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′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}, defined for an object moving in a fluid to be

where L size 12{L} {} is a characteristic length of the object (a sphere’s diameter, for example), ρ size 12{ρ} {} the fluid density, η size 12{η} {} its viscosity, and v size 12{v} {} the object’s speed in the fluid. If N′R size 12{ { {N}} sup { ' } rSub { size 8{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 size 12{ { {N}} sup { ' } rSub { size 8{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 size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} between 10 and 106 size 12{"10" rSup { size 8{6} } } {}, the flow may be either laminar or turbulent and may oscillate between the two. For N′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} greater than about 106 size 12{"10" rSup { size 8{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 FV size 12{F rSub { size 8{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 size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} less than about one) viscous drag is proportional to speed, whereas for N′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} between about 10 and 106 size 12{"10" rSup { size 8{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′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} greater than 106 size 12{"10" rSup { size 8{6} } } {}, drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, FV size 12{F rSub { size 8{V} } } {} is proportional to fluid viscosity η size 12{η} {}, the object’s characteristic size L size 12{L} {}, and its speed v size 12{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 FS=6πrηv size 12{F rSub { size 8{S} } =6πrηv} {}. For the special case of a small sphere of radius R size 12{R} {} moving slowly in a fluid of viscosity η size 12{η} {}, the drag force FS size 12{F rSub { size 8{S} } } {} is given by

(a) Motion of this sphere to the right is equivalent to fluid flow to the left. Here the flow is laminar with N′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} less than 1. There is a force, called viscous drag FV size 12{F rSub { size 8{V} } } {}, to the left on the ball due to the fluid’s viscosity. (b) At a higher speed, the flow becomes partially turbulent, creating a wake starting where the flow lines separate from the surface. Pressure in the wake is less than in front of the sphere, because fluid speed is less, creating a net force to the left F′V size 12{ { {F}} sup { ' } rSub { size 8{V} } } {} that is significantly greater than for laminar flow. Here N′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} is greater than 10. (c) At much higher speeds, where N′R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} is greater than 106 size 12{"10" rSup { size 8{6} } } {}, flow becomes turbulent everywhere on the surface and behind the sphere. Drag increases dramatically.

An interesting consequence of the increase in FV size 12{F rSub { size 8{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.

There are three forces acting on an object falling through a viscous fluid: its weight w size 12{w} {}, the viscous drag FV size 12{F rSub { size 8{V} } } {}, and the buoyant force FB size 12{F rSub { size 8{B} } } {}.