# Viscosity and Turbulence

*Viscosity*is a measure of the stickiness of a fluid. Molasses is stickier, for instance, than water, and water is stickier than air. A precise definition of viscosity looks like this: Say we have a floor covered with a fluid to a depth

*d*. A hockey puck of area

*A*is traveling along at velocity

*v*. The drag force on the hockey puck due to the stickiness of the fluid is given by

...(14) |

*η*is the viscosity in kg/ms (that is, kilograms per meter per second, not kilograms per millisecond). You should understand this equation but need not memorize it. Just remember that viscosity is stickiness.

Viscosity often calms down a flow. Consider a stream of water flowing past a rock or the flow of air past a weather vane. The flow can break off into wild swirls and chaotic patterns, called

*turbulence*. Smooth flow is called

*streamline*or

*laminar*. If we pull a spoon through a bowl of molasses, the molasses is mostly undisturbed (laminar flow around the spoon), whereas if we pull the spoon through a cup of tea with cream at the bottom, the tea and cream undergo turbulent motion. What's the difference? The more viscosity, the less turbulence.

An important parameter for determining the type of flow is the Reynolds number:

…..(15) |

*l*is the size (in m) of the obstacle in the flow (spoon or whatever),

*ρ*is the density of the fluid (in kg/m

^{3}),

*v*is the velocity of the flow (in m/s), and

*η*is viscosity (in kg/m s).

The flow starts getting rough when

*Re*is around 40, and it is usually turbulent if*Re*is greater than 20,000. Do not memorize this equation, but do realize that the presence of*η*in the denominator means that higher viscosity reduces turbulence. It should also make sense to you that*v*is in the numerator.Example

What is the Reynolds number for a spoon moving through tea with cream in it? (Use *η* = 1.0 x 10^{–3} kg/ms and estimate other values.

Solution

We can use information corresponding to water, so the density is 10^{3} kg/m^{3}, and the viscosity is 1.0 x 10^{–3} kg/m s. A good rate to stir tea is 0.1 m/s, and a typical spoon has size 0.03 m. Thus