12.6 Motion of an object in a viscous fluid

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

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.

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