In fact, fluid motion is so common – think about breathing, blood circulation or the weather – that exploring is worthwhile.
F IGURE 233 Examples of fluid motion: a vertical water jet striking a horizontal impactor, two jets of a glycerol–water mixture colliding at an oblique angle, a water jet impinging on a reservoir (all © John Bush, MIT) and a dripping water tap (© Andrew Davidhazy).
The state of a fluid
To describe motion means to describe the state of a system. For most fluids, the state at every point in space is described by composition, velocity, temperature and pressure. We will explore temperature below. We thus have one new observable: The pressure at a point in a fluid is the force per area that a body of negligible size feels at that point. Pressure is measured with the help of barometers or similar instruments. The unit of pressure is the pascal: 1 Pa is 1N/m2.
Pressure is not a simple property. Can you explain the observations of Figure 235?
F IGURE 235 The hydrostatic and the hydrodynamic paradox (© IFE).
If the hydrostatic paradox – an effect of the so-called communicating vases – would not be valid, it would be easy to make perpetuum mobiles. Can you think about an example?
Another puzzle about pressure is given in Figure 236.
FIGURE 236 A puzzle: Challenge 545 s Which method of emptying a container is fastest? Does the method at the right hand side work at all?
Air has a considerable pressure, of the order of 100 kPa. As a result, it is not easy to make a vacuum; indeed, everyday forces are often too weak to overcome air pressure. This is known since several centuries, as Figure 237 shows. Your favorite physics laboratory should posess a vacuum pump and a pair of (smaller) Magdeburg hemispheres; enjoy performing the experiment yourself.
F IGURE 237 The pressure of air leads to surprisingly large forces, especially for large objects that enclose a vacuum. This was regularly demonstrated in the years from 1654 onwards by Otto von Guericke with the help of his so-called Magdeburg hemispheres and, above all, the various vacuum pumps that he invented (© Deutsche Post, Otto-von-Guericke-Gesellschaft, Deutsche Fotothek).
Laminar and turbulent flow
Like all motion, fluid motion obeys energy conservation. In the case that no energy is transformed into heat, the conservation of energy is particularly simple. Motion that does not generate heat is motion without vortices; such fluid motion is called laminar. If, in addition, the speed of the fluid does not depend on time at all positions, it is called stationary.
For motion that is both laminar and stationary, energy conservation can be expressed with the help of speed v and pressure p:
½ pv2 + p + qgh = const
where h is the height above ground. This is called Bernoulli’s equation. *
* Daniel Bernoulli (b. 1700 Bale, d. 1782 Bale), important mathematician and physicist. His father Johann and his uncle Jakob were famous mathematicians, as were his brothers and some of his nephews. Daniel Bernoulli published many mathematical and physical results. In physics, he studied the separation of compound motion into translation and rotation. In 1738 he published the Hydrodynamique, in which he deduced all results from a single principle, namely the conservation of energy.The so-called Bernoulli equation states that (and how) the pressure of a fluid decreases when its speed increases. He studied the tides and many complex mechanical problems, and explained the Boyle–Mariotte gas ‘law’. For his publications he won the prestigious prize of the French Academy of Sciences – a forerunner of the Nobel Prize – ten times.
F IGURE 234 Daniel Bernoulli (1700–1782)
In this equation, the last term is only important if the fluid rises against ground. The first term is the kinetic energy (per volume) of the fluid, and the other two terms are potential energies (per volume). Indeed, the second term is the potential energy (per volume) resulting from the compression of the fluid.This is due to a second way to define pressure:
Pressure is potential energy per volume
Energy conservation implies that the lower the pressure is, the larger the speed of a fluid becomes.
We can use this relation to measure the speed of a stationary water flow in a tube. We just have to narrow the tube somewhat at one location along the tube, and measure the pressure difference before and at the tube restriction.The speedv far from the constriction is then given as
A device using this method is called a Venturi gauge. If the geometry of a system is kept fixed and the fluid speed is increased – or the relative speed of a body in fluid – at a certain speed we observe a transition: the liquid loses its clarity, the flow is not laminar anymore. We can observe the transition whenever we open a water tap: at a certain speed, the flow changes from laminar to turbulent. At this point, Bernoulli’s equation is not valid any more.
The description of turbulence might be the toughest of all problems in physics. When the young Werner Heisenberg was asked to continue research on turbulence, he refused – rightly so – saying it was too difficult; he turned to something easier and he discovered and developed quantum mechanics instead. Turbulence is such a vast topic, with many of its concepts still not settled, that despite the number and importance of its applications, only now, at the beginning of the twenty-first century, are its secrets beginning to be unravelled.
It is thought that the equations of motion describing fluids, the so called Navier–Stokes equations, are sufficient to understand turbulence.** But the mathematics behind them is mind-boggling. There is even a prize of one million dollars offered by the Clay Mathematics Institute for the completion of certain steps on the way to solving the equations.
** They are named after Claude Navier (b. 1785 Dijon, d. 1836 Paris), important engineer and bridge builder, and Georges Gabriel Stokes (b. 1819 Skreen, d. 1903 Cambridge), important physicist and mathematician.
FIGURE 238 Left: non-stationary and stationary laminar flows; right: an example of turbulent flow (© Martin Thum, Steve Butler).
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