11.6 Motion of an object in a viscous fluid  (Page 2/4)

An interesting consequence of the increase in $F_{\text{V}}$ 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. [link] 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 ${N\prime }_{\text{R}}^{}=\frac{\rho \text{vL}}{\eta }\text{(object in fluid),}$ which indicates whether flow is laminar or turbulent.
• For ${N\prime }_{\text{R}}^{}$ less than about one, flow is laminar.
• For ${N\prime }_{\text{R}}^{}$ greater than ${\text{10}}^6$ , flow is entirely turbulent.

Conceptual questions