In a steady flow with constant density, the continuity equation reduces to which relation?

In a steady flow with constant density, the continuity equation reduces to which relation?

• A. v1A1 = v2A2
• B. p1A1v1 = p2A2v2
• C. ρ1A1v1 = ρ2A2v2
• D. Q = m cp ΔT

Answer: (A)

The main idea is mass conservation in steady, incompressible flow. Since nothing accumulates, the mass flow entering a section must equal the mass flow leaving it. The mass flow rate is ṁ = ρ A v. With constant density, ρ is the same on both sides, so ṁ1 = ṁ2 becomes ρ A1 v1 = ρ A2 v2, and canceling ρ gives A1 v1 = A2 v2. This is why the product of cross-sectional area and velocity remains the same across sections in steady incompressible flow. The other forms either mix in pressure (not part of mass conservation), represent the general compressible form with density changing (not needed when density is constant), or relate to heat transfer rather than fluid continuity.

The main idea is mass conservation in steady, incompressible flow. Since nothing accumulates, the mass flow entering a section must equal the mass flow leaving it. The mass flow rate is ṁ = ρ A v. With constant density, ρ is the same on both sides, so ṁ1 = ṁ2 becomes ρ A1 v1 = ρ A2 v2, and canceling ρ gives A1 v1 = A2 v2. This is why the product of cross-sectional area and velocity remains the same across sections in steady incompressible flow.

The other forms either mix in pressure (not part of mass conservation), represent the general compressible form with density changing (not needed when density is constant), or relate to heat transfer rather than fluid continuity.