Dimensionless groups in fluid–particle systems: which of the following is NOT primarily used to characterize fluid–particle interaction phenomena?

Introduction:
Fluid–particle interactions (settling, fluidization, drop/bubble breakup) are often analyzed using appropriate dimensionless numbers. Knowing which groups are relevant helps select proper correlations for drag, terminal velocity, and regime transitions.

Given Data / Assumptions:

• Single particle or dispersed phase in a continuous fluid.
• Interest in drag, buoyancy, viscous, and surface tension effects.

Concept / Approach:
Galileo (and Archimedes) numbers combine gravity, density difference, viscosity, and size to correlate particle motion and drag. Weber number compares inertial to surface tension forces and is crucial for droplets/bubbles. The drag coefficient C_d directly characterizes particle–fluid momentum exchange. The Froude number compares inertia to gravity for free-surface or open-channel flows and is not central to single-particle interaction correlations in quiescent fluids.

Step-by-Step Solution:
List particle-relevant groups: Ga, Ar, Re_p, C_d, We (for interfacial phenomena).Identify Fr: more relevant to wave speed and open-channel flow rather than single-particle drag/settling.Therefore, Fr is the least relevant among the options.

Verification / Alternative check:
Empirical drag and terminal velocity charts use Re_p with C_d and Ga/Ar. Studies on drop breakup use We and Ohnesorge numbers; Fr is uncommon for isolated particle correlations.

Why Other Options Are Wrong:

• C_d, Ga, Ar: standard in particle drag correlations.
• We: standard for droplets/bubbles dealing with surface tension effects.

Common Pitfalls:
Assuming any dimensionless group involving gravity applies equally; context matters (free surface versus dispersed particle systems).

Final Answer:
Froude number (Fr)