Open the kitchen tap a little. The water comes out as a clear, smooth rod that looks almost frozen, like a glass stick. Now open the tap fully. The stream turns rough and white, it wobbles, and it splashes noisily into the sink. The same water, from the same tap, flows in two quite different ways.
Two ways to watch a moving fluid
A body in Kinematics was one particle, and we followed it. A fluid in motion has a huge number of particles, each moving in its own way. We can still follow one small drop of water as it moves. We can also stand at one fixed point and watch the velocity of whichever drop passes that point. The second view is often simpler.
Steady flow
Look at a fixed point in the stream from a tap turned a little. Every drop that passes the point has the same velocity as the drop before it. The same is true at every other point, though the velocity may differ from point to point. Such a flow is called steady flow, or streamline flow. The pattern of the flow does not change with time, even though the water itself keeps moving.
A flow is steady when the velocity of the fluid at every fixed point stays the same in magnitude and direction as time passes. Every particle that passes a given point then follows the same path as the particles before it.
Steady does not mean that a drop keeps a constant velocity. A drop may speed up as it moves into a narrow part of a pipe, and slow down again later. What stays fixed is the velocity at each place. The next drop to arrive at that place will speed up in exactly the same way.
Turbulent flow
Now open the tap fully. The velocity at a fixed point jumps about from instant to instant. Small whirls, called eddies, form, grow and break up. Two drops that pass the same point a moment apart follow different paths. This is turbulent flow. It is irregular, it mixes the fluid, and it wastes energy as heat and sound.
Watch the two chips of Fig 1.1. In the slow flow the dye traces one straight line, and the line stays put while the water flows along it. That line is a picture of the flow pattern, and the pattern does not change. In the fast flow the thread breaks up a little way down the pipe, and the pattern is different in every frame. A flow changes from steady to turbulent when its speed passes a critical speed.
The critical speed is not the same for every flow. It is lower in a wider pipe, and lower for a fluid that is less sticky, such as water compared with honey. The chapter on Viscosity puts a number to this, the Reynolds number. Here it is enough to know that slow, smooth flows are steady and fast ones in wide pipes tend to be turbulent.
At Manchester, Osborne Reynolds let water from a large tank flow out through a long glass tube. A fine nozzle fed a thread of coloured water into the entrance. At low speeds the colour stretched through the tube as a straight line. As he opened the valve further, the thread suddenly began to waver and then mixed through the whole tube. He showed that the change depends on the speed, the width of the tube and the fluid, taken together.
Water flows steadily through a pipe that narrows. What does "steady" tell you?
A slow, smooth flow of water through a pipe is made faster and faster. What happens?
Streamlines
In steady flow, follow one particle of fluid. Its path is a streamline. Every particle that later passes through any point of that path follows the same path. As for any moving body in the chapter on Motion in Two Dimensions, the velocity is along the tangent to the path. So the velocity of the fluid at every point is along the streamline through that point.
A streamline is a curve whose tangent at every point gives the direction of the velocity of the fluid at that point. In steady flow it is the path followed by every particle that passes through it.
Two streamlines can never cross. Suppose they did. A particle arriving at the crossing point would have two directions to choose from, one along each streamline. But the velocity at a point in steady flow has only one direction. So a crossing is impossible.
A tube of flow
Take a small closed loop across the flow, and draw the streamline through every point of it. These streamlines form a tube of flow. The fluid inside the tube moves along the streamlines, so it never crosses the side of the tube. A tube of flow behaves like a pipe with invisible walls. Whatever enters at one end must leave at the other.
Notice two things in Fig 1.2. First, the streamlines crowd together where the pipe narrows. Second, the tracers move faster there. Each band between two streamlines carries the same share of water every second. A narrow band must therefore carry its share faster. So crowded streamlines mean fast flow, and the next lesson turns this into an equation.
Why can two streamlines of a steady flow never cross?
In a picture of a steady flow, the streamlines are drawn so that each band between neighbours carries the same volume of fluid each second. Explain why the flow is fastest where the streamlines are closest together.
No fluid crosses a streamline, so each band is a tube of flow and carries the same volume each second all along its length. Where the band is narrow, the same volume must pass through a smaller cross-section in the same time. That can happen only if the fluid moves faster there. So closely spaced streamlines mean a large speed, and widely spaced ones a small speed.
The ideal fluid
Real fluids are complicated. So this chapter works with an ideal fluid, and the lessons that follow say when it is a good model. An ideal fluid has four properties. Each one drops an effect that real fluids show, and each is close to the truth for water flowing gently.
- It is incompressible: its density is the same everywhere and at all times.
- It is non-viscous: there is no internal friction between layers that slide past each other, so no energy becomes heat.
- Its flow is steady: the velocity at each point does not change with time.
- Its flow is irrotational: a small element of fluid moves along without spinning about its own centre.
Water is very nearly incompressible. From the chapter on Stress, Strain and Hooke's Law, its bulk modulus is \(2 \times 10^9\,\text{Pa}\). An extra pressure of \(10^5\,\text{Pa}\) squeezes it by only one part in \(20\,000\). Gases are easy to squeeze, but even air behaves as incompressible when it flows well below the speed of sound.
The ideal fluid is a model, not a claim about water. Honey pouring from a spoon is strongly viscous, and the smoke of a flame is turbulent. Use the results of this chapter only where the flow is smooth and friction is small. Expect real answers to come out a little lower.
Which of these are assumptions of the ideal-fluid model?
Water has a bulk modulus of \(2 \times 10^9\,\text{Pa}\). By what fraction does its volume shrink when the pressure on it rises by \(4 \times 10^5\,\text{Pa}\), the pressure in a city water main? Give the answer as a pure number.
Light an agarbatti in a still room and watch its smoke. For the first few centimetres the smoke rises as a thin, smooth ribbon: a streamline you can see. Higher up, the rising smoke has gathered speed and width, and it suddenly breaks into curls and whirls that change every moment. The same smoke shows steady flow below and turbulent flow above.
Now return to the tap at the start of the lesson. Opened a little, the water flows slowly, below its critical speed. Every drop follows the path of the drop before, so the stream is steady and looks like a still glass rod. Opened fully, the water passes the critical speed, eddies form, and the stream turns rough and white. This chapter's equations describe the first stream.
- Thinking that steady flow means every drop moves at a constant velocity.Steady flow fixes the velocity at each point. A drop can still speed up as it enters a narrow part.
- Drawing two streamlines that cross or touch.A particle at a crossing would need two velocities. Streamlines of a steady flow never meet.
- Using the equations of ideal flow for honey, smoke or a fast turbulent jet.The ideal fluid is a model. It suits smooth flow with little friction.