where L is a characteristic length of the object (a sphere’s diameter, for example), ρ the fluid density, η its viscosity, and v the object’s speed in the fluid. If N′R is less than about 1, flow around the object can be laminar, particularly if the object has a smooth shape. The transition to turbulent flow occurs for N′R between 1 and about 10, depending on surface roughness and so on. Depending on the surface, there can be a turbulent wake behind the object with some laminar flow over its surface. For an N′R between 10 and 106, the flow may be either laminar or turbulent and may oscillate between the two. For N′R greater than about 106, the flow is entirely turbulent, even at the surface of the object. (See Figure 1.) Laminar flow occurs mostly when the objects in the fluid are small, such as raindrops, pollen, and blood cells in plasma.
Calculate the Reynolds number N′R for a ball with a 7.40-cm diameter thrown at 40.0 m/s.
We can use
Substituting values into the equation for N′R yields
This value is sufficiently high to imply a turbulent wake. Most large objects, such as airplanes and sailboats, create significant turbulence as they move. As noted before, the Bernoulli principle gives only qualitatively-correct results in such situations.
One of the consequences of viscosity is a resistance force called viscous drag FV that is exerted on a moving object. This force typically depends on the object’s speed (in contrast with simple friction). Experiments have shown that for laminar flow (N′R less than about one) viscous drag is proportional to speed, whereas for N′R between about 10 and 106, viscous drag is proportional to speed squared. (This relationship is a strong dependence and is pertinent to bicycle racing, where even a small headwind causes significantly increased drag on the racer. Cyclists take turns being the leader in the pack for this reason.) For N′R greater than 106, drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, FV is proportional to fluid viscosity η, the object’s characteristic size L, and its speed v. All of which makes sense—the more viscous the fluid and the larger the object, the more drag we expect. Recall Stoke’s law FS = 6πrηv. For the special case of a small sphere of radius R moving slowly in a fluid of viscosity η, the drag force FS is given by
FS = 6πRηv.
An interesting consequence of the increase in FV with speed is that an object falling through a fluid will not continue to accelerate indefinitely (as it would if we neglect air resistance, for example). Instead, viscous drag increases, slowing acceleration, until a critical speed, called the terminal speed, is reached and the acceleration of the object becomes zero. Once this happens, the object continues to fall at constant speed (the terminal speed). This is the case for particles of sand falling in the ocean, cells falling in a centrifuge, and sky divers falling through the air. Figure 2 shows some of the factors that affect terminal speed. There is a viscous drag on the object that depends on the viscosity of the fluid and the size of the object. But there is also a buoyant force that depends on the density of the object relative to the fluid. Terminal speed will be greatest for low-viscosity fluids and objects with high densities and small sizes. Thus a skydiver falls more slowly with outspread limbs than when they are in a pike position—head first with hands at their side and legs together.
Knowledge of terminal speed is useful for estimating sedimentation rates of small particles. We know from watching mud settle out of dirty water that sedimentation is usually a slow process. Centrifuges are used to speed sedimentation by creating accelerated frames in which gravitational acceleration is replaced by centripetal acceleration, which can be much greater, increasing the terminal speed.
1. What direction will a helium balloon move inside a car that is slowing down—toward the front or back? Explain your answer.
2. Will identical raindrops fall more rapidly in 5º C air or 25º C air, neglecting any differences in air density? Explain your answer.
3. If you took two marbles of different sizes, what would you expect to observe about the relative magnitudes of their terminal velocities?
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