Micro-hydrodynamics — drag coefficient at very low Reynolds number A bacterium of size 1 µm swims in water at 1 mm/s (kinematic viscosity ν = 1×10^-6 m^2/s). What is the order of magnitude of its drag coefficient?

Introduction / Context:At microscopic scales, flows are dominated by viscosity (low Reynolds number). The drag coefficient for small particles or organisms scales inversely with Reynolds number in the creeping-flow regime, illustrating why swimming at micro-scales is fundamentally different from macro-scales.

Given Data / Assumptions:

• Characteristic length L ≈ 1 µm = 1×10^-6 m.
• Speed V = 1 mm/s = 1×10^-3 m/s.
• Kinematic viscosity ν = 1×10^-6 m^2/s.
• Reynolds number Re = V * L / ν.

Concept / Approach:For very small Re (creeping flow), an approximate relation for bluff-body drag coefficient is Cd ≈ 24/Re (Stokes regime). We compute Re and then estimate Cd order of magnitude.

Step-by-Step Solution:Compute Re: Re = (1×10^-3 m/s) * (1×10^-6 m) / (1×10^-6 m^2/s) = 1×10^-3.Estimate Cd: Cd ≈ 24 / Re = 24 / (1×10^-3) = 24,000.Round to order of magnitude as required → 24,000.

Verification / Alternative check:The creeping-flow assumption holds since Re ≪ 1; empirical corrections (Oseen) are minor for this order-of-magnitude question.

Why Other Options Are Wrong:

• 24: corresponds to Re ≈ 1.
• 0.44 and 0.24: typical of high-Re streamlined bodies, not micro-swimmers.

Common Pitfalls:

• Using high-Re intuition; micro-swimmers operate where inertia is negligible.

Final Answer:24000