Q10.25(a):

In deriving Bernoulli’s equation, we equated the work done on the fluid in the tube to its change in the potential and kinetic energy. (a) What is the largest average velocity of blood flow in an artery of diameter $2 \times 10^{-3} m$ if the flow must remain laminar ?

1. Given Variables & Physical Constants

To determine the largest average velocity for laminar flow, we utilize the Reynolds number ($R_e$), which characterizes the transition from laminar to turbulent flow in a pipe or vessel. For blood flow in human arteries, the critical Reynolds number ($R_c$) is typically taken as $2000$.

• Diameter of the artery ($d$): $2 \times 10^{-3} \, \text{m}$
• Radius of the artery ($r = d/2$): $1 \times 10^{-3} \, \text{m}$
• Critical Reynolds Number ($R_c$): $2000$ [Dimensionless empirical constant]
• Density of blood ($\rho$): $\approx 1.06 \times 10^3 \, \text{kg/m}^3$
• Viscosity of blood ($\eta$): $\approx 2.084 \times 10^{-3} \, \text{Pa}\cdot\text{s}$ (at body temperature)

2. Theoretical Foundation: The Reynolds Number Equation

The Reynolds number is defined as the ratio of inertial forces to viscous forces in a fluid flow [Per the definition of the Reynolds number in Fluid Dynamics]:

$R_e = \frac{\rho v d}{\eta}$

Where:

• $\rho$ is the density of the fluid ($\text{kg/m}^3$)
• $v$ is the average velocity ($\text{m/s}$)
• $d$ is the hydraulic diameter ($\text{m}$)
• $\eta$ is the coefficient of dynamic viscosity ($\text{Pa}\cdot\text{s}$)

3. Step-by-Step Algebraic Derivation

Step 1: Rearrange the formula to solve for velocity ($v$):

$v = \frac{R_e \cdot \eta}{\rho \cdot d}$

Step 2: Substitution of known values:

$v = \frac{2000 \cdot (2.084 \times 10^{-3} \, \text{Pa}\cdot\text{s})}{(1.06 \times 10^3 \, \text{kg/m}^3) \cdot (2 \times 10^{-3} \, \text{m})}$

Step 3: Simplify the numerator and denominator:

$v = \frac{4.168}{2.12}$

$v \approx 1.966 \, \text{m/s}$

4. Visual Representation of Laminar vs. Turbulent Transition

5. Analysis of Results

The calculation demonstrates that for an artery with a diameter of $2 \, \text{mm}$, the average velocity must remain below approximately $1.97 \, \text{m/s}$ to ensure the blood flow does not transition into turbulence. Turbulent flow in the circulatory system is generally pathological (e.g., associated with stenosis or heart murmurs) and increases the workload on the heart [Based on the energy dissipation principles of fluid mechanics].

Final Solution: The largest average velocity of blood flow to maintain laminar conditions is approximately $1.97 \, \text{m/s}$.