Viscosity

In continuum mechanics, viscosity is a property of a fluid that quantifies the resistance force acting on fluids when there is relative motion between fluid parcels. This resistance force is caused by the stress in fluid parcels, which ideally is directly proportional to the strain rate (the time derivative of strain) that arises when fluid parcels are in relative motion, and the relative speed between the boundary between adjacent fluid parcels is zero. In liquids, viscosity arises from cohesive molecular forces, while in gases it results from molecular collisions. Except for the case of superfluidity, there is no fluid with zero viscosity, and thus all fluid flows involve viscous effects to some degree. For liquids, it corresponds to the informal concept of thickness; for example, syrup has a higher viscosity than water. Viscosity is defined scientifically as a force multiplied by a time divided by an area. Thus its SI units are newton-seconds per metre squared, or pascal-seconds. For instance, when a viscous fluid is forced through a tube, it flows more quickly near the tube's center line than near its walls. Some stress (such as a pressure difference between the two ends of the tube) is needed to sustain the flow. This is because a force is required to overcome the friction between the layers of the fluid which are in relative motion. For a tube with a constant rate of flow, the strength of the compensating force is proportional to the fluid's viscosity. In general, viscosity depends on a fluid's state, such as its temperature, pressure, and rate of deformation. However, the dependence on some of these properties is negligible in certain cases. For example, the viscosity of a Newtonian fluid does not vary significantly with the rate of deformation. Zero viscosity (no resistance to shear stress) is observed only at very low temperatures in superfluids; otherwise, the second law of thermodynamics requires all fluids to have positive viscosity. A fluid that has zero viscosity (non-viscous) is called ideal or inviscid. For non-Newtonian fluids' viscosity, there are pseudoplastic, plastic, and dilatant flows that are time-independent, and there are thixotropic and rheopectic flows that are time-dependent.

Etymology The word "viscosity" is derived from the Latin viscum ("mistletoe"). Viscum also referred to a viscous glue derived from mistletoe berries.

Definitions

Dynamic viscosity

In materials science and engineering, there is often interest in understanding the forces or stresses involved in the deformation of a material. For instance, if the material were a simple spring, the answer would be given by Hooke's law, which says that the force experienced by a spring is proportional to the distance displaced from equilibrium. Stresses which can be attributed to the deformation of a material from some rest state are called elastic stresses. In other materials, stresses are present which can be attributed to the deformation rate over time. These are called viscous stresses. For instance, in a fluid such as water the stresses which arise from shearing the fluid do not depend on the distance the fluid has been sheared; rather, they depend on how quickly the shearing occurs. Viscosity is the material property which relates the viscous stresses in a material to the rate of change of a deformation (the strain rate). Although it applies to general flows, it is easy to visualize and define in a simple shearing flow, such as a planar Couette flow. In the Couette flow, a fluid is trapped between two infinitely large plates, one fixed and one in parallel motion at constant speed u {\displaystyle u} (see illustration to the right). If the speed of the top plate is low enough (to avoid turbulence), then in steady state the fluid particles move parallel to it, and their speed varies from 0 {\displaystyle 0} at the bottom to u {\displaystyle u} at the top. Each layer of fluid moves faster than the one just below it, and friction between them gives rise to a force resisting their relative motion. In particular, the fluid applies on the top plate a force in the direction opposite to its motion, and an equal but opposite force on the bottom plate. An external force is therefore required in order to keep the top plate moving at constant speed. In many fluids, the flow velocity is observed to vary linearly from zero at the bottom to u {\displaystyle u} at the top. Moreover, the magnitude of the force, F {\displaystyle F} , acting on the top plate is found to be proportional to the speed u {\displaystyle u} and the area A {\displaystyle A} of each plate, and inversely proportional to their separation y {\displaystyle y} :

F = μ A u y . {\displaystyle F=\mu A{\frac {u}{y}}.}

The proportionality factor is the dynamic viscosity of the fluid, often simply referred to as the viscosity. It is denoted by the Greek letter mu (μ). The dynamic viscosity has the dimensions ( m a s s / l e n g t h ) / t i m e {\displaystyle \mathrm {(mass/length)/time} } , therefore resulting in the SI units and the derived units:

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