Question

Stokes' law is NOT valid when:

• A. Reynolds number $<$ 0.1
• B. Particles are spherical
• C. Flow is laminar
• D. Particle concentration is high

Hint:

Stokes' Law assumes "free settling" of an isolated sphere in an infinite fluid pool. Once particle concentration increases, fluid displacement forces and particle-particle interactions restrict movement, causing "hindered settling."

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks to identify the boundary condition or physical state under which the standard Stokes' Law equation becomes invalid for predicting particle settling or creaming velocities in food suspensions.

Step 2: Key Formula or Approach:
Stokes' Law defines the terminal settling velocity ($v_t$) of a spherical particle in a fluid medium:
\[ v_t = \frac{g d^2 (\rho_p - \rho_f)}{18 \mu} \] where:
$g$ = gravitational acceleration ($\text{m/s}^2$),
$d$ = particle diameter ($\text{m}$),
$\rho_p$ = density of the particle ($\text{kg/m}^3$),
$\rho_f$ = density of the fluid ($\text{kg/m}^3$),
$\mu$ = dynamic viscosity of the fluid ($\text{Pa}\cdot\text{s}$).

Step 3: Detailed Explanation:

• Assumptions of Stokes' Law: The derivation of Stokes' Law relies on several critical simplifying physical assumptions:

• The flow must be strictly laminar, which is satisfied when the particle Reynolds number ($Re_p$) is very small ($Re_p < 0.1$ or $Re_p < 1$), matching options (A) and (C).

• The particles must be rigid, smooth, and perfectly spherical, matching option (B).

• The settling must be "free settling", meaning particles are spaced far enough apart that they do not physically interact or influence each other's flow fields.

• High Particle Concentration (Hindered Settling): When particle concentration is high, the particles are closely packed. As a particle falls, it displaces fluid upward, and this upward flow hinders the settling of neighboring particles. Physical collisions and hydrodynamic interactions between particles become dominant.

• Invalidation: Under these conditions of hindered settling, Stokes' Law significantly overestimates the true settling velocity, and empirical correction factors (such as the Richardson-Zaki equation) must be applied.

Step 4: Final Answer:
Stokes' Law fails under high particle concentrations due to hindered settling effects, making option (D) the correct choice.