance from the center. The maximum velocity occurs along the central axis, decreasing progressively toward the vessel walls, where it reaches zero due to viscous drag.
Approximating Blood Flow
To estimate the total volumetric flow, or flux, the vessel’s circular cross-section is conceptually divided into a series of thin concentric rings. Each ring contributes a portion of the total flow, determined by its area—calculated as the product of its circumference and thickness—and the fluid velocity at that specific radius. Summing the contributions from all rings gives an approximation of the total flow through the vessel.
Deriving the Flow Using Integration
As the number of rings increases and their thickness decreases, the summation of flow contributions approaches a definite integral over the vessel’s radius. The law of laminar flow provides the velocity profile as a function of radial distance, incorporating the pressure gradient along the vessel, the fluid's viscosity, and the vessel's length. Integrating this velocity function from the center to the outer wall yields the exact value of total blood flow.
This result is formalized in Poiseuille’s Law, which states that the volumetric flow rate is directly proportional to the pressure difference and the fourth power of the vessel’s radius, and inversely proportional to the viscosity and length of the vessel. This pronounced dependence on the radius underscores the critical influence of even small changes in vessel diameter on circulatory efficiency.
Blood flow in a blood vessel can be described using the law of laminar flow, where velocity varies with radial distance from the center.
The total blood flow can be approximated by dividing the cross-section into thin concentric rings with varying radii.
Each ring’s area is given by its circumference multiplied by its small thickness.
The flow through each ring can be approximated as the product of its area and the velocity at that radius.
Summing these flows for all rings gives an approximate total flow, called the Riemann sum.
As the number of rings approaches infinity, the Riemann sum approaches the exact values, represented by the definite integral from the center to the vessel wall.
The velocity function in the integral comes from the law of laminar flow. It states that the velocity at a distance r depends on the pressure difference, the fluid’s viscosity, and the vessel’s length.
Substituting the velocity expression into the integral and integrating over the vessel radius yields the exact blood flow.
This result, called Poiseuille’s Law, shows that flow scales with the fourth power of the vessel’s radius. This is why even a small narrowing of blood vessels sharply reduces blood flow.
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