Dimensional analysis – definition of Weber number (We) Select the correct definition of the Weber number used in free-surface and multiphase flows.
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ARatio of inertia force to elastic force
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BRatio of gravity force to surface tension force
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CRatio of viscous force to pressure force
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DRatio of inertia force to surface tension force
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ERatio of viscous force to gravity force
Answer
Correct Answer: Ratio of inertia force to surface tension force
Explanation
Introduction / Context:Nondimensional numbers organize complex fluid phenomena. The Weber number is particularly important when surface tension competes with inertia, such as in droplet formation, jets, splashing, and wave breaking.
Given Data / Assumptions:
- Characteristic velocity V, length L, density ρ, and surface tension σ.
- Definition pertains to flows where interfaces and capillarity matter.
Concept / Approach:Weber number compares inertia to surface tension effects: We = (ρ V^2 L) / σ. Large We implies inertia dominates and interfaces deform or break; small We indicates surface tension resists deformation and stabilizes the interface.
Step-by-Step Solution:
Identify inertia scaling: F_inertia ∝ ρ V^2 L^2 (up to constant factors).Identify capillary scaling: F_surface ∝ σ L.Form the ratio → We ∝ (ρ V^2 L) / σ.Verification / Alternative check:Related groups: Reynolds number Re = inertia/viscous, Froude number Fr = inertia/gravity. By analogy, Weber links inertia and surface tension, matching option D.
Why Other Options Are Wrong:
- Elastic, gravity, or pressure-force ratios correspond to other dimensionless groups, not the Weber number.
Common Pitfalls:Confusing We with Re or Fr; remembering the numerator ρ V^2 L (inertia) divided by σ (surface tension) helps avoid errors.
Final Answer:Ratio of inertia force to surface tension force