What is Bernoulli's Principle?
Bernoulli's principle is a cornerstone of fluid dynamics: it links a fluid's pressure, speed, and height along a streamline. It explains how airplane wings generate lift, why a shower curtain billows inward, and how a venturi meter measures flow.
Bernoulli's principle states that for an ideal, incompressible, non-viscous fluid flowing steadily, the sum of static pressure, dynamic pressure, and hydrostatic pressure is constant along a streamline — so as speed increases, pressure decreases.
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Step-by-step worked examples
Water flows through a horizontal pipe that narrows from a wide section (v1 = 2 m/s, P1 = 200,000 Pa) to a narrow section (v2 = 6 m/s). Find P2 (ρ = 1000 kg/m³).
P1 + ½ρv1² = P2 + ½ρv2² 200,000 + 0.5×1000×2² = P2 + 0.5×1000×6² 200,000 + 2,000 = P2 + 18,000 P2 = 202,000 − 18,000 = 184,000 Pa
Air flows over an airplane wing at 250 m/s on top and 200 m/s below. If P_below = 101,000 Pa and ρ_air = 1.2 kg/m³, find the pressure above.
P_above = P_below + ½ρ(v_below² − v_above²) P_above = 101,000 + 0.5×1.2×(200² − 250²) P_above = 101,000 + 0.6×(40,000 − 62,500) P_above = 101,000 + 0.6×(−22,500) = 101,000 − 13,500 = 87,500 Pa
A venturi meter has a wide section (v1 = 1 m/s) and throat (v2 = 4 m/s), both at the same height, water flowing (ρ = 1000 kg/m³). Find the pressure drop P1 − P2.
P1 − P2 = ½ρ(v2² − v1²) P1 − P2 = 0.5×1000×(4² − 1²) P1 − P2 = 500×(16 − 1) = 500×15 = 7,500 Pa
Flashcards
Quick quiz
Q1.As fluid speed increases in a horizontal pipe, what happens to pressure?
Q2.Bernoulli's equation assumes the fluid is...
Q3.In P + ½ρv² + ρgh = constant, the term ρgh represents...
Q4.Why does an airplane wing generate lift?
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Common mistakes
Applying Bernoulli's equation to viscous or turbulent flow without caution. — Correct: Bernoulli's equation strictly applies to ideal, non-viscous, steady flow — real fluids have losses.
Thinking higher velocity always means higher pressure. — Correct: It's the opposite: along a streamline, higher velocity means LOWER pressure.
Forgetting the height term ρgh when elevation changes. — Correct: Include ρgh whenever the two points are at different heights.
Confusing static pressure with total (stagnation) pressure. — Correct: Total pressure = static + dynamic pressure; Bernoulli's constant is the total pressure.
FAQ
What is Bernoulli's principle?
It states that for ideal fluid flow along a streamline, P + ½ρv² + ρgh is constant — so pressure drops where speed increases.
What is the formula for Bernoulli's principle?
P + ½ρv² + ρgh = constant, where P is pressure, ρ is density, v is velocity, g is gravity, and h is height.
What are examples of Bernoulli's principle?
Airplane lift, a venturi meter, a curveball's curved path, and a shower curtain being pulled inward.
How do you calculate pressure using Bernoulli's principle?
Solve P2 = P1 + ½ρ(v1² − v2²) for horizontal flow, or include ρgh if height changes.