127. Motion of an Object in a Viscous FluidLearning Objectives
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 10m/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 $${N\prime}_{\text{R}}^{}=\frac{\rho \text{vL}}{\eta}\text{(object in fluid),}$$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 Figure 1.) Laminar flow occurs mostly when the objects in the fluid are small, such as raindrops, pollen, and blood cells in plasma. Example 1: Does a Ball Have a Turbulent Wake?Calculate the Reynolds number ${N\prime}_{\text{R}}^{}$ for a ball with a 7.40cm diameter thrown at 40.0 m/s. Strategy We can use ${N\prime}_{\text{R}}^{}=\frac{\rho \text{vL}}{\eta}$ to calculate ${N\prime}_{\text{R}}^{}$, 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 ${N\prime}_{\text{R}}^{}$ yields $$\begin{array}{lll}{N\prime}_{R}^{}& =& \frac{\rho \text{vL}}{\eta}=\frac{(1\text{.}\text{29}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3})(\text{40.0 m/s})(\text{0.0740 m})}{1.81\times {\text{10}}^{5}\phantom{\rule{0.25em}{0ex}}\mathrm{1.00\; Pa}\cdot \text{s}}\\ & =& 2.11\times {\text{10}}^{5}\text{.}\end{array}$$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 qualitativelycorrect results in such situations. One of the consequences of viscosity is a resistance force called 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 TakeHome 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 Section 125 Table 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. Section Summary
Conceptual QuestionsExercise 1What direction will a helium balloon move inside a car that is slowing down—toward the front or back? Explain your answer. Exercise 2Will identical raindrops fall more rapidly in $\mathrm{5\xba\; C}$ air or $\text{25\xba C}$ air, neglecting any differences in air density? Explain your answer. Exercise 3If you took two marbles of different sizes, what would you expect to observe about the relative magnitudes of their terminal velocities?
