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Fluid dynamics: Class 11 Physics
Fluid pressure, flow line and turbulent flow, the equation of continuity, Bernoulli's equation and its applications, Torricelli's theorem, viscosity, Stokes' law, terminal velocity and drag.
Key points
A fluid is a substance that can flow, so liquids and gases are fluids. Pressure is the force acting normally per unit area, P = F/A, in pascal (1 Pa = 1 N/m²). The pressure at depth h in a liquid of density ρ is P = P₀ + ρgh, where P₀ is the pressure at the surface. Pascal's law says that a pressure change applied to an enclosed fluid is passed on equally in all directions. This is used in the hydraulic jack and brakes.
Fluid dynamics studies fluids in motion. To keep things simple, we first study an ideal fluid. An ideal fluid is incompressible (constant density), non-viscous (no internal friction) and flows steadily without turbulence.
Types of flow:
- Laminar (steady) flow: every particle passing a point follows the same path and has the same velocity as the particles before it. The path is called a flow line, and flow lines never cross.
- Turbulent flow: the motion is irregular, with eddies and whirlpools. It occurs at high speeds or when the fluid meets obstacles.
- The Reynolds number decides the type of flow. A low value means laminar flow and a high value means turbulent flow.
The equation of continuity expresses conservation of mass. For an incompressible fluid flowing through a pipe of cross section area A at speed v, the volume flow rate Q = Av stays the same at every section. So A₁v₁ = A₂v₂. The fluid moves faster where the pipe is narrow and slower where it is wide. This is why water from a hose speeds up when you partly close the opening with your thumb.
Bernoulli's principle follows from conservation of energy. For steady, non-viscous, incompressible flow along a flow line, P + ½ρv² + ρgh = constant. The three terms are the pressure energy, the kinetic energy and the potential energy per unit volume. For a horizontal pipe, P + ½ρv² = constant, so where the speed is high the pressure is low.
Applications of Bernoulli's equation:
- Torricelli's theorem: water leaves a small hole at depth h below the surface with speed v = √(2gh), the same as a freely falling body.
- Venturi meter: a narrow throat in a pipe has a high speed and a low pressure. The difference in pressure measures the flow speed.
- Aerofoil lift: air moves faster over the curved upper surface of a wing, giving lower pressure above and a net upward force.
- Atomizer and spray pump: fast air over a tube lowers the pressure, so liquid rises and is sprayed.
- A roof can be lifted in a strong wind because the pressure outside is lower than the pressure inside.
Viscosity is the internal friction of a fluid, a property that opposes relative motion between its layers. Honey is more viscous than water. The coefficient of viscosity η is defined by F = ηA(Δv/Δx), where A is the area of the layers and Δv/Δx is the velocity gradient. Its SI unit is the pascal second (Pa s). The viscosity of a liquid decreases when it is heated, but the viscosity of a gas increases with temperature.
A body moving through a fluid feels a resisting force called drag. For a small sphere of radius r moving slowly at speed v through a fluid of viscosity η, Stokes' law gives the drag force as F = 6πηrv. At higher speeds the flow becomes turbulent and the drag grows roughly with the square of the speed.
Terminal velocity: when a sphere falls through a fluid, three forces act on it: weight downwards, and buoyancy and drag upwards. As the speed increases, the drag increases, until the net force is zero. Then the sphere falls with a constant velocity, called the terminal velocity. For a sphere of density ρ in a fluid of density σ, v_t = 2r²(ρ − σ)g / 9η. Raindrops reach terminal velocity, so they do not hurt as they would if they fell freely from the clouds.
The buoyant force on a body in a fluid equals the weight of the fluid displaced (Archimedes' principle). A body floats if its average density is less than that of the fluid. Fluid dynamics matters in the design of ships, aeroplanes, pipelines, blood flow in arteries, and the flow of canal water in Pakistan's irrigation system.
Study tip: Remember that Bernoulli's equation is an energy equation, and the continuity equation is a mass equation. Many numericals use both: first find the speed with continuity, then the pressure with Bernoulli. Write the assumptions of an ideal fluid when you state the principle.
Definitions and flashcards
Say the meaning first, then open the card to check.
Ideal fluid
A fluid that is incompressible, non-viscous and steady in flow.
Flow line
The path followed by a fluid particle in steady flow.
Turbulent flow
Irregular flow with eddies.
Equation of continuity
A₁v₁ = A₂v₂, from conservation of mass.
Viscosity
The internal friction between layers of a fluid.
Stokes' law
The drag on a small sphere in a viscous fluid, F = 6πηrv.
Terminal velocity
The constant velocity reached when drag and buoyancy balance the weight.
Venturi meter
A device that measures flow speed from the pressure drop in a narrow section.
Formula sheet
| Name | Formula | What it means |
|---|---|---|
| Pressure at depth | P = P₀ + ρgh | ρ is the density of the liquid. |
| Continuity | A₁v₁ = A₂v₂ | Volume flow rate Q = Av. |
| Bernoulli | P + ½ρv² + ρgh = constant | Along a flow line. |
| Torricelli | v = √(2gh) | Speed of outflow at depth h. |
| Stokes' law | F = 6πηrv | Drag on a small sphere. |
| Terminal velocity | v_t = 2r²(ρ − σ)g / 9η | ρ is the density of the sphere and σ of the fluid. |
Short questions with model answers
These are the kind of questions asked for 2, 3 and 5 marks. Try each one before opening the answer.
State the equation of continuity and explain it. (3 marks)
For an incompressible fluid in steady flow, the product of the cross section area and the speed is the same at all points, A₁v₁ = A₂v₂. This follows from conservation of mass: the mass entering a section in a given time equals the mass leaving. Hence the fluid moves faster in a narrow part and slower in a wide part.
State Bernoulli's principle. (2 marks)
For a steady, non-viscous and incompressible flow, the sum P + ½ρv² + ρgh is constant along a flow line. In horizontal flow, a rise in speed is matched by a fall in pressure.
How does an aeroplane wing produce lift? (3 marks)
The wing is curved on top and flatter below, and it is tilted slightly into the air stream. Air moves faster over the upper surface than the lower surface. By Bernoulli's principle, the pressure above is lower than the pressure below, so there is a net upward force called lift.
Derive Torricelli's theorem. (5 marks)
Consider a large tank with a small hole at depth h below the water surface. Both the surface and the hole are at atmospheric pressure. Applying Bernoulli's equation between the surface (speed almost zero) and the hole (speed v): P₀ + ρgh = P₀ + ½ρv². The terms P₀ cancel, so ρgh = ½ρv², giving v = √(2gh). This is the speed that a body would gain by falling freely through the height h. The speed does not depend on the density of the liquid.
Define coefficient of viscosity and give its unit. (2 marks)
The coefficient of viscosity η is the tangential force per unit area needed to maintain a unit velocity gradient between layers of a fluid, η = F/(A Δv/Δx). Its SI unit is the pascal second (Pa s).
Why does a small steel ball falling through glycerine reach a constant speed? (3 marks)
At first the ball speeds up. As the speed grows, the viscous drag, 6πηrv, grows with it. The buoyant force is constant. When drag plus buoyancy equals the weight, the net force is zero, so the acceleration becomes zero and the ball falls at a constant terminal velocity.
Numericals with worked steps
Water flows at 2 m/s through a pipe of cross section 4 cm². The pipe narrows to 1 cm². Find the speed in the narrow part.
- A₁v₁ = A₂v₂.
- 4 × 2 = 1 × v₂.
- v₂ = 8 m/s.
Answer: 8 m/s.
A tank has a small hole 5 m below the water surface. Find the speed of water leaving it. Take g = 9.8 m/s².
- v = √(2gh).
- v = √(2 × 9.8 × 5) = √98 = 9.90 m/s.
Answer: 9.9 m/s.
In a horizontal pipe, water speed is 2 m/s in the wide part and 6 m/s in the narrow part. Find the pressure difference. Density of water is 1000 kg/m³.
- P₁ + ½ρv₁² = P₂ + ½ρv₂².
- P₁ − P₂ = ½ρ(v₂² − v₁²) = ½ × 1000 × (36 − 4).
- = 500 × 32 = 16000 Pa.
Answer: 1.6 × 10⁴ Pa (higher in the wide part).
A sphere of radius 1 mm moves at 0.05 m/s through oil of viscosity 0.2 Pa s. Find the drag force.
- F = 6πηrv.
- r = 1 mm = 1 × 10⁻³ m.
- F = 6 × 3.14 × 0.2 × 1 × 10⁻³ × 0.05 = 18.85 × 10⁻⁵.
- = 1.885 × 10⁻⁴ N.
Answer: 1.9 × 10⁻⁴ N.
Water flows at 3 m/s through a pipe of radius 2 cm. Find the volume flow rate.
- r = 2 cm = 0.02 m.
- A = πr² = 3.14 × 0.0004 = 1.257 × 10⁻³ m².
- Q = Av = 1.257 × 10⁻³ × 3 = 3.77 × 10⁻³ m³/s.
Answer: 3.8 × 10⁻³ m³/s (3.8 litres per second).
Common mistakes
- Using the diameter instead of the radius when finding the pipe area.
- Forgetting to convert cm² to m² or cm to m before using SI formulas.
- Believing that fast flowing fluid has high pressure. Bernoulli's principle says the opposite along a horizontal flow line.
- Applying Bernoulli's equation to viscous or compressible flow without stating that it is an approximation.
- Confusing the density of the sphere with the density of the fluid in the terminal velocity formula.