When you pour yourself a glass of juice, the liquid flows freely and quickly. But when you pour syrup on your pancakes, that liquid flows slowly and sticks to the pitcher. The difference is fluid friction, both within the fluid itself and between the fluid and its surroundings. We call this property of fluids *viscosity*. Juice has low viscosity, whereas syrup has high viscosity. In the previous sections we have considered ideal fluids with little or no viscosity. In this section, we will investigate what factors, including viscosity, affect the rate of fluid flow.

The precise definition of viscosity is based on *laminar*, or nonturbulent, flow. Before we can define viscosity, then, we need to define laminar flow and turbulent flow. Figure 1 shows both types of flow. Laminar flow is characterized by the smooth flow of the fluid in layers that do not mix. Turbulent flow, or turbulence, is characterized by eddies and swirls that mix layers of fluid together.

Figure 2 shows schematically how laminar and turbulent flow differ. Layers flow without mixing when flow is laminar. When there is turbulence, the layers mix, and there are significant velocities in directions other than the overall direction of flow. The lines that are shown in many illustrations are the paths followed by small volumes of fluids. These are called *streamlines*. Streamlines are smooth and continuous when flow is laminar, but break up and mix when flow is turbulent. Turbulence has two main causes. First, any obstruction or sharp corner, such as in a faucet, creates turbulence by imparting velocities perpendicular to the flow. Second, high speeds cause turbulence. The drag both between adjacent layers of fluid and between the fluid and its surroundings forms swirls and eddies, if the speed is great enough. We shall concentrate on laminar flow for the remainder of this section, leaving certain aspects of turbulence for later sections.

**Making Connections: Take-Home Experiment: Go Down to the River: **

Try dropping simultaneously two sticks into a flowing river, one near the edge of the river and one near the middle. Which one travels faster? Why?

Figure 3 shows how viscosity is measured for a fluid. Two parallel plates have the specific fluid between them. The bottom plate is held fixed, while the top plate is moved to the right, dragging fluid with it. The layer (or lamina) of fluid in contact with either plate does not move relative to the plate, and so the top layer moves at while the bottom layer remains at rest. Each successive layer from the top down exerts a force on the one below it, trying to drag it along, producing a continuous variation in speed from

*to 0 as shown. Care is taken to insure that the flow is laminar; that is, the layers do not mix. The motion in Figure 3 is like a continuous shearing motion. Fluids have zero shear strength, but the*v v size 12{v} {}

*rate*at which they are sheared is related to the same geometrical factors

*and*A A size 12{A} {}

*as is shear deformation for solids.*L L size 12{L} {}

A force is required to keep the top plate in Figure 3 moving at a constant velocity

*, and experiments have shown that this force depends on four factors. First,*v v size 12{v} {}

*is directly proportional to*F F size 12{F} {}

*(until the speed is so high that turbulence occurs—then a much larger force is needed, and it has a more complicated dependence on*v v size 12{v} {}

*). Second,*v v size 12{v} {}

*is proportional to the area*F F size 12{F} {}

*of the plate. This relationship seems reasonable, since*A A size 12{A} {}

*is directly proportional to the amount of fluid being moved. Third,*A A size 12{A} {}

*is inversely proportional to the distance between the plates*F F size 12{F} {}

*. This relationship is also reasonable;*L L size 12{L} {}

*is like a lever arm, and the greater the lever arm, the less force that is needed. Fourth,*L L size 12{L} {}

*is directly proportional to*F F size 12{F} {}

*the coefficient of viscosity*,

*. The greater the viscosity, the greater the force required. These dependencies are combined into the equation*η η size 12{η} {}

which gives us a working definition of fluid viscosity* *. Solving for

*gives*η η size 12{η} {}

which defines viscosity in terms of how it is measured. The SI unit of viscosity is

Viscosity varies from one fluid to another by several orders of magnitude. As you might expect, the viscosities of gases are much less than those of liquids, and these viscosities are often temperature dependent. The viscosity of blood can be reduced by aspirin consumption, allowing it to flow more easily around the body. (When used over the long term in low doses, aspirin can help prevent heart attacks, and reduce the risk of blood clotting.)