# Estimating the Reynolds Number for a Blood Cell in the Circulatory System The Reynolds number (\(Re\)) characterizes the flow regime around an object, indicating whether it is laminar or turbulent. It is given by: \[ Re = \frac{\rho v L}{\mu} \] where: - \(\rho\) = density of the fluid (blood) - \(v\) = velocity of the blood cell - \(L\) = characteristic length (size of the cell) - \(\mu\) = dynamic viscosity of blood --- ## Step 1: Typical Values ### Blood Cell Size (\(L\)) - Typical diameter of a red blood cell: **~8 μm (micrometers)** - Convert to meters: \[ L = 8 \times 10^{-6} \text{ m} \] ### Blood Velocity (\(v\)) - Typical blood flow velocity in capillaries: **~.1 mm/s to 1 mm/s** - Use an approximate value: \[ v = 1 \times 10^{-3} \text{ m/s} \] ### Blood Properties - Density (\(\rho\)): **~106 kg/m\(^3\)** - Dynamic viscosity (\(\mu\)): **~3.5 cP (centipoise)**, which is \[ \mu \approx 3.5 \times 10^{-3} \text{ Pa·s} \] --- ## Step 2: Calculate Reynolds Number \[ Re = \frac{\rho v L}{\mu} \] Plugging in the values: \[ Re = \frac{(106\, \text{kg/m}^3) \times (1 \times 10^{-3}\, \text{m/s}) \times (8 \times 10^{-6}\, \text{m})}{3.5 \times 10^{-3}\, \text{Pa·s}} \] Calculate numerator: \[ 106 \times 1 \times 10^{-3} \times 8 \times 10^{-6} = 106 \times 8 \times 10^{-9} = 848 \times 10^{-9} = 8.48 \times 10^{-6} \] Divide by viscosity: \[ Re = \frac{8.48 \times 10^{-6}}{3.5 \times 10^{-3}} \approx 2.42 \times 10^{-3} \] --- ## **Final Estimate** \[ \boxed{ Re \approx 2.4 \times 10^{-3} } \] **Conclusion:** The Reynolds number for a blood cell moving through the bloodstream is approximately **.0024**, indicating **laminar flow** at the cellular level within blood vessels.