Water flows steadily through a completely filled horizontal pipe. At section 1, area A₁ = 8.000 × 10^-4 m², speed v₁ = 1.2 m s⁻¹ and absolute pressure P₁ = 1.620 × 10^5 Pa. At section 2, A₂ = 4.000 × 10^-4 m². Model water as incompressible and non-viscous with density 1000 kg m⁻³. Find v₂, volume flow rate and P₂. Explain the pressure change using energy per unit volume.

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v₂ = 2.4 m s⁻¹; flow rate 9.600 × 10^-4 m³ s⁻¹; P₂ = 1.598 × 10^5 Pa. Pressure falls in the narrower section. A steady incompressible flow carries the same volume through every cross-section each second. Thus A₁v₁ = A₂v₂; a smaller area requires a larger speed, not a smaller flow rate. Because A₂ = A₁/2, v₂ = 2v₁ = 2.4 m s⁻¹. The volume flow rate is A₁v₁ = 9.600 × 10^-4 m³ s⁻¹. Bernoulli's equation along a streamline is P + ½ρv² + ρgy = constant. The height terms cancel here because both sections are horizontal and at the same elevation. Therefore P₂ = P₁ + ½ρ(v₁² − v₂²) = 1.598 × 10^5 Pa. The pressure drop is 2160 Pa. The increase in kinetic energy per unit volume is balanced by a decrease in pressure energy per unit volume. The positive absolute pressure is physically sensible; the result would require modification for significant viscous losses.

Continuity and pressure in a narrowing pipe · set 2

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