Tip a spoon of water over a cup, and the water is gone in a moment. Tip a spoon of honey, and it pours as a slow, thick rope. A thin coat of honey stays on the spoon for a long time. Both are liquids, and both flow under the smallest push. Yet honey resists the flow far more than water.
A liquid flows in layers
Think of a liquid flowing over a fixed surface as a stack of very thin layers, like the pages of a book. The layer that touches the surface sticks to it and does not move at all. The next layer slides over this resting layer, and the layer above slides over that one. Each layer moves a little faster than the one below it.
Where two layers slide past each other, each drags on the other. The faster layer pulls the slower one forward, and the slower layer holds the faster one back. This friction between the layers of a fluid is called viscosity. It is like the kinetic friction between two sliding solids, with one difference: it acts everywhere inside the fluid, not only at a surface.
Watch the coloured column in Fig 1.1. At the start it stands upright. The dot nearest the bottom hardly moves, because the layer touching the fixed surface sticks to it. The dot at the top moves almost as fast as the plate, because the top layer sticks to the plate. The column tilts into a straight sloping line. The layer halfway up moves at \(0.25\,\text{m/s}\), half the speed of the plate.
The velocity gradient
Let \(y\) be the height above the fixed surface. The speed \(v\) of a layer changes with \(y\). The rate of that change, \(dv/dy\), is called the velocity gradient. It says how quickly the speed changes as you move across the flow, from layer to layer. In the film of Fig 1.1 the speed grows steadily, so the gradient is the same everywhere: the change in speed divided by the thickness. \[ \frac{dv}{dy} = \frac{0.5\,\text{m/s}}{10^{-3}\,\text{m}} = 500\,\text{s}^{-1} \] The unit is metres per second per metre, which is \(\text{s}^{-1}\).
In a flowing fluid, the velocity gradient is the rate \(dv/dy\) at which the speed of the layers changes with distance \(y\). The distance is measured at right angles to the flow, and the unit is \(\text{s}^{-1}\).
A glass plate slides at \(0.3\,\text{m/s}\) over a film of glycerine \(2\,\text{mm}\) thick on a fixed table. The speed grows steadily across the film. What is the velocity gradient in the film, in \(\text{s}^{-1}\)?
Newton's law of viscosity
How hard must you pull to keep the plate of Fig 1.1 moving? Experiments with plates and films give two results. Double the area of the plate, and the force doubles. Double the velocity gradient, by moving the plate twice as fast or by halving the film, and the force doubles again. The force is proportional to both, and the constant of proportion depends only on the fluid.
The tangential force between two layers of a fluid in streamline flow is proportional to the area of the layers and to the velocity gradient. In symbols, \(F = \eta A\,dv/dy\), where the constant \(\eta\) is the coefficient of viscosity of the fluid.
The constant \(\eta\) (the Greek letter eta) is the coefficient of viscosity, often simply called the viscosity. A large \(\eta\) means a large force for the same gradient: a thick, sticky fluid such as honey. A small \(\eta\) means a thin, runny fluid such as water. The force per unit area, \(F/A\), is a tangential stress, and it equals \(\eta\,dv/dy\).
The plate of Fig 1.1 has an area of \(0.1\,\text{m}^2\). It slides at \(0.5\,\text{m/s}\) on a film of oil \(1\,\text{mm}\) thick, with \(\eta = 0.2\,\text{Pa s}\). Find the force needed to keep it moving at a steady speed. Find the force again if the oil is replaced by water, with \(\eta = 10^{-3}\,\text{Pa s}\). What force does the oil exert on the fixed surface?
(a) The gradient is \(500\,\text{s}^{-1}\), so \(F = \eta A\,dv/dy = 0.2 \times 0.1 \times 500 = 10\,\text{N}\). At steady speed the pull balances the viscous force of the oil.
(b) With water, \(F = 10^{-3} \times 0.1 \times 500 = 0.05\,\text{N}\), two hundred times less.
(c) The oil drags the fixed surface forward with \(10\,\text{N}\) as well. Each layer passes the force on to the next, and the surface holds the film back.
Play with the sim above. Halve the thickness of the film, and the dashed line through the arrow tips grows flatter. The same change in speed now happens over half the distance. The gradient doubles, and so does the force. The speed of the plate does the same thing from the other side. On honey the same plate needs \(250\,\text{N}\), five thousand times the pull on water.
The plate of the example is moved at the same speed on a film of the same oil that is twice as thick. The force needed to keep it moving
The same plate is moved at the same speed over films of the same thickness of two liquids, \(A\) and \(B\). Liquid \(B\) needs four times the force. Find \(\eta_B/\eta_A\).
A solid and a fluid under shear
The chapter on Stress, Strain and Hooke's Law met a shearing stress on a solid. A rubber block glued to a table and pushed along its top face tilts by an angle, and then stops. A fluid behaves differently. Push the top of a liquid sideways and it never stops moving. A solid resists how far it is sheared; a fluid resists how fast.
The unit of viscosity
Rearrange Newton's law as \(\eta = F/(A\,dv/dy)\). The unit is newtons divided by square metres and by \(\text{s}^{-1}\): \(\text{N s/m}^2\). A newton per square metre is a pascal, so the SI unit of viscosity is the pascal second, \(\text{Pa s}\). The older CGS unit, the poise, is a tenth of this: \(1\,\text{poise} = 0.1\,\text{Pa s}\). Water at room temperature has \(\eta = 0.01\,\text{poise}\), which is \(10^{-3}\,\text{Pa s}\).
The dimensions follow in the same way. Force is \([\text{MLT}^{-2}]\), area \([\text{L}^2]\) and the gradient \([\text{T}^{-1}]\). \[ [\eta] = \frac{[\text{MLT}^{-2}]}{[\text{L}^2][\text{T}^{-1}]} = [\text{ML}^{-1}\text{T}^{-1}] \] Lesson 3 uses these dimensions to find the drag on a falling ball.
Some viscosities near room temperature, rounded:
- air, \(2 \times 10^{-5}\,\text{Pa s}\) (the value this chapter uses; nearer \(1.8 \times 10^{-5}\) at \(20\,°\text{C}\))
- water, \(10^{-3}\,\text{Pa s}\); blood, about \(3 \times 10^{-3}\,\text{Pa s}\)
- olive oil, about \(0.08\,\text{Pa s}\); castor oil, about \(1\,\text{Pa s}\)
- glycerine, about \(1.4\,\text{Pa s}\); honey, from \(2\) to \(10\,\text{Pa s}\)
Honey is a few thousand times as viscous as water, and air is fifty times less viscous than water.
The viscosity of an oil is \(2\,\text{poise}\). In SI units it is
Which quantity has the same dimensions as the coefficient of viscosity?
The motor of a ceiling fan turns on a steel shaft that runs in a bronze sleeve. A film of oil fills the thin gap between them, so metal never rubs on metal. Take a shaft of radius \(5\,\text{mm}\) in a sleeve \(2\,\text{cm}\) long, and a film \(0.05\,\text{mm}\) thick of oil with \(\eta = 0.1\,\text{Pa s}\). At \(300\) turns a minute the shaft's surface moves at about \(0.16\,\text{m/s}\). Newton's law gives a drag of about \(0.2\,\text{N}\), and the fan loses only about \(0.03\,\text{W}\) to it. A dry bearing would grind itself hot. When the oil dries up after years, the fan starts to hum and slows down.
In his Principia, Isaac Newton studied how a fluid resists a body turning inside it. He made a bold guess about the fluid itself. The resistance arising from a want of slipperiness between the parts of a fluid, he wrote, is proportional to the speed with which they separate. That guess is the law of this lesson. Fluids that obey it, like water, oil and air, are still called Newtonian fluids.
- Taking the viscous force to be proportional to the speed of the plate alone.It is proportional to the velocity gradient, the speed divided by the thickness of the film. A thinner film needs more force.
- Thinking the layer of liquid touching a wall slides along it.The layer touching a surface moves with that surface. Against a fixed wall it is at rest.
- Converting poise to pascal seconds by multiplying by ten.One poise is 0.1 Pa s. Water is 0.01 poise, or \(10^{-3}\,\text{Pa s}\).