Question

In Poiseuille flow, the flow rate is directly proportional to:

• A. Length of the vessel, pressure, coefficient of viscosity
• B. Pressure drop, length of the vessel, coefficient of viscosity
• C. Pressure drop, diameter of the vessel
• D. Radius of the vessel, coefficient of viscosity

Hint:

The Power of the 4th Power in Hemodynamics: Because flow rate ($Q$) is directly proportional to $d^4$, even tiny adjustments to a blood vessel's diameter have a massive impact on blood flow. - For example, doubling a vessel's diameter ($2^4$) increases the volumetric flow rate by 16 times, assuming pressure remains constant! This relationship explains how the body uses vasoconstriction and vasodilation to easily regulate localized blood distribution.

Answer 1

The Correct Option is C

Concept: The steady, laminar flow of an incompressible, Newtonian fluid (such as simplified blood flow) through a rigid, cylindrical tube of constant cross-section is governed mathematically by Hagen-Poiseuille's Law. The fundamental equation states: \[ Q = \frac{\pi \cdot \Delta P \cdot r^4}{8 \cdot \eta \cdot L} \] where:
• $Q$ = Volumetric fluid flow rate
• $\Delta P$ = Hydrostatic pressure drop across the vessel length
• $r$ = Internal radius of the vessel
• $\eta$ = Dynamic coefficient of fluid viscosity
• $L$ = Total axial length of the vessel segment

Step 1: Translating radius into diameter terms.
The internal radius ($r$) is equal to exactly half of the total vessel diameter ($d$), or $r = \frac{d}{2}$. Substituting this into our fluid equations gives: \[ Q = \frac{\pi \cdot \Delta P \cdot \left(\frac{d}{2}\right)^4}{8 \cdot \eta \cdot L} = \frac{\pi \cdot \Delta P \cdot d^4}{128 \cdot \eta \cdot L} \]

Step 2: Evaluating proportional relationships.
From this equation, we can separate the variables into direct and inverse proportionalities:
• Direct Proportionality (Numerator): The flow rate $Q$ is directly proportional to the pressure drop ($\Delta P$) and to the fourth power of the diameter of the vessel ($d^4$).
• Inverse Proportionality (Denominator): The flow rate $Q$ is inversely proportional to the vessel length ($L$) and fluid viscosity ($\eta$). Evaluating the choices shows that Option (C) correctly identifies the parameters that share a direct proportional relationship with flow rate.