Continuity equation applicability: required conditions for using the simple form A₁V₁ = A₂V₂ For the commonly used one-streamtube continuity form A₁V₁ = A₂V₂, the flow should be steady, incompressible, effectively one-dimensional, and have a uniform velocity profile across each section. Select the correct combined statement.

Introduction / Context:
The general continuity equation expresses mass conservation. The simplified duct formula A₁V₁ = A₂V₂ is widely used in pipes and channels but requires specific assumptions to be valid without density terms or profile corrections.

Given Data / Assumptions:

• Single fluid, no phase change across the section pair.
• No lateral inflows between the two cross-sections.
• Objective: conditions for A₁V₁ = A₂V₂.

Concept / Approach:
The integral continuity equation reduces to ρA₁V₁ = ρA₂V₂ for steady, one-dimensional flow. If the fluid is incompressible (ρ constant), this becomes A₁V₁ = A₂V₂. A uniform velocity profile across each section is implied when using the area-mean velocity V without additional correction factors.

Step-by-Step Solution:

Verification / Alternative check:
For compressible flow, the correct form is ρ₁A₁V₁ = ρ₂A₂V₂. With non-uniform profiles, include a profile factor or use the integral form directly.

Why Other Options Are Wrong:
Each single statement is necessary but not sufficient alone. The complete and correct choice is “all the above”.

Common Pitfalls:
Using A V = constant in compressible gases with significant density changes; neglecting profile correction factors in high-Re turbulent ducts.

Final Answer:
all the above