Viscosity

Type of physical science: Classical physics

Field of study: Fluids

The internal friction of fluids is called viscosity. Viscosity causes resistance to flow and is important in every problem involving fluid motion from painting to the curve ball in baseball.

Overview

Viscosity is a property of fluids (gases and liquids) by which the flow motion is gradually damped (slowed) and dissipated into heat. Viscosity is a familiar phenomenon in daily life. An opened bottle of wine can be easily poured; the wine flows easily under the influence of gravity into a glass. Maple syrup, on the other hand, cannot be poured so easily; under the action of gravity, it flows sluggishly. The syrup has a higher viscosity than the wine. Almost all fluids have viscosity; the only exceptions are the two isotopes of helium, helium 3 and helium 4, which at extremely low temperatures exhibit no viscosity at all. (In such a state, they are known as superfluids; quantum mechanics is required to understand superfluids.)

A viscous fluid has internal friction. This frictional force exists between the flowing fluid and the surface of the container, such as a tube (the boundary effect), as well as between different parts of the fluid (the bulk effect). Quantitatively, the bulk effect can be illustrated in the following way. Imagine a fluid flowing east. If the flow is uniform, that is, if different parts of the fluid flow with the same velocity, then there can be no frictional force between different parts of the fluid. This results from the fact that if the flow is uniform, then the fluid looks static from the point of view of an observer moving with the fluid. Sir Isaac Newton's first law of motion states that it should stay that way. Consequently, the flow will continue and no viscous damping is possible, at least in bulk.

To observe viscous effect in the bulk, it is essential that the flow be nonuniform. If the easterly flow is assumed to be nonuniform, the magnitude of the flow velocity increases in the northern direction; the flow is more rapid in the north than in the south. To determine how nonuniform the flow is, it is measured in terms of a velocity gradient, which measures the change in flow velocity (meter per second) per unit distance (meter) in the direction of that change (north in the example). In a viscous fluid, such a nonuniform flow generally implies a nonzero viscous force, which in the example is exerted by the relatively slow fluid to the south on the relatively fast fluid to the north to slow down the latter. Following Newton's third law of motion, there is a force equal in magnitude and opposite in direction, which is exerted by the fast fluid to the north to speed up the fluid to the south. This force is proportional to the area of contact between the two parts of the fluid. The frictional force per unit area is known as the viscous stress, which causes the flow to be more uniform. In the example, it slows down the fast flowing fluid and speeds up the slower flowing fluid.

The boundary effect of viscosity forces the fluid that is in direct contact with the boundary to have the same velocity as the boundary. If one considers the flow of a fluid in a tube, and if the tube is stationary, then the viscosity has the effect of slowing down the flow. In the absence of any externally applied pressure (gravity, pump), this will stop the flow completely; with applied pressure the flow may continue, but the viscosity causes resistance to the flow.

Exactly how the viscous stress varies as a function of the velocity gradient depends on the fluid. In many simple fluids, such as water and air, the viscous stress is simply proportional to the velocity gradient. The proportionality coefficient is called the viscosity of the fluid. Thus, viscous stress equals viscosity times velocity gradient. Such a fluid is known as a Newtonian fluid. Viscosity is measured in units of poise, in honor of the French physician, Jean-Louis-Marie Poiseuille. In a fluid with a viscosity of 1 poise, if the velocity gradient is 1 (meter/second/meter), the viscous stress will be 0.1 newton per square meter. Under normal conditions, water has a viscosity of 0.01 poise and air has about 0.00017 poise.

Many polymers are non-Newtonian fluids in that their viscosity depends on the velocity gradient (so that the viscous stress is not simply proportional to the velocity gradient--the functional relationship is nonlinear) or even how long the velocity gradient has been maintained, among other things. Paint, for example, is a non-Newtonian fluid whose viscosity decreases (the fluid becomes thinner) after it has been stirred.

Usually, the viscosity of a fluid depends sensitively on the ambient temperature. The viscosity of gases generally increases with increasing temperature. In contrast, most liquids become less viscous at high temperatures. The viscosity of some substances can vary by a factor of many billions as a function of temperature. Pitch, for example, has a viscosity in excess of 10 billion poises at room temperature but is poured quite easily at elevated temperatures. In the process of glass formation, the viscosity increases from less than several hundred poises at high temperatures to more than 10¹⁴ poises below the glass transition temperature, which is the temperature at which the viscosity is equal to 10¹³ poises.

One can get a sense of where viscosity comes from by considering the following microscopic model. Imagine many molecules forming a gas. Initially, it is assumed that the gas is at rest. Although the macroscopic velocity of the gas is zero, the molecules are in constant random motion at any nonvanishing temperature. The velocity is zero only on average. Imagine such a gas is set up in a flow. Microscopically, the molecules are still undergoing random motion. The difference is that now the average velocity is no longer zero but rather equals the flow velocity. One can now assume that the flow is nonuniform, with the flow velocity being higher on the left. The molecules on the left have a higher average velocity than the molecules on the right. At this point, the random motion of the molecules will tend to mix the fast and the slow. Some of the fast-moving molecules on the left may be carried by random motion to the right, where they will collide with the molecules already there and that originally had a lower average velocity. In this process, the molecules exchange momentum; the fast ones give up some of their momentum to the slow ones, causing the latter to speed up. A similar process, in which the slow molecules on the right are carried by random motion to the left, causes the fast ones to slow down. This manifests itself on the macroscopic scale as viscosity.

The viscosity of a gas can be understood on the basis of this model. The thermal random motion of the molecules mixes the velocity of molecules in a nonuniform flow. The farther a molecule can travel between two collisions, the more effective this mixing will be. At higher temperatures, the random motion of molecules is faster and can carry a molecule farther between two collisions. This is the reason that the viscosity of a gas increases with temperature.

A simple model is not possible for liquids. In a liquid, the molecules are very close together and the motions of individual molecules are highly correlated. This presents a formidable theoretical problem. The microscopic calculation of the viscosity of a liquid has met with only limited success.

Applications

With the exception of superfluids, all fluids have viscosity. It is impossible to understand the flow of fluids without a proper understanding of viscous effects. Because of the viscous effect, the velocity of fluid in direct contact with the wall is zero. In a tube, the flow velocity is not uniform but, rather, has a parabolic distribution, going to zero at the boundary and reaching a maximum at the center. The steady flow of viscous fluids in a tube is known as Poiseuille flow. The total flux of fluid is proportional to the fourth power of the tube radius. All valves depend on viscosity. Closing the valve reduces the opening and increases the resistance, which slows down the flow. Without viscosity, devices ranging from a water faucet to the gas pedal in a car would fail.

To drive the flow in a tube, the viscous resistance of the fluid must be balanced by a pressure difference on the two ends of the tube, whether it is the Alaskan oil pipeline or the human artery system. The viscosity slows down the falling raindrops, causing them to reach some terminal velocity. Without viscosity, the rain drops would shoot down from the sky with the velocity of a bullet.

The thin layer of fluid near a solid object where the flow velocity rapidly decreases to zero relative to the solid is known as the boundary layer. In the boundary layer, the fluid is largely dragged by the object. This is important in a number of sports. A spinning ball drags the air around it and causes the air to circulate. If such a ball is thrown through the air, the circulation and the flow of the air rushing past the ball will be in the same direction on one side of the ball and in opposite directions on the other side. Consequently, the net air velocity on the two sides of the ball will be different. Using Bernoulli's principle, this leads to a difference in pressure on the two sides, which drives the ball to one side of the trajectory. This phenomenon is familiar in baseball and volleyball. To pitch a curve ball, the pitcher gives the ball a spin around a vertical axis. The pressure difference will therefore be between the left and the right side of the trajectory, causing the ball to be deflected to one side. The surface of a golf ball is intentionally roughened with dimples. This enhances the dragging effect and gives the ball a significant lift to an upward spinning ball, greatly increasing its range.

Non-Newtonian fluids are important technologically. For example, paint must be easy to spread with a brush and, at the same time, sufficiently viscous once it is applied to avoid dripping. These two conditions can both be satisfied because of its non-Newtonian viscous properties. The viscosity of a paint depends on how fast the paint is stirred. It decreases when the paint is stirred and sheared by the brush but increases once the stir ceases.

Context

Viscosity can be studied macroscopically and microscopically. The macroscopic study of viscosity belongs to hydrodynamics. In hydrodynamics, the value of the viscosity is assumed to be known; the theory concentrates on working out the dynamical consequences of viscous flow. The significance of viscosity was known in the early days of hydrodynamics. Not until the twentieth century, however, were effective mathematical tools developed to treat viscous flow problems of any complexity. The complicated problem of turbulent flow with viscosity is only beginning to be understood in the last quarter of the twentieth century. The application of hydrodynamics to the viscous flow of unconventional fluids, such as water and oil, in sediments and in biological systems such as human organs and the body, will continue to be of great interest for many years.

The microscopic study of viscosity belongs to the field of statistical mechanics. Here, one is interested in understanding the origin of viscosity, the non-Newtonian behavior, and viscoelastic properties of a fluid on the basis of its molecular properties, such as the intermolecular forces and the deformation of the molecules under stress. Extensive work was done in the last one hundred years. The understanding of the viscosity of gases under normal conditions is essentially complete. One can now calculate the viscosity quantitatively with considerable confidence. The theory for the viscosity of liquids is at a more primitive stage; quantitative results can be expected only for simple liquids. Still less well understood are the viscous properties of concentrated polymer systems. These fluids are often non-Newtonian.

Despite their technological importance, no fundamental theory is available. It is likely that qualitatively new ideas are needed before any progress can be made on this problem.

The fascinating behavior of fluids, the ubiquity of viscosity, its technological significance, and the theoretical challenge of its understanding from a microscopic point of view will continue to be a major driving force in statistical mechanics and hydrodynamics.

Principal terms

BOUNDARY LAYER: a layer of viscous fluid immediately adjacent to a solid; in it, the velocity of fluid rapidly approaches zero relative to the solid

NEWTONIAN FLUID: a fluid in which the viscous stress is proportional to the velocity gradient

POISEUILLE FLOW: the steady flow of viscous fluid in a tube driven by an external pressure difference

VELOCITY GRADIENT: the rate at which the flow velocity changes spatially

VISCOUS STRESS: the internal frictional force per unit contact area between two parts of a fluid in nonuniform flow

Bibliography

Feynman, Richard P., Robert B. Leighton, and Matthew Sands. THE FEYNMAN LECTURES ON PHYSICS. Vol. 2. Reading, Mass.: Addison-Wesley, 1963. Feynman's lectures are insightful and entertaining. Traditional textbooks on hydrodynamics tend to emphasize the elegant mathematical theory of highly idealized models of fluids (which Feynman called "dry water"). Feynman, on the other hand, emphasizes the basic physics.

National Committee for Fluid Mechanics Films. ILLUSTRATED EXPERIMENT IN FLUID MECHANICS. Cambridge, Mass.: MIT Press, 1972. Contains pictures of fluids in motion, with accompanying text.

Shapiro, A. H. SHAPE AND FLOW. Garden City, N.Y.: Doubleday, 1961. An introduction to hydrodynamics with illustrations. No mathematics.

Smith, N. F. "Bernoulli and Newton in Fluid Mechanics." PHYSICS TEACHER 10 (1972): 451. Explains the seemingly strange trajectory of a spinning ball.

Trevena, D. H. THE LIQUID PHASE. New York: Springer-Verlag, 1975. Gives a well-balanced treatment of liquids at the beginning college level. Chapters 9 and 10, dealing respectively with Newtonian and non-Newtonian viscous fluids, are of particular interest.