Laminar Flow and Viscosity

Laminar Flow and Viscosity

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. The figure below 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.

Smoke rises smoothly for a while and then begins to form swirls and eddies. The smooth flow is called laminar flow, whereas the swirls and eddies typify turbulent flow. If you watch the smoke (being careful not to breathe on it), you will notice that it rises more rapidly when flowing smoothly than after it becomes turbulent, implying that turbulence poses more resistance to flow. (credit: Creativity103)

The figure below 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.

(a) Laminar flow occurs in layers without mixing. Notice that viscosity causes drag between layers as well as with the fixed surface. (b) An obstruction in the vessel produces turbulence. Turbulent flow mixes the fluid. There is more interaction, greater heating, and more resistance than in laminar flow.

The figure below 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 \(v\) 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 \(v\) to 0 as shown. Care is taken to insure that the flow is laminar; that is, the layers do not mix. The motion in the figure below is like a continuous shearing motion. Fluids have zero shear strength, but the rate at which they are sheared is related to the same geometrical factors \(A\) and \(L\) as is shear deformation for solids.

The graphic shows laminar flow of fluid between two plates of area \(A\). The bottom plate is fixed. When the top plate is pushed to the right, it drags the fluid along with it.

A force \(F\) is required to keep the top plate in the figure above moving at a constant velocity \(v\), and experiments have shown that this force depends on four factors. First, \(F\) is directly proportional to \(v\) (until the speed is so high that turbulence occurs—then a much larger force is needed, and it has a more complicated dependence on \(v\)). Second, \(F\) is proportional to the area \(A\) of the plate. This relationship seems reasonable, since \(A\) is directly proportional to the amount of fluid being moved. Third, \(F\) is inversely proportional to the distance between the plates \(L\). This relationship is also reasonable; \(L\) is like a lever arm, and the greater the lever arm, the less force that is needed. Fourth, \(F\) is directly proportional to the coefficient of viscosity, \(\eta \). The greater the viscosity, the greater the force required. These dependencies are combined into the equation

\(F=\eta \cfrac{\text{vA}}{L}\text{,}\)

which gives us a working definition of fluid viscosity \(\eta \). Solving for \(\eta \) gives

\(\eta =\cfrac{\text{FL}}{\text{vA}}\text{,}\)

which defines viscosity in terms of how it is measured. The SI unit of viscosity is \(\text{N}\cdot \text{m/}[(\text{m/s}){\text{m}}^{2}]=({\text{N/m}}^{2})\text{s or Pa}\cdot \text{s}\). This table lists the coefficients of viscosity for various fluids.

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.)