Stokes settling of a small sphere: the terminal velocity in a viscous fluid varies primarily with which parameter (holding others constant)?

Introduction:
Very small particles settling slowly in a viscous fluid obey Stokes’ law, an analytical solution valid at low Reynolds numbers. Recognizing how terminal velocity depends on material and fluid properties is essential for clarifier design and fine particle separations.

Given Data / Assumptions:

• Spherical particle, creeping flow (Re < 1).
• Uniform fluid, isothermal, quiescent.
• Terminal (steady) settling velocity v_t.

Concept / Approach:
Stokes’ law gives v_t = g (ρ_p − ρ_f) d^2 / (18 μ). Thus v_t is directly proportional to d^2 and to the density difference, and inversely proportional to dynamic viscosity μ. Among the options provided, the correct single-statement dependence highlighted is the inverse proportionality to μ.

Step-by-Step Solution:
Write Stokes formula: v_t = [g (ρ_p − ρ_f) d^2] / (18 μ).Identify dependences: v_t ∝ d^2; v_t ∝ (ρ_p − ρ_f); v_t ∝ 1/μ.Match with options: only the viscosity inverse dependence is stated correctly.

Verification / Alternative check:
Dimensional analysis of drag in creeping flow (D = 3 π μ d v) plus force balance D = W − B leads directly to v_t inversely proportional to μ.

Why Other Options Are Wrong:

• First power of diameter: should be second power (d^2) under Stokes law.
• Inverse square of diameter: incorrect trend and dimensionally inconsistent.
• Square of density difference: dependence is first power, not squared.
• Fourth power of diameter: not applicable in creeping flow.

Common Pitfalls:
Confusing Stokes (v_t ∝ d^2) with Newton’s law regime (v_t ∝ d^0.5) at high Reynolds numbers. Always check regime validity first.

Final Answer:
Inverse of fluid viscosity