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  • 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.

Strategy

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.

Solution

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 .

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.

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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

resistance of thermometer in relation to temperature
Ifeanyi Reply
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Bernard
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Paul
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Gift
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grace Reply
Divide with 3.6
Mateo
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Muhammad
2 how heat loss is prevented in a vacuum flask
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Helen
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due to non in contact mean no conduction and no convection bec of non conducting base and walls and also their is a grape between the layer like to take the example of thermo flask
Abdul
dimensions v²=u²+2at
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what if time is not given in finding the average velocity?
Alan Reply
the magnetic circuit of a certain of the flux paths in each of the long and short sides being 25cm and 20cm reprectielectrove. there is an air gap of 2mm long in one the long sides if a flux density of 0.8weber/m is to produce in the magnet of 1500 turns..
Daniel Reply
How do you calculate precision
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Fillemon
Chemisty 1A?
Fillemon
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Sacky
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Fillemon
Friday bro... But the topics we did are in this app... Just try to master them quickly before the test dates... Are you done with the Maths sheet
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Anderson
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Emi
There are very large numbers of charged particles in most objects. Why, then, don’t most objects exhibit static electricity?
Bilkisu Reply
Because there's an equal number of negative and positive charges... objects are neutral in nature
NELSON
when a ball rolls on a smooth level ground,the motion of its centre is?
Mary Reply
what is electro magnetic field?
Mary
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NELSON
Electromagnetic field is caused by moving electric charge
Muhammad
when a ball rolls on a smooth level ground,the motion of its centre is?
Mumeh
what's the relationship btw displacement and position
Declan Reply
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Anderson
what is the meaning of elasticity
Pele Reply
is the ability of a material to or any object to expand to a limit point
king
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Emi
Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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