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We can also use p 2 p 1 = R Q to analyze pressure drops occurring in more complex systems in which the tube radius is not the same everywhere. Resistance is much greater in narrow places, such as in an obstructed coronary artery. For a given flow rate Q , the pressure drop is greatest where the tube is most narrow. This is how water faucets control flow. Additionally, R is greatly increased by turbulence, and a constriction that creates turbulence greatly reduces the pressure downstream. Plaque in an artery reduces pressure and hence flow, both by its resistance and by the turbulence it creates.

Measuring turbulence

An indicator called the Reynolds number     N R can reveal whether flow is laminar or turbulent. For flow in a tube of uniform diameter, the Reynolds number is defined as

N R = 2 ρ v r η (flow in tube)

where ρ is the fluid density, v its speed, η its viscosity, and r the tube radius. The Reynolds number is a dimensionless quantity. Experiments have revealed that N R is related to the onset of turbulence. For N R below about 2000, flow is laminar. For N R above about 3000, flow is turbulent.

For values of N R between about 2000 and 3000, flow is unstable—that is, it can be laminar, but small obstructions and surface roughness can make it turbulent, and it may oscillate randomly between being laminar and turbulent. In fact, the flow of a fluid with a Reynolds number between 2000 and 3000 is a good example of chaotic behavior. A system is defined to be chaotic when its behavior is so sensitive to some factor that it is extremely difficult to predict. It is difficult, but not impossible, to predict whether flow is turbulent or not when a fluid’s Reynold’s number falls in this range due to extremely sensitive dependence on factors like roughness and obstructions on the nature of the flow. A tiny variation in one factor has an exaggerated (or nonlinear) effect on the flow.

Using flow rate: turbulent flow or laminar flow

In [link] , we found the volume flow rate of an air conditioning system to be Q = 3.84 × 10 −3 m 3 /s . This calculation assumed laminar flow. (a) Was this a good assumption? (b) At what velocity would the flow become turbulent?

Strategy

To determine if the flow of air through the air conditioning system is laminar, we first need to find the velocity, which can be found by

Q = A v = π r 2 v .

Then we can calculate the Reynold’s number, using the equation below, and determine if it falls in the range for laminar flow

R = 2 ρ v r η .

Solution

  1. Using the values given:
    v = Q π r 2 = 3.84 × 10 −3 m 3 s 3.14 ( 0.09 m ) 2 = 0.15 m s R = 2 ρ v r η = 2 ( 1.23 kg m 3 ) ( 0.15 m s ) ( 0.09 m ) 0.0181 × 10 −3 Pa s = 1835 .

    Since the Reynolds number is 1835<2000, the flow is laminar and not turbulent. The assumption that the flow was laminar is valid.
  2. To find the maximum speed of the air to keep the flow laminar, consider the Reynold’s number.
    R = 2 ρ v r η 2000 v = 2000 ( 0.0181 × 10 −3 Pa s ) 2 ( 1.23 kg m 3 ) ( 0.09 m ) = 0.16 m s .

Significance

When transferring a fluid from one point to another, it desirable to limit turbulence. Turbulence results in wasted energy, as some of the energy intended to move the fluid is dissipated when eddies are formed. In this case, the air conditioning system will become less efficient once the velocity exceeds 0.16 m/s, since this is the point at which turbulence will begin to occur.

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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