Continuity equation: what it expresses In a steady flow field along a streamline, the continuity equation fundamentally expresses which conservation statement?

Introduction / Context:
The continuity equation is one of the fundamental equations governing fluid flow. It enforces conservation of mass and is applicable to both compressible and incompressible flows, in integral and differential forms. Along a streamline in steady incompressible flow, the mass flow rate remains constant.

Given Data / Assumptions:

• Steady flow condition (no explicit time dependence in control volume).
• Single inlet–outlet streamline description.
• Incompressible fluid when using A V constancy; compressible form uses ρ A V.

Concept / Approach:
For a streamtube, continuity gives ρ A V = constant. In incompressible flow, density ρ is constant, so A V = constant. This expresses that mass cannot accumulate in a differential element in steady flow; whatever enters must exit.

Step-by-Step Solution:

Verification / Alternative check:
For two sections 1 and 2: ρ1 A1 V1 = ρ2 A2 V2. If ρ1 = ρ2, then A1 V1 = A2 V2. Measurements in venturi meters and nozzles use this directly.

Why Other Options Are Wrong:

• Work–energy relation is tied to Bernoulli/energy equation, not continuity.
• Momentum equation handles forces and momentum change, not mass conservation.
• Newton’s second law and angular momentum are separate governing principles (momentum and moment-of-momentum equations).

Common Pitfalls:
Applying A V = constant to compressible flows without density correction; confusing energy and mass conservation; ignoring boundary-layer displacement thickness effects on effective area.

Final Answer:
Conservation of mass (mass rate of flow is constant along a streamline)