Sci-Tech Encyclopedia: Viscosity

The material property that measures a fluid's resistance to flowing. For example, water flows from a tilted jar more quickly and easily than honey does. Honey is more viscous than water, so although gravity creates nearly the same stresses in honey and water, the more viscous fluid flows more slowly.

v_x = fluid velocity at distance y above the stationary plate, γ ˙ = velocity gradient or shear rate, δ = distance between plates.">
Planar Couette flow. v_x = fluid velocity at distance y above the stationary plate, γ ˙ = velocity gradient or shear rate, δ = distance between plates.

The viscosity can be measured where the fluid of interest is sheared between two flat plates which are parallel to one another (see illustration). This is known as planar Couette flow. The shear stress is the ratio of the tangential force F needed to maintain the moving plate at a constant velocity V to the plate area A. The shear flow created between the plates has the velocity profile given by Eq. (1),
1.
where v_x is the velocity parallel to the plates at a perpendicular distance y above the stationary plate. The coefficient γ ˙, called the velocity gradient or shear rate, is given by V/δ, where δ is the distance between the plates. It is expected that the shear stress increases with increasing shear rate but that the ratio of these two quantities depends only on the fluid between the plates. This ratio is used to define the shear viscosity, η, as in Eq. (2). The shear viscosity may depend on temperature, pressure,
2.
and shear rate.

Isaac Newton is credited with first suggesting a model for the viscous property of fluids in 1687. Newton proposed that the resistance to flow caused by viscosity is proportional to the velocity at which the parts of the fluid are being separated from one another because of the flow. Although Newton's law of viscosity is an empirical idealization, many fluids, such as low-molecular-weight liquids and dilute gases, are well characterized by it over a large range of conditions. However, many other fluids, such as polymer solution and melts, blood, ink, liquid crystals, and colloidal suspensions, are not described well by Newton's law. Such fluids are referred to as non-newtonian.

For planar Couette flow, Newton's law of viscosity is given mathematically by
3.
Eq. (3), where is the shear stress, and μ, a function of temperature and pressure, is the coefficient of viscosity or simply the viscosity. Therefore, by comparing Eqs. (2) and (3) the shear viscosity is equal to the coefficient of viscosity (that is, η = μ) for a newtonian fluid. Because of this re­lation the shear viscosity is also often referred to as the viscosity. However, it should be clear that the two quantities are not equivalent; μ is a newtonian-model parameter, which varies only with temperature and pressure, while η is a more general material property which may vary nonlinearly with shear rate. See also Fluid flow; Newtonian fluid.

From Eqs. (2) and (3), the units of viscosity are given by force per area per inverse time. If in planar Couette flow, for example, 1 dyne of tangential force is applied for every 1 cm² area of plate to create a velocity gradient of 1 s⁻¹, then the fluid between the plates has a viscosity of 1 poise (=1 dyne · s/cm²). Several viscosity units are in common use (see table). Comparison of the viscosities of different fluids demonstrates some general trends. For example, the viscosity of gases is generally much less than that of liquids. Whereas gases tend to become more viscous as temperature is increased, the opposite is true of liquids. Other data also show that increasing pressure tends to increase the viscosity of dense gases, but pressure has only a small effect on the viscosity of dilute gases and liquids.

Viscosity conversions

Unit	poise	cp	Pa · s	lbm/(ft · s)	lbf · s/ft2
1 poise*	1	100	0.1	6.72 × 10−2	2.089 ×10−3
1 centipoise	0.01	1	0.001	6.72 × 10−4	2.089 × 10−5
1 pascal-second†	10	1000	1	0.672	2.089 × 10−2
1 lbm/(ft · s)	14.88	1488	1.488	1	3.108 × 10−2
1 lbf · s/ft2	478.8	4.788 × 104	47.88	32.17	1

^*1 poise = 1 dyne · s/cm² = 1 g/(cm · s).

^†1 Pa · s = 1 kg/(m · s).

Whereas dilute gas molecules interact primarily in pairs as they collide, molecules in the liquid phase are in continuous interaction with many neighboring molecules. The concepts of average velocity and mean free path have little meaning for liquids. It is clear, however, that increasing temperature increases the mobility of molecules, thus allowing neighboring molecules to more easily overcome energy barriers and slip past one another. Such arguments lead to an exponential relation for the dependence of viscosity on temperature. See also Gas; Liquid.

Many non-newtonian fluids not only exhibit a viscosity which depends on shear rate (pseudoplastic or dilantant) but also exhibit elastic properties. These viscoelastic fluids require a large number of strain-rate-dependent material properties in addition to the shear viscosity to characterize them. The situation can become more complex when the material properties are time dependent (thixotropic or rheopectic). Fluids that are nonhomogeneous or nonisotropic require even more sophisticated analysis. The field of rheology attempts to deal with these complexities.

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