Viscosity & Flow Through Tubes
Internal friction: why honey pours slowly and pipes need pressure.
Lesson at a glanceShow the key points
- Layers of a fluid moving relative to each other exert tangential forces opposing the slip: viscosity.
- F = −ηA dv/dz (14.19): proportional to area and velocity gradient; η is the coefficient of viscosity, SI unit N·s/m² (1 poise = 0.1 N·s/m²), dimensions ML⁻¹T⁻¹ (14.20).
- Viscosity of liquids falls sharply with temperature; that of gases rises.
- Poiseuille's equation: volume flow rate V/t = πPr⁴/(8ηl) (14.21), found up to a constant by dimensions.
- Reynolds number ρvD/η (14.24): below about 2000 flow is steady, above about 3000 turbulent, in between unstable. The largest speed for steady flow is the critical velocity.
Pour water and honey side by side. Both are liquids, but honey crawls. The difference is internal friction between layers of the fluid: viscosity.
Internal friction
When one layer of a fluid slides (or tries to slide) over another, the two layers exert tangential forces on each other that oppose the relative motion. This property is viscosity, and the forces are viscous forces.
A solid surface moving through a fluid drags the layer touching it along. A boat moving slowly on a calm river carries the water in contact with it, while the water at the river bed stays still; layers in between move at intermediate speeds. If the speed changes by dv across a distance dz perpendicular to the flow, is the velocity gradient.
The SI unit of η is N·s/m². The CGS unit, dyne·s/cm², is called the poise (after Poiseuille): 1 poise = 0.1 N·s/m². Its dimensions:
A flat plate of area 0.50 m² slides at 0.20 m/s over a 1.0 mm layer of glycerine (η = 1.5 N·s/m²) on a fixed surface. What force keeps it moving?
| Temperature (°C) | Castor oil (poise) | Water (centipoise) | Air (micropoise) |
|---|---|---|---|
| 0 | 53 | 1.79 | 171 |
| 20 | 9.86 | 1.01 | 181 |
| 40 | 2.31 | 0.66 | 190 |
| 60 | 0.80 | 0.47 | 200 |
| 100 | 0.17 | 0.28 | 218 |
Flow through a narrow tube: Poiseuille’s equation
In steady flow through a tube, the fluid touching the wall is at rest and the centre moves fastest. Poiseuille found how the volume flowing per second depends on the pressure difference P across a tube of length l and radius r. Assume the rate depends on the pressure gradient P/l, on η and on r as a product of powers:
Matching M, L, T: a + b = 0, −2a − b + c = 3, −2a − b = −1, so a = 1, b = −1, c = 4. With the full theory giving k = π/8:
Deposits reduce the inner radius of a blood vessel by half. If the pressure difference stays the same, by what factor does the blood flow fall?
Critical velocity and the Reynolds number
At low speeds the flow through a tube is steady; above a certain speed it becomes turbulent. The largest speed that still allows steady flow is the critical velocity. Whether flow is steady depends on the fluid’s density ρ, speed v and viscosity η and on the tube diameter D, all combined in a dimensionless number:
Water (η = 1.0 × 10⁻³ N·s/m²) flows through a 2.0 cm diameter pipe. Classify the flow at 0.10 m/s and at 0.50 m/s.
- 1Viscosity: internal friction between fluid layers; ; [η] = ML⁻¹T⁻¹; 1 poise = 0.1 N·s/m².
- 2Liquid viscosity falls steeply with temperature; gas viscosity rises.
- 3Poiseuille: .
- 4Reynolds number : < 2000 steady, > 3000 turbulent.
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