Stokes' law....
... gives the drag on a spherical object passing through a fluid in terms of the viscosity of the fluid h, the radius of the sphere r, and the velocity v (often the terminal velocity of a falling object). The appropriate application of Stokes law in a practical situation is poorly understood in high school physics classes.
Stokes' law applies only under conditions of laminar flow. It is important when describing Millikan's experiment with oil drops. It is commonly assumed that the drag force is given by Stokes' law at low speed and by the force due to the 'fire hose' formula (rAv2) at higher speeds. The difficulty is knowing what counts as low speed.
Above some critical speed turbulent flow sets in and the drag force becomes proportional to velocity squared according the the equation....
... where r .is the density of the fluid and A is the cross sectional area of the object. To take account of variations in drag due to different shapes, surface textures etc. a drag coefficient (Cd) is defined so that....
The value of Cd for a smooth sphere is 0.47, for a donkey cart is about 0.9 and for a streamlined modern car is about 0.16-0.2.
| Why the constant in the turbulent flow drag equation is defined as 1/2Cd is a mystery. If a reader is able to provide an explanation it would be received with thanks. |
Paper muffin cups falling in air at about one m/s are in one sense slow, but air resistance is found to be proportional to v2. The same v2 dependence is found for metal muffin cups falling in water and cooking oil.
Reynolds number is a defined quantity often used to predict whether laminar or turbulent flow will result in a particular situation. Reynolds number Re for a sphere is given by....
... where r is the fluid density, d is the diameter of the sphere, v is the (terminal) velocity and h. is the viscosity.
Paper muffin cups in air
The viscosity of air at STP is 1.83x10-5 Nsm-2. The density of air is 1.26 kg/m3. The muffin cups have a diameter of ~5 cm and a terminal velocity of ~1 m/s
Reynolds number for falling paper cups in air (taking them to be approximately spherical) is given by....
Turbulent flow typically occurs when Re is above about 1000 and laminar flow is usually established when Re is less than 1. The muffin cups fall with turbulent flow, as do autumn leaves.
Metal muffin cups in water
For falling metal cups in water the density is 1x103 kg/m3 and the terminal velocity is again around one meter per second. The viscosity of water at room temperature is 1.0x10-3 Nsm-2.
... and the flow is again turbulent.
A 0-10 Newton force probe (mounted and calibrated to measure small forces) is attached to a sphere suspended in a cylinder of water, sugar solution, oil etc. Liquids with higher viscosity will have smaller Reynolds numbers.
The cylinder is slowly lowered. The drag force is measured as the increment in the force probe measurement.
The answers to many questions could be sought by this method.
1 Is measurable additional drag introduced by the thin nylon suspension fiber? If so does this depend on the length of the fiber submerged and can a correction factor be developed?
2 What percentage of additional drag is introduced by the presence of the walls and what is the dependence of this 'wall effect' on the ratio of the sphere radius to the cylinder radius?
3 What is the velocity-drag relationship for a range of speeds from zero upwards and in what range is there a change from proportionality to v2 dependence.
*How does the change-over velocity, (if it can be found), depend on Reynold's number?
Note: a force probe and a motion detector can be used in combination to plot drag versus velocity graphs in real time.
4 Can this method be used to measure drag at low speeds in water, oil, glycerol, etc. as a function of temperature and hence to determine the viscosity of fluids as a function of temperature?
5 Could this method be used to measure drag force in finely granulated materials like quick-sand, or in materials that are thixatropic like corn starch solutions?
6 Could this method be used to measure drag in finely granulated materials with the addition of vibration, as a function of frequency and amplitude?
7 The onset of turbulence in air
Filling a wide vertical transparent tube with smoke may allow the airflow over a falling paper cup to be filmed. If this is the case it may be possible to determine the speed at which laminar flow breaks down by making paper cups with thinner paper, less weight, and consequently progressively lower terminal velocities.
It may be possible to make very light paper cups by pressing paper between metal cups (which we have). Possibly the metal cups could be warmed. Pressing damp paper which is then dried as it is pressed may hold its shape or possibly a film of starch may preserve the shape without adding appreciable mass? Some experimentation is required.
... or it may be possible to do the same thing by dropping inflated balloons with walls of different thicknesses and hence different terminal velocities.
Note: edge effects could be important and it may not be possible to drop balloons or cups down the middle of the tube.