Dimensionless groups – defining Euler’s number (Eu) The ratio of the inertia force to the __________ force is termed Euler’s number (or its reciprocal, depending on convention). Choose the quantity that completes the definition used in many textbooks:

Introduction / Context:
Dimensionless numbers classify flow regimes by comparing dominant forces. Euler’s number (Eu) is associated with pressure effects relative to inertia, widely used in turbomachinery and internal flow analysis.

Given Data / Assumptions:

• Characteristic velocity V, length L, density rho.
• Pressure difference Δp represents the driving or resisting pressure forces.
• We focus on the common MCQ convention used in many introductory texts.

Concept / Approach:
In classical definition, Eu = Δp / (0.5 * rho * V^2), which is pressure force divided by inertia force. Some authors present the reciprocal form (inertia/pressure) but keep the name “Euler number.” Within the pattern of force-ratio definitions (Reynolds: inertia/viscous; Froude: inertia/gravity; Weber: inertia/surface tension), matching “inertia to pressure” identifies Euler’s group. Hence, completing the blank with “pressure” is consistent.

Step-by-Step Solution:
Step 1: Recognize the target forces: inertia force scale ≈ rho * V^2 * L^2 over area.Step 2: Pressure force scale ≈ Δp * L^2.Step 3: Form the ratio inertia/pressure or its reciprocal. The blank should be “pressure.”Step 4: Note that many texts use Eu as pressure/inertia; the MCQ phrasing mirrors the recurring “inertia to X” pattern.

Verification / Alternative check:
Check consistency: if pressure dominates, Eu (pressure/inertia) is large; the reciprocal (inertia/pressure) is small. Both communicate the same physical balance.

Why Other Options Are Wrong:

• Elastic: relates to compressibility and wave speed, not Eu’s standard definition.
• Surface tension: associated with Weber number.
• Viscous: associated with Reynolds number.
• Gravitational: associated with Froude number.

Common Pitfalls:
Memorizing only one convention and getting confused when the inverse appears; always check which way a source defines Eu.

Final Answer:
pressure