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A moving object in a viscous fluid is equivalent to a stationary object in a flowing fluid stream. (For example, when you ride a bicycle at 10 m/s in still air, you feel the air in your face exactly as if you were stationary in a 10-m/s wind.) Flow of the stationary fluid around a moving object may be laminar, turbulent, or a combination of the two. Just as with flow in tubes, it is possible to predict when a moving object creates turbulence. We use another form of the Reynolds number , defined for an object moving in a fluid to be
where is a characteristic length of the object (a sphere’s diameter, for example), the fluid density, its viscosity, and the object’s speed in the fluid. If 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 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 between 10 and , the flow may be either laminar or turbulent and may oscillate between the two. For greater than about , the flow is entirely turbulent, even at the surface of the object. (See [link] .) 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 for a ball with a 7.40-cm diameter thrown at 40.0 m/s.
Strategy
We can use to calculate , since all values in it are either given or can be found in tables of density and viscosity.
Solution
Substituting values into the equation for yields
Discussion
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 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 ( less than about one) viscous drag is proportional to speed, whereas for between about 10 and , 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 greater than , drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, is proportional to fluid viscosity , the object’s characteristic size , and its speed . All of which makes sense—the more viscous the fluid and the larger the object, the more drag we expect. Recall Stoke’s law . For the special case of a small sphere of radius moving slowly in a fluid of viscosity , the drag force is given by
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