11.11

*Viscous Flow

In an ideal fluid there is no viscosity to hinder the fluid layers as they slide past one another. Within a pipe of uniform cross section, every layer of an ideal fluid moves with the same velocity, even the layer next to the wall, as Figure 11-37a shows. When viscosity is present, the fluid layers have different velocities, as part b of the drawing illustrates. The fluid at the center of the pipe has the greatest velocity. In contrast, the fluid layer next to the wall surface does not move at all because it is held tightly by intermolecular forces. So strong are these forces that if a solid surface moves, the adjacent fluid layer moves along with it and remains at rest relative to the moving surface.

Figure 11-37 (a) In ideal (nonviscous) fluid flow, all fluid particles across the pipe have the same velocity. (b) In viscous flow, the speed of the fluid is zero at the surface of the pipe and increases to a maximum along the center axis.

To help introduce viscosity in a quantitative fashion, Figure 11-38a shows a viscous fluid between two parallel plates. The top plate is free to move while the bottom one is stationary. If the top plate is to move with a velocity relative to the bottom plate, a force is required. For a highly viscous fluid, like thick honey, a large force is needed; for a less viscous fluid, like water, a smaller one will do. As part b of the drawing suggests, we may imagine the fluid to be composed of many thin horizontal layers. When the top plate moves, the intermediate fluid layers slide over each other. The velocity of each layer is different, changing uniformly from at the top plate to zero at the bottom plate. The resulting flow is called laminar flow, since a thin layer is often referred to as a lamina. As each layer moves, it is subjected to viscous forces from its neighbors. The purpose of the force is to compensate for the effect of these forces, so that any layer can move with a constant velocity.

Figure 11-38 (a) A force is applied to the top plate, which is in contact with a viscous fluid. (b) Because of the force , the top plate and the adjacent layer of fluid move with a constant velocity .

The amount of force required in Figure 11-38a depends on several factors. Larger areas A, being in contact with more fluid, require larger forces, so that the force is proportional to the contact area . For a given area, greater speeds require larger forces, with the result that the force is proportional to the speed . The force is also inversely proportional to the perpendicular distance y between the top and bottom plates . The larger the distance y, the smaller is the force required to achieve a given speed with a given contact area. These three proportionalities can be expressed simultaneously in the following way: . Equation 11.13 expresses this relationship with the aid of a proportionality constant (Greek letter eta), which is called the coefficient of viscosity or simply the viscosity.

| Force Needed to Move a Layer of Viscous Fluid With a Constant Velocity
| The magnitude of the tangential force required to move a fluid layer at a constant speed v, when the layer has an area A and is located a perpendicular distance y from an immobile surface, is given by

| (11.13)

where is the coefficient of viscosity.
SI Unit of Viscosity: Pa·s
Common Unit of Viscosity: poise (P)

By solving Equation 11.13 for the viscosity, , it can be seen that the SI unit for viscosity is . Another common unit for viscosity is the poise (P), which is used in the CGS system of units and is named after the French physician Jean Poiseuille (1799–1869; pronounced, approximately, as Pwah-zoy). The following relation exists between the two units:

Values of viscosity depend on the nature of the fluid. Under ordinary conditions, the viscosities of liquids are significantly larger than those of gases. Moreover, the viscosities of either liquids or gases depend markedly on temperature. Usually, the viscosities of liquids decrease as the temperature is increased. Anyone who has heated honey or oil, for example, knows that these fluids flow much more freely at an elevated temperature. In contrast, the viscosities of gases increase as the temperature is raised. An ideal fluid has .
Viscous flow occurs in a wide variety of situations, such as oil moving through a pipeline or a liquid being forced through the needle of a hypodermic syringe. Figure 11-39 identifies the factors that determine the volume flow rate Q (in m3/s) of the fluid. First, a difference in pressures must be maintained between any two locations along the pipe for the fluid to flow. In fact, Q is proportional to , a greater pressure difference leading to a larger flow rate. Second, a long pipe offers greater resistance to the flow than a short pipe does, and Q is inversely proportional to the length L. Because of this fact, long pipelines, such as the Alaskan pipeline, have pumping stations at various places along the line to compensate for a drop in pressure (see Figure 11-40). Third, high-viscosity fluids flow less readily than low-viscosity fluids, and Q is inversely proportional to the viscosity . Finally, the volume flow rate is larger in a pipe of larger radius, other things being equal. The dependence on the radius R is a surprising one, Q being proportional to the fourth power of the radius, or R4. If, for instance, the pipe radius is reduced to one-half of its original value, the volume flow rate is reduced to one-sixteenth of its original value, assuming the other variables remain constant. The mathematical relation for Q in terms of these parameters was discovered by Poiseuille and is known as Poiseuille’s law.

The physics of pipeline pumping stations.

| Poiseuille’s Law
| A fluid whose viscosity is , flowing through a pipe of radius R and length L, has a volume flow rate Q given by

| (11.14)

where P1 and P2 are the pressures at the ends of the pipe.

Figure 11-39 For viscous flow, the difference in pressures , the radius R and length L of the tube, and the viscosity of the fluid influence the volume flow rate.

Figure 11-40 As oil flows along the Alaskan pipeline, the pressure drops because oil is a viscous fluid. Pumping stations are located along the pipeline to compensate for the drop in pressure. (Ken Graham/Stone/Getty Images)

Example 17 illustrates the use of Poiseuille’s law.

| Example 17 | | | Giving an Injection

A hypodermic syringe is filled with a solution whose viscosity is . As Figure 11-41 shows, the plunger area of the syringe is , and the length of the needle is 0.025 m. The internal radius of the needle is . The gauge pressure in a vein is 1900 Pa (14 mm of mercury). What force must be applied to the plunger, so that of solution can be injected in 3.0 s?

Figure 11-41 The difference in pressure required to sustain the fluid flow through a hypodermic needle can be found with the aid of Poiseuille’s law.

Reasoning The necessary force is the pressure applied to the plunger times the area of the plunger. Since viscous flow is occurring, the pressure is different at different points along the syringe. However, the barrel of the syringe is so wide that little pressure difference is required to sustain the flow up to point 2, where the fluid encounters the narrow needle. Consequently, the pressure applied to the plunger is nearly equal to the pressure P2 at point 2. To find this pressure, we apply Poiseuille’s law to the needle. Poiseuille’s law indicates that . We note that the pressure P1 is given as a gauge pressure, which, in this case, is the amount of pressure in excess of atmospheric pressure. This causes no difficulty, because we need to find the amount of force in excess of that applied to the plunger by the atmosphere. The volume flow rate Q can be obtained from the time needed to inject the known volume of solution.

The physics of a hypodermic syringe.

Solution The volume flow rate is . According to Poiseuille’s law (Equation 11.14), the required pressure difference is

Since , the pressure P2 must be . The force that must be applied to the plunger is this pressure times the plunger area:

Test your knowledge of the important ideas in Sections 11.7–11.11:

· | The Equation of Continuity
· | Bernoulli’s Equation
· | Viscous Flow

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