Chapter 12 Fluid Dynamics and Its Biological and Medical Applications
Chapter 12 Fluid Dynamics and Its Biological and Medical Applications
12.6 Motion of an Object in a Viscous Fluid
Summary
- 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 [latex]{N^{\prime}_{\text{R}}},[/latex] defined for an object moving in a fluid to be
where [latex]{L}[/latex] is a characteristic length of the object (a sphere’s diameter, for example), [latex]{\rho}[/latex] the fluid density, [latex]{\eta}[/latex] its viscosity, and [latex]{v}[/latex] the object’s speed in the fluid. If [latex]{N^{\prime}_{\text{R}}}[/latex] 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 [latex]{N^{\prime}_{\text{R}}}[/latex] 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 [latex]{N^{\prime}_{\text{R}}}[/latex] between 10 and [latex]{10^6},[/latex] the flow may be either laminar or turbulent and may oscillate between the two. For [latex]{N^{\prime}_{\text{R}}}[/latex] greater than about [latex]{10^6},[/latex] 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.
Example 1: Does a Ball Have a Turbulent Wake?
Calculate the Reynolds number [latex]{N^{\prime}_{\text{R}}}[/latex] for a ball with a 7.40-cm diameter thrown at 40.0 m/s.
Strategy
We can use [latex]{N^{\prime}_{\text{R}}=\frac{\rho{v}L}{\eta}}[/latex] to calculate [latex]{N^{\prime}_{\text{R}}},[/latex] 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 [latex]{N^{\prime}_{\text{R}}}[/latex] 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 [latex]{F_{\text{V}}}[/latex] that is exerted on a moving object. This force typically depends on the object’s speed (in contrast with simple f