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Learning objectives

By the end of this section, you will be able to:

  • Calculate the Reynolds number for an object moving through a fluid.
  • Explain whether the Reynolds number indicates laminar or turbulent flow.
  • Describe the conditions under which an object has a terminal speed.

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 N R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} , defined for an object moving in a fluid to be

N R = ρ vL η (object in fluid), size 12{ { {N}} sup { ' } rSub { size 8{R} } = { {ρ ital "vL"} over {η} } } {}

where L size 12{L} {} is a characteristic length of the object (a sphere's diameter, for example), ρ size 12{ρ} {} the fluid density, η size 12{η} {} its viscosity, and v size 12{v} {} the object's speed in the fluid. If N R size 12{ { {N}} sup { ' } rSub { size 8{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 size 12{ { {N}} sup { ' } rSub { size 8{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 size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} between 10 and 10 6 size 12{"10" rSup { size 8{6} } } {} , the flow may be either laminar or turbulent and may oscillate between the two. For N R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} greater than about 10 6 size 12{"10" rSup { size 8{6} } } {} , 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.

Does a ball have a turbulent wake?

Calculate the Reynolds number N R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} for a ball with a 7.40-cm diameter thrown at 40.0 m/s.


We can use N R = ρ vL η size 12{ { {N}} sup { ' } rSub { size 8{R} } = { {ρ ital "vL"} over {η} } } {} to calculate N R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} , since all values in it are either given or can be found in tables of density and viscosity.


Substituting values into the equation for N R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} yields

N R = ρ vL η = ( 1 . 29 kg/m 3 ) ( 40.0 m/s ) ( 0.0740 m ) 1.81 × 10 5 1.00 Pa s = 2.11 × 10 5 .


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.

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One of the consequences of viscosity is a resistance force called viscous drag     F V size 12{F rSub { size 8{V} } } {} 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 size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} less than about one) viscous drag is proportional to speed, whereas for N R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} between about 10 and 10 6 size 12{"10" rSup { size 8{6} } } {} , 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 size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} greater than 10 6 size 12{"10" rSup { size 8{6} } } {} , drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, F V size 12{F rSub { size 8{V} } } {} is proportional to fluid viscosity η size 12{η} {} , the object's characteristic size L size 12{L} {} , and its speed v size 12{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 F S = 6 πrηv size 12{F rSub { size 8{S} } =6πrηv} {} . For the special case of a small sphere of radius R size 12{R} {} moving slowly in a fluid of viscosity η size 12{η} {} , the drag force F S size 12{F rSub { size 8{S} } } {} is given by

Questions & Answers

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