Question

Which of the following equation expresses the velocity of settling particles as described by Stokes law ?

• A. $v = 2g r^2 (D_p - D_l) / 9\eta$
• B. $v = 2g r (D_p - D_l) / 9\eta$
• C. $v = 2g r^2 (D_l - D_p) / 9\eta$
• D. $v = g r (D_l - D_p) / 4.5\eta$

Hint:

Always remember that settling velocity is proportional to the square of the radius ($r^2$), and depends on the density difference $(D_p - D_l)$. This helps you eliminate incorrect options.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Stokes' Law describes the settling velocity of spherical particles falling through a viscous fluid under the influence of gravity.
It forms the physical basis for soil texture analysis methods like the pipette or hydrometer method.
Key Formula or Approach:
The mathematical expression for Stokes' Law is: \[ v = \frac{2}{9} \frac{g r^2 (\rho_p - \rho_f)}{\eta} \] Using the notation in the options: - $v$ = settling velocity of the particle
- $g$ = acceleration due to gravity
- $r$ = radius of the spherical particle
- $D_p$ = density of the particle ($\rho_p$)
- $D_l$ = density of the liquid ($\rho_f$)
- $\eta$ = dynamic viscosity of the liquid

Step 2: Detailed Explanation:
Under terminal velocity conditions, the upward drag force plus buoyant force equals the downward gravitational force: \[ v = \frac{2 g r^2 (D_p - D_l)}{9 \eta} \] - If the particle density is greater than the liquid density ($D_p > D_l$), the velocity $v$ is positive, indicating the particle settles downward.
- This formula represents the settling velocity.

Step 3: Final Answer:
The equation matches Option (A).