Dimensionless group representing inertia-to-gravity force ratio The dimensionless number that compares inertial forces to gravitational forces in a flow (neglecting other forces) is the:

Introduction / Context:
Dimensionless numbers classify dynamic similarity in fluid flows. Choosing the right group helps scale models and predict dominant forces. When gravity plays a major role (open-channel flows, ship hydrodynamics), the inertia-to-gravity ratio is key.

Given Data / Assumptions:

• We compare inertial and gravitational forces only.
• Characteristic velocity V and length L are defined.
• Fluid density is constant for scaling purposes.

Concept / Approach:

The Froude number Fr is defined as Fr = V / √(g L). It can also be viewed from force scaling as Fr^2 ∼ inertia/gravity. It governs phenomena with free surfaces and wave-making, such as spillways, ship waves, and open-channel transitions.

Step-by-Step Solution:

Verification / Alternative check:

Other groups: Reynolds compares inertia to viscous (ρ V L / μ); Weber compares inertia to surface tension (ρ V^2 L / σ); Euler links pressure to inertia; Mach compares inertia to compressibility (V / a_sound). None are inertia/gravity except Froude.

Why Other Options Are Wrong:

Euler, Reynolds, Weber, and Mach address different force balances and are not suited for gravity-dominated similarity.

Common Pitfalls:

Confusing Froude with Reynolds; overlooking that Fr controls wave behavior and critical/supercritical flows in open channels.

Final Answer:

Froude number