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

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

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

Section summary

  • When an object moves in a fluid, there is a different form of the Reynolds number N ′ R = ρ vL η (object in fluid), size 12{ { {N}} sup { ' } rSub { size 8{R} } = { {ρ ital "vL"} over {η} } } {} which indicates whether flow is laminar or turbulent.
  • For N ′ R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} less than about one, flow is laminar.
  • For N ′ R size 12{ { {N}} sup { ' } rSub { size 8{R} } } {} greater than 10 6 size 12{"10" rSup { size 8{6} } } {} , flow is entirely turbulent.

Conceptual questions

What direction will a helium balloon move inside a car that is slowing down—toward the front or back? Explain your answer.

Will identical raindrops fall more rapidly in 5º C size 12{5 rSup { size 12{ circ } } C} {} air or 25º C size 12{"25" rSup { size 12{ circ } } C} {} air, neglecting any differences in air density? Explain your answer.

If you took two marbles of different sizes, what would you expect to observe about the relative magnitudes of their terminal velocities?
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Source:  OpenStax, Introduction to physics for vanguard high school (derived from college physics). OpenStax CNX. Oct 15, 2014 Download for free at http://legacy.cnx.org/content/col11715/1.1
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