On a winter morning, cooking oil pours from the bottle slowly and clings to the spoon. Poured into a hot kadhai, the same oil runs thin and spreads across the pan at once. Ghee goes further: solid in the tin, it flows like water in the pan. Heat clearly makes these liquids less viscous.
Heat thins a liquid
In a liquid the molecules are packed close together, and each one attracts its neighbours. For one layer to slide over another, molecules must keep breaking away from their neighbours and joining new ones. That attraction is the main source of a liquid's viscosity. Heat makes the molecules jiggle faster, so they break free more easily. The layers slide with less resistance, and the viscosity falls.
The fall is large, as Fig 2.1 shows for water. At \(0\,°\text{C}\) water has \(\eta \approx 1.8 \times 10^{-3}\,\text{Pa s}\). At \(20\,°\text{C}\) it has \(1.0 \times 10^{-3}\), and at \(100\,°\text{C}\) only \(0.28 \times 10^{-3}\). Oils change even more. This is why the viscosity of a liquid is always quoted with its temperature.
Heat thickens a gas
A gas behaves the other way. Its molecules are far apart and hardly attract each other. Its viscosity comes from a different cause. Molecules keep darting from one layer to the next. A molecule from a fast layer carries its extra forward momentum into a slower layer and speeds it up. A molecule from a slow layer drifts into a fast layer and holds it back.
Heat a gas, and its molecules move faster. They cross between layers more often, and each carries its momentum across more quickly. So the layers drag on each other harder, and the viscosity rises. For air, Fig 2.1 shows about \(1.7 \times 10^{-5}\,\text{Pa s}\) at \(0\,°\text{C}\) and about \(2.2 \times 10^{-5}\) at \(100\,°\text{C}\).
A bottle of glycerine and a cylinder of nitrogen gas are both warmed from \(20\,°\text{C}\) to \(60\,°\text{C}\). What happens to their viscosities?
Explain in two or three sentences why air becomes more viscous when it is heated.
In a gas, viscosity comes from molecules crossing between layers that move at different speeds. A molecule from a faster layer carries its extra momentum into a slower layer, and one from a slower layer holds a faster layer back. Heating makes the molecules move faster, so they cross more often and transfer momentum more quickly, and the layers drag on each other more strongly.
An auto-rickshaw engine starts cold in the morning and runs at over \(100\,°\text{C}\) in the afternoon traffic. Its oil must be thin enough to reach every part at the cold start. It must also stay thick enough to keep a film between the hot moving parts. A plain oil fails one test or the other. So engines use a multigrade oil, marked for example 10W-30. It carries long chain molecules that stay coiled in cold oil and uncoil when hot, holding the viscosity up as the oil warms.
The layers in a pipe
In a pipe, the layers are thin cylinders, one inside the next, like the layers of an onion. The layer touching the wall sticks to it and is at rest. The next cylinder in slides past it and moves a little faster. Each cylinder is held back by the slower one outside it and pulled forward by the faster one inside. So the speed grows from zero at the wall to its largest value, \(v_{max}\), on the axis.
The dye in Fig 2.2 records the speeds. After a while, each part of the dye line has moved in proportion to the speed of its layer. At the moment it passes the tips of the arrows, the dye has the shape of the velocity profile. That shape is a parabola, and Lesson 5 derives it from Newton's law of viscosity. The speed at a distance \(s\) from the axis of a pipe of radius \(R\) is \[ v = v_{max}\left(1 - \frac{s^2}{R^2}\right) \]
The slope of the profile is the velocity gradient. The parabola is flat at the axis, so the gradient there is zero. It is steepest at the wall. By Newton's law the viscous stress \(\eta\,dv/ds\) is therefore largest at the wall, where the pipe itself holds the liquid back. Averaged over the whole cross-section, the liquid moves at exactly half of \(v_{max}\).
Water flows steadily through a pipe in streamline flow. Which statements are true?
Water flows through a pipe with a speed of \(0.2\,\text{m/s}\) on the axis. What is its speed halfway between the axis and the wall, in m/s?
When the layers break up
The chapter on Fluid Flow met two kinds of flow. At low speed the flow is smooth, the streamline flow of this lesson. At high speed the layers break up into swirls and eddies: a turbulent flow. What decides which one we get? The liquid's inertia tends to carry each bit of it onward, and any small wobble grows. Viscosity drags neighbouring layers together and damps the wobbles out. Turbulence wins when inertia is strong compared with viscosity.
The ratio of the two is a pure number, the Reynolds number: \[ R_e = \frac{\rho v D}{\eta} \] Here \(\rho\) is the density of the fluid, \(v\) its mean speed and \(D\) the diameter of the pipe. A fast flow of a dense, runny fluid in a wide pipe has a large \(R_e\). A slow flow of a sticky fluid in a narrow pipe has a small one.
For flow in a pipe, experiments give rough limits. Below about \(2000\) the flow stays streamline. Above about \(3000\) it is turbulent. In between it can switch from one to the other. The limits are not sharp: a very smooth pipe and a still inlet keep the flow steady for longer. Some books quote \(1000\) and \(2000\) as the limits instead.
Water, of density \(1000\,\text{kg/m}^3\) and viscosity \(10^{-3}\,\text{Pa s}\), flows at \(0.1\,\text{m/s}\) through a pipe of diameter \(2\,\text{cm}\). (a) Find the Reynolds number. Is the flow streamline? (b) Taking turbulence to start at \(R_e = 3000\), find the speed at which it starts.
(a) \(R_e = 1000 \times 0.1 \times 0.02 / 10^{-3} = 2000\). This is at the upper limit of streamline flow, so the flow is just streamline.
(b) \(R_e\) is proportional to \(v\), so \(v = 0.1 \times 3000/2000 = 0.15\,\text{m/s}\).
Water (\(1000\,\text{kg/m}^3\), \(10^{-3}\,\text{Pa s}\)) flows through a pipe of diameter \(1\,\text{cm}\). Taking \(R_e = 2000\) as the limit of streamline flow, what is the greatest speed for streamline flow, in m/s?
Honey (\(1400\,\text{kg/m}^3\), \(\eta = 5\,\text{Pa s}\)) flows at \(0.1\,\text{m/s}\) through a pipe \(2\,\text{cm}\) across. Its flow is
A kitchen tap is fully open. Its water leaves a pipe about \(1.5\,\text{cm}\) across at about \(1\,\text{m/s}\). Estimate the Reynolds number of the flow in the pipe.
In 1860, James Clerk Maxwell worked out the viscosity of a gas from molecules crossing between layers. His result looked wrong: the viscosity should not depend on the pressure at all. Squeeze the gas, and more molecules cross, but each carries its momentum a shorter way. The two effects cancel. Maxwell did not believe it until he tested it. In 1866 he timed the swing of discs in air at many pressures, and the damping stayed the same.
- Saying that heating lowers the viscosity of every fluid.Heating lowers a liquid's viscosity but raises a gas's, because the two have different causes.
- Taking the velocity gradient in a pipe to be largest where the liquid is fastest.The gradient is the slope of the profile. It is zero on the axis and largest at the wall.
- Using the radius of the pipe in the Reynolds number.The Reynolds number of a pipe uses its diameter, \(D = 2R\).