Question

Which of the following is true with respect to the Stoke's Law?
(A) It relates to the terminal settling velocity of particles in a viscous medium.
(B) It relates the terminal settling velocity to particle radius and density.
(C) It is applied in the determination of the particle size distribution.
(D) It is applicable only to the spherical particles.
Choose the correct answer from the options given below:

• A. (A), (B), (C) and (D).
• B. (A), (B) & (C) only.
• C. (B), (C) and (D) only.
• D. (A), (C) and (D) only.

Hint:

Stokes' Law assumes:
1. Spherical, rigid, and smooth particles.
2. No interaction between settling particles (low concentration suspension, $< 5\%$).
3. Laminar flow (Reynolds number $< 0.1$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Stokes' Law describes the settling behavior of a spherical particle falling through a viscous fluid under the influence of gravity.
It is the fundamental physical principle used in soil mechanical analysis (sedimentation methods).
Key Formula or Approach:
The terminal settling velocity ($v$) of a spherical particle is given by:
\[ v = \frac{2}{9} \frac{g \cdot r^2 \cdot (\rho_p - \rho_f)}{\eta} \] where:
- $r$ is the particle radius.
- $\rho_p$ is the particle density.
- $\rho_f$ is the fluid density.
- $g$ is the acceleration due to gravity.
- $\eta$ is the dynamic viscosity of the fluid.

Step 2: Detailed Explanation:
Let us analyze each of the given statements:
- Statement (A): Stokes' Law calculates the constant maximum (terminal) settling velocity reached by a particle when the gravitational force is balanced by the buoyant and drag forces in a viscous medium. This is true.
- Statement (B): As seen in the formula, the velocity ($v$) is directly proportional to the square of the radius ($r^2$) and the density difference ($\rho_p - \rho_f$). This is true.
- Statement (C): In soil science, the hydrometer and pipette methods measure density changes in a soil suspension over time.
Using Stokes' Law, the time required for particles of a specific size to settle past a given depth is calculated, allowing the determination of the sand, silt, and clay fractions. This is true.
- Statement (D): The mathematical derivation of Stokes' Law assumes smooth, rigid, spherical particles.
While soil particles (especially clay) are often plate-like, the law is applied by defining an "equivalent spherical diameter." This is true.
Since all four statements are true, the correct combination includes (A), (B), (C), and (D).

Step 3: Final Answer:
The correct option is (A).