According to Stokes’ law, what is the terminal settling velocity V (in still water at temperature T°C) for a small spherical particle of diameter d and specific gravity G?

Introduction / Context:
In water and wastewater engineering and sediment transport, estimating the settling velocity of fine, spherical particles is fundamental. Stokes’ law applies for laminar settling (very small Reynolds numbers), linking velocity with particle size, density contrast, gravity, and fluid viscosity.

Given Data / Assumptions:

• Particle: small, spherical with diameter d.
• Specific gravity of particle: G (relative to water).
• Fluid: water at T°C (kinematic viscosity ν depends on temperature).
• Laminar regime: particle Reynolds number is small (Re <~ 1).

Concept / Approach:
Stokes’ law balances gravitational force minus buoyancy with viscous drag for a small sphere settling at terminal velocity V. The final expression explicitly shows the quadratic dependence on particle diameter and inverse dependence on viscosity.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional analysis confirms V has dimensions of length/time. Also, increasing temperature lowers ν, increasing V, which matches observation.

Why Other Options Are Wrong:

• d or d^3 dependence is wrong for Stokes regime; the correct dependence is d^2.
• Inverting the expression (option d) gives units and trends inconsistent with physics.

Common Pitfalls:
Using Stokes’ law beyond laminar range; ignoring flocculation; using dynamic instead of kinematic viscosity without proper conversion; neglecting temperature dependence of ν.

Final Answer:
V = g * (G - 1) * d^2 / (18 * ν)