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

N_{R}^{'} = \frac{ρ vL}{η} (object in fluid), N_{R}^{'} = \frac{ρ vL}{η} (object in fluid), size 12{ { {N}} sup { ' } rSub { size 8{R} } = {  {ρ ital "vL"}  over  {η} } } {}

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where $L L size 12{L} {}$ is a characteristic length of the object (a sphere’s diameter, for example), $ρ ρ size 12{ρ} {}$ the fluid density, $η η size 12{η} {}$ its viscosity, and $v v size 12{v} {}$ the object’s speed in the fluid. If $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ 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 $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ 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 $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ between 10 and $10^{6} 10^{6} size 12{"10" rSup { size 8{6} } } {}$ , the flow may be either laminar or turbulent and may oscillate between the two. For $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ greater than about $10^{6} 10^{6} size 12{"10" rSup { size 8{6} } } {}$ , the flow is entirely turbulent, even at the surface of the object. (See Figure 12.18.) Laminar flow occurs mostly when the objects in the fluid are small, such as raindrops, pollen, and blood cells in plasma.

Example 12.10

Does a Ball Have a Turbulent Wake?

Calculate the Reynolds number $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ for a ball with a 7.40-cm diameter thrown at 40.0 m/s.

Strategy

We can use $N_{R}^{'} = \frac{ρ vL}{η} N_{R}^{'} = \frac{ρ vL}{η} size 12{ { {N}} sup { ' } rSub { size 8{R} } = { {ρ ital "vL"} over {η} } } {}$ to calculate $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ , 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 $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ yields

\begin{array}{l} N_{R}^{'} & = & \frac{ρ vL}{η} = \frac{(1.29 {kg/m}^{3}) (40.0 m/s) (0.0740 m)}{1.81 \times 10^{- 5} 1.00 Pa \cdot s} \\ = & 2.11 \times 10^{5} . \end{array} \begin{array}{l} N_{R}^{'} & = & \frac{ρ vL}{η} = \frac{(1.29}{1.81 \times 10^{- 5}} \\ = 2.11 \times 10^{5} . \end{array}

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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 $F_{V} F_{V} size 12{F rSub { size 8{V} } } {}$ 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 ( $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ less than about one) viscous drag is proportional to speed, whereas for $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ between about 10 and $10^{6} 10^{6} size 12{"10" rSup { size 8{6} } } {}$ , 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 $N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}$ greater than $10^{6} 10^{6} size 12{"10" rSup { size 8{6} } } {}$ , drag increases dramatically and behaves with greater complexity. For laminar flow around a sphere, $F_{V} F_{V} size 12{F rSub { size 8{V} } } {}$ is proportional to fluid viscosity $η η size 12{η} {}$ , the object’s characteristic size $L L size 12{L} {}$ , and its speed $v v size 12{v} {}$ . All of which makes sense—the more viscous the fluid and the larger the object, the more drag we expect. Recall Stoke’s law $F_{S} = 6 πrηv F_{S} = 6 πrηv size 12{F rSub { size 8{S} } =6πrηv} {}$ . For the special case of a small sphere of radius $R R size 12{R} {}$ moving slowly in a fluid of viscosity $η η size 12{η} {}$ , the drag force $F_{S} F_{S} size 12{F rSub { size 8{S} } } {}$ is given by

F_{S} = 6 πRηv . F_{S} = 6 πRηv . size 12{F rSub { size 8{S} } =6πRηv} {}

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Part a of the figure shows a sphere moving in a fluid. The fluid lines are shown to move toward the left. The viscous force on the sphere is also toward the left given by F v as shown by the arrow. The flow is shown as laminar as shown by linear bending lines. Part b of the figure shows a sphere moving with higher speed in a fluid. The fluid lines are shown to move toward the left. The viscous force on the sphere is also toward the left given by F v prime as shown by the arrow. The flow is shown as laminar above and below the sphere shown by linear lines of flow and turbulent on left of the sphere shown by curly lines of flow. Part c of the figure shows a sphere still moving with higher speed in a fluid. The fluid lines are shown to move toward the left at the edges of flow away from the sphere. The viscous force on the sphere is also toward the left given by F v double prime as shown by the arrow. The flow is turbulent all around the sphere as shown by curly lines of flow. The viscous drag F v double prime is shown to be still greater by longer length of arrows.

Figure 12.18 (a) Motion of this sphere to the right is equivalent to fluid flow to the left. Here the flow is laminar with

N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}

less than 1. There is a force, called viscous drag

F_{V} F_{V} 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}^{'} F_{V}^{'} size 12{ { {F}} sup { ' } rSub { size 8{V} } } {}

that is significantly greater than for laminar flow. Here

N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}

is greater than 10. (c) At much higher speeds, where

N_{R}^{'} N_{R}^{'} size 12{ { {N}} sup { ' } rSub { size 8{R} } } {}

is greater than

10^{6} 10^{6} 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 $F_{V} F_{V} size 12{F rSub { size 8{V} } } {}$ 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. Figure 12.19 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.

Take-Home Experiment: Don’t Lose Your Marbles

By measuring the terminal speed of a slowly moving sphere in a viscous fluid, one can find the viscosity of that fluid (at that temperature). It can be difficult to find small ball bearings around the house, but a small marble will do. Gather two or three fluids (syrup, motor oil, honey, olive oil, etc.) and a thick, tall clear glass or vase. Drop the marble into the center of the fluid and time its fall (after letting it drop a little to reach its terminal speed). Compare your values for the terminal speed and see if they are inversely proportional to the viscosities as listed in Table 12.1. Does it make a difference if the marble is dropped near the side of the glass?

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.

The figure shows the forces acting on an oval shaped object falling through a viscous fluid. An enlarged view of the object is shown toward the left to analyze the forces in detail. The weight of the object w acts vertically downward. The viscous drag F v and buoyant force F b acts vertically upward. The length of the object is given by L. The density of the object is given by rho obj and density of the fluid by rho fl.

Figure 12.19 There are three forces acting on an object falling through a viscous fluid: its weight

w w size 12{w} {}

, the viscous drag

F_{V} F_{V} size 12{F rSub { size 8{V} } } {}

, and the buoyant force

F_{B} F_{B} size 12{F rSub { size 8{B} } } {}