☰ Contents · Physics

Liquid flow and Bernoulli’s equation

Lessons 20 · 1 lessons · N. Sh. Turdiyev, K. A. Tursunmetov, A. G. Ganiyev, K. T. Suyarov, J. E. Usarov, A. K. Avliyoqulov. Physics Grade 10, 1st edition. Niso Poligraf Publishing House, Tashkent, 2017
20

Motion of liquids and gases, the continuity theorem. Bernoulli’s equation

Textbook: pp. 68–70
GoalKnow the continuity equation and Bernoulli’s equation and apply them to liquid flowing in pipes.
New words
laminar flow · laminar oqimturbulent flow · turbulent oqimcontinuity of flow · oqimning uzluksizligistatic and dynamic pressure · statik va dinamik bosim
Explanation

If a flow of liquid or gas is smooth and in layers, it is laminar; if it is fast and forms eddies, it is turbulent. When an incompressible ideal liquid flows steadily through a pipe of changing cross-section, the same volume passes every second, so S₁υ₁ = S₂υ₂ (the continuity equation): the speed is large in the narrow part and small in the wide part. Bernoulli’s equation (1738) follows from conservation of mechanical energy: p + ρgh + ρυ²/2 = const. Here p is the static pressure, ρυ²/2 the dynamic pressure and ρgh the hydrostatic pressure. In a horizontal pipe (h = const) p + ρυ²/2 = const, so where the speed is high the pressure is low, and where the speed is low the pressure is high. Bernoulli’s equation holds for an ideal (frictionless, incompressible) liquid; in a real liquid friction lowers the pressure further along the flow.

Worked examples
A pipe has a wide section of 12 cm² and a narrow section of 3 cm². If water flows at 0.5 m/s in the wide part, S₁υ₁ = S₂υ₂ gives υ₂ = 12·0.5/3 = 2 m/s in the narrow part.
In a horizontal pipe water speeds up from υ₁ = 1 m/s to υ₂ = 5 m/s (ρ = 1000 kg/m³). The pressure difference is p₁ – p₂ = ρ(υ₂² – υ₁²)/2 = 1000·(25 – 1)/2 = 12 000 Pa, so the pressure in the narrow part is lower by 12 kPa.
Class activity

In the school yard, only with the teacher: pinch the end of a water hose slightly with a finger and watch the jet travel farther; explain it with the continuity equation. Do not do this near electrical equipment.

Practice
1
Write the continuity equation and say what it means.
2
A pipe narrows from 20 cm² to 5 cm². If the speed in the wide part is 0.4 m/s, what is it in the narrow part (m/s)?
3
In a horizontal pipe the speed rises from 2 m/s to 6 m/s (ρ = 1000 kg/m³). By how many Pa does the pressure drop?
4
Why does water flow faster in a narrow part of a river?