# 12.6 Motion of an object in a viscous fluid

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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 ${N\prime }_{\text{R}}^{}$ , defined for an object moving in a fluid to be

${N\prime }_{\text{R}}^{}=\frac{\rho \text{vL}}{\eta }\text{(object in fluid),}$

where $L$ is a characteristic length of the object (a sphere's diameter, for example), $\rho$ the fluid density, $\eta$ its viscosity, and $v$ the object's speed in the fluid. If ${N\prime }_{\text{R}}^{}$ 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 ${N\prime }_{\text{R}}^{}$ 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 ${N\prime }_{\text{R}}^{}$ between 10 and ${\text{10}}^{6}$ , the flow may be either laminar or turbulent and may oscillate between the two. For ${N\prime }_{\text{R}}^{}$ greater than about ${\text{10}}^{6}$ , 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 ${N\prime }_{\text{R}}^{}$ for a ball with a 7.40-cm diameter thrown at 40.0 m/s.

Strategy

We can use ${N\prime }_{\text{R}}^{}=\frac{\rho \text{vL}}{\eta }$ to calculate ${N\prime }_{\text{R}}^{}$ , 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 ${N\prime }_{\text{R}}^{}$ yields

$\begin{array}{lll}{N\prime }_{R}^{}& =& \frac{\rho \text{vL}}{\eta }=\frac{\left(1\text{.}\text{29}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}\right)\left(\text{40.0 m/s}\right)\left(\text{0.0740 m}\right)}{1.81×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}1.00 Pa\cdot \text{s}}\\ & =& 2.11×{\text{10}}^{5}\text{.}\end{array}$

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     ${F}_{\text{V}}$ 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 ( ${N\prime }_{\text{R}}^{}$ less than about one) viscous drag is proportional to speed, whereas for ${N\prime }_{\text{R}}^{}$ between about 10 and ${\text{10}}^{6}$ , 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 ${N\prime }_{\text{R}}^{}$ greater than ${\text{10}}^{6}$ , drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, ${F}_{\text{V}}$ is proportional to fluid viscosity $\eta$ , the object's characteristic size $L$ , and its speed $v$ . All of which makes sense—the more viscous the fluid and the larger the object, the more drag we expect. Recall Stoke's law ${F}_{\text{S}}=6\mathrm{\pi r\eta v}$ . For the special case of a small sphere of radius $R$ moving slowly in a fluid of viscosity $\eta$ , the drag force ${F}_{\text{S}}$ is given by

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