Develop the Weber number by starting with estimates for the inertia and surface tension forces.

In fluid dynamics, the inertia force is the force that opposes changes in motion. It can be calculated by multiplying the mass of the fluid element with the change in velocity per unit time. Therefore, it's written as \( F_i = m \cdot \frac{Δv}{Δt} \), where: \( F_i \) is the inertia force, \( m \) is the mass of the fluid element, and \( \frac{Δv}{Δt} \) is the rate of change of velocity.

Surface tension is the force that makes the surface of liquids behave like a stretched elastic sheet. It results from the imbalance in the cohesive forces between molecules at the surface of a fluid. It can be defined as: \( F_s = σ \cdot L \), where \( F_s \) is the surface tension force, \( σ \) is the surface tension, and \( L \) is characteristic length or perimeter over which the force is acting.

The Weber Number (\( We \)) is a dimensionless number that provides a measure of the relative importance of inertia forces over surface tension forces. It is defined as the ratio of inertia force to surface tension force, \( We = \frac{F_i}{F_s} \). By substituting our original equations \( F_i = m \cdot \frac{Δv}{Δt} \) and \( F_s = σ \cdot L \), we have: \( We = \frac {m \cdot \frac{Δv}{Δt}} {σ \cdot L} \). By recalling that \( ρ = \frac{m}{V} \) where \( ρ \) is the density and rearranging, we can finally express the Weber number as: \( We = \frac {ρ \cdot V \cdot \frac{Δv}{Δt}} {σ / L} = \frac {ρ \cdot L \cdot V^2} {σ} \). This is often simplified in terms of velocity to: \( We = \frac {ρ \cdot L \cdot v^2} {σ} \), where \( v \) is the flow velocity.