103 Motion of an Object in a Viscous Fluid

[latex]{N′}_{\text{R}}=\frac{\rho vL}{\eta}\text{(object in fluid)}\\[/latex],

where L is a characteristic length of the object (a sphere’s diameter, for example), ρ the fluid density, η its viscosity, and v the object’s speed in the fluid. If N′_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 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 between 10 and 10⁶, the flow may be either laminar or turbulent and may oscillate between the two. For N′_R greater than about 10⁶, 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.

One of the consequences of viscosity is a resistance force called viscous drag F_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 less than about one) viscous drag is proportional to speed, whereas for N′_R between about 10 and 10⁶, 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 greater than 10⁶, drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, F_V is proportional to fluid viscosity η, 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_S = 6πrηv. For the special case of a small sphere of radius R moving slowly in a fluid of viscosity η, the drag force F_S is given by

F_S = 6πRηv.

An interesting consequence of the increase in F_V with speed is that an object falling through a fluid will not continue to accelerate indefinitely (as it would if we neglect air resistance, for example). Instead, viscous drag increases, slowing acceleration, until a critical speed, called the terminal speed, is reached and the acceleration of the object becomes zero. Once this happens, the object continues to fall at constant speed (the terminal speed). This is the case for particles of sand falling in the ocean, cells falling in a centrifuge, and sky divers falling through the air. Figure 2 shows some of the factors that affect terminal speed. There is a viscous drag on the object that depends on the viscosity of the fluid and the size of the object. But there is also a buoyant force that depends on the density of the object relative to the fluid. Terminal speed will be greatest for low-viscosity fluids and objects with high densities and small sizes. Thus a skydiver falls more slowly with outspread limbs than when they are in a pike position—head first with hands at their side and legs together.

Knowledge of terminal speed is useful for estimating sedimentation rates of small particles. We know from watching mud settle out of dirty water that sedimentation is usually a slow process. Centrifuges are used to speed sedimentation by creating accelerated frames in which gravitational acceleration is replaced by centripetal acceleration, which can be much greater, increasing the terminal speed.

Section Summary

• When an object moves in a fluid, there is a different form of the Reynolds number [latex]{N′}_{\text{R}}^{}=\frac{\rho \text{vL}}{\eta }\text{(object in fluid)}\\[/latex], which indicates whether flow is laminar or turbulent.
• For N′_R less than about one, flow is laminar.
• For N′_R greater than 10⁶, flow is entirely turbulent.