Streamline and turbulent flow
A moving fluid is described by flow lines. In streamline (laminar) flow each part of the fluid follows a smooth path without crossing others. When the speed is too high or the passage too rough, the flow becomes turbulent and swirls in eddies. An ideal fluid is incompressible and has no viscosity.
The equation of continuity and Bernoulli’s principle
If an ideal fluid flows through a pipe, the volume passing each section every second is the same. So where the pipe is narrower the fluid moves faster.
Choose the speed of the water in the wide pipe and the pipe areas. Find the speed in the narrow section and the drop in pressure there. (ρ = 1 000 kg/m³)
| Application | Bernoulli idea |
|---|---|
| Aeroplane wing | Air moves faster over the curved top, so the pressure there is lower: lift |
| Spray gun and carburettor | Fast air in a narrow section has low pressure and draws up liquid |
| Bunsen burner | Fast gas jet draws in air |
| Two ships close together | Fast water between them has lower pressure and pushes the ships together |
| Curved ball | A spinning ball drags air round and curves |
Viscosity and Stokes’ law
Viscosity is the internal friction of a fluid that resists the relative motion of its layers. Honey is more viscous than water. The viscosity η (eta) falls as a liquid is heated and rises for gases. A small sphere moving slowly through a viscous fluid experiences a drag force given by Stokes’ law.
A sphere falling through a fluid is acted on by weight (down), upthrust (up) and viscous drag (up). The drag grows with speed until the forces balance and the sphere falls at a constant terminal velocity.
Choose the radius of a steel ball and the viscosity of the oil it falls through. (Steel ρ = 7 800 kg/m³, oil ρ = 900 kg/m³, g = 9.8 m/s²)
Laminar flow in pipes
The flow of a viscous fluid through a narrow tube depends strongly on the radius of the tube. Poiseuille’s equation: the volume flow rate is Q = πPr⁴ ÷ (8ηL), where P is the pressure difference and L the length. Halving the radius reduces the flow sixteen times. This is why narrowed arteries seriously reduce blood flow.
Key terms
- Streamline flow
- Smooth flow in which fluid layers do not mix.
- Viscosity
- The internal friction of a fluid, resisting relative motion of its layers.
- Terminal velocity
- The constant velocity reached when drag balances the driving force.
- Bernoulli's principle
- In a steady flow, an increase in speed is accompanied by a decrease in pressure.
Summary
- Continuity: A₁v₁ = A₂v₂; Bernoulli: P + ½ρv² + ρgh = constant.
- Faster fluid, lower pressure: lift, spray guns.
- Stokes' law F = 6πηrv; terminal velocity v = 2r²(ρₛ − ρ_f)g ÷ 9η.
- Poiseuille: flow in a tube ∝ r⁴.