Settling under Stokes’ regime: For a spherical particle of diameter d < 0.1 mm settling in a fluid, which expression gives the terminal settling velocity according to Stokes’ law?

Introduction / Context:
Primary sedimentation and grit removal design rely on predicting particle settling velocities. For very small spherical particles at low Reynolds number (laminar regime), Stokes’ law provides a simple analytical expression relating size, fluid properties, and density contrast to settling velocity.

Given Data / Assumptions:

• Spherical particle with diameter d < 0.1 mm (small enough for laminar settling).
• Specific gravity of particle = Gs; fluid kinematic viscosity = nu.
• Flow conditions satisfy Stokes’ regime (Re <≈ 1).

Concept / Approach:

Stokes’ law (laminar settling) states the drag force equals weight minus buoyancy at terminal velocity. Solving the force balance yields v_s proportional to d^2 and inversely proportional to viscosity. The density term appears as (Gs − 1) because buoyancy subtracts the fluid density from particle density.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional check: numerator has L/T^2 * L^2 = L^3/T^2; denominator has L^2/T; result is L/T (velocity), consistent.

Why Other Options Are Wrong:

Option (b) inverts the relation; (c) uses d^1 not d^2; (d) uses d^3, which applies to buoyant weight but not to velocity form.

Common Pitfalls:

Applying Stokes’ law beyond its Re limit; using kinematic vs dynamic viscosity inconsistently; neglecting flocculation that changes effective particle size and density.

Final Answer:

v_s = [ g * (Gs − 1) * d^2 ] / (18 * nu)