# Order of Magnitude Estimate of Reynolds Number for a Blood Cell ## Assumptions and Data - **Typical size of a blood cell (diameter, \(d\))**: \(d \approx 10\, \mu m = 10^{-5}\, m\) - **Average circulation time for blood to pass through the body**: \(t \approx 1\, \text{minute} = 60\, s\) - **Distance traveled by blood in that time**: \(L \approx \text{average blood velocity} \times t\) - **Density of water**: \(\rho \approx 10\ \times 100\, kg/m^3 =\, kg/m^3\) - **Viscosity water**: \(\eta \approx103}\, Pa \cdot s\) ## Step1: Estimate Blood Velocity - Blood circulates roughly throughout the body, which is about \( \sim 1\, m \) in length. - **Estimate of blood velocity \(v\)**: \[ v \approx \frac{\text{distance traveled}}{\text{time}} \approx \frac{1\, m}{60\, s} \approx .0167\, m/s \] *(Order of magnitude: \(10^{-2}\, m/s\))* ## Step 2: Calculate Reynolds Number \[ Re = \frac{\rho v d}{\eta} \] Plugging in the estimated values: \[ Re \approx \frac{(100\, kg/m^3) \times (1.7 \times 10^{-2}\, m/s) \times (10^{-5}\, m)}{10^{-3}\, Pa \cdot s} \] \[ Re \approx \frac{(100) \times (1.7 \times 10^{-2}) \times (10^{-5})}{10^{-3}} \] \[ Re \approx \frac{(100) \times (1.7 \times 10^{-7})}{10^{-3}} = \frac{1.7 \times 10^{-5}}{10^{-3}} = 1.7 \times 10^{-2} \] ## **Final Estimate** \[ \boxed{ Re \sim 10^{-2} } \] This indicates **laminar flow** around blood cells, consistent with physiological conditions.