Viscosity

When two fluid layers move relative to each other, a friction force develops between them and the slower layer acts to slow down the faster layer. This internal resistance to flow is described by the fluid property called viscosity, which reflects the internal stickiness of the fluid. 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 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 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 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. 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. 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 . If the speed of the top plate is low enough, then in steady state the fluid particles move parallel to it, and their speed varies from at the bottom to 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 at the top. Moreover, the magnitude of the force,, acting on the top plate is found to be proportional to the speed and the area of each plate, and inversely proportional to their separation :
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. The dynamic viscosity has the dimensions, therefore resulting in the SI units and the derived units:
The aforementioned ratio is called the rate of shear deformation or shear velocity, and is the derivative of the fluid speed in the direction parallel to the normal vector of the plates. If the velocity does not vary linearly with, then the appropriate generalization is:
where, and is the local shear velocity. This expression is referred to as Newton's law of viscosity. In shearing flows with planar symmetry, it is what defines. It is a special case of the general definition of viscosity, which can be expressed in coordinate-free form.
Use of the Greek letter mu for the dynamic viscosity is common among mechanical and chemical engineers, as well as mathematicians and physicists. However, the Greek letter eta is also used by chemists, physicists, and the IUPAC. The viscosity is sometimes also called the shear viscosity. However, at least one author discourages the use of this terminology, noting that can appear in non-shearing flows in addition to shearing flows.

Kinematic viscosity

In fluid dynamics, it is sometimes more appropriate to work in terms of kinematic viscosity, defined as the ratio of the dynamic viscosity over the density of the fluid. It is usually denoted by the Greek letter nu :
and has the dimensions, therefore resulting in the SI units and the derived units:

General definition

In very general terms, the viscous stresses in a fluid are defined as those resulting from the relative velocity of different fluid particles. As such, the viscous stresses must depend on spatial gradients of the flow velocity. If the velocity gradients are small, then to a first approximation the viscous stresses depend only on the first derivatives of the velocity. In Cartesian coordinates, the general relationship can then be written as
where is a viscosity tensor that maps the velocity gradient tensor onto the viscous stress tensor. Since the indices in this expression can vary from 1 to 3, there are 81 "viscosity coefficients" in total. However, assuming that the viscosity rank-2 tensor is isotropic reduces these 81 coefficients to three independent parameters,, :
and furthermore, it is assumed that no viscous forces may arise when the fluid is undergoing simple rigid-body rotation, thus, leaving only two independent parameters. The most usual decomposition is in terms of the standard viscosity and the bulk viscosity such that and. In vector notation this appears as:
where is the unit tensor. This equation can be thought of as a generalized form of Newton's law of viscosity.
The bulk viscosity expresses a type of internal friction that resists the shearless compression or expansion of a fluid. Knowledge of is frequently not necessary in fluid dynamics problems. For example, an incompressible fluid satisfies and so the term containing drops out. Moreover, is often assumed to be negligible for gases since it is in a monatomic ideal gas. One situation in which can be important is the calculation of energy loss in sound and shock waves, described by Stokes' law of sound attenuation, since these phenomena involve rapid expansions and compressions.
The defining equations for viscosity are not fundamental laws of nature, so their usefulness, as well as methods for measuring or calculating the viscosity, must be established using separate means. A potential issue is that viscosity depends, in principle, on the full microscopic state of the fluid, which encompasses the positions and momenta of every particle in the system. Such highly detailed information is typically not available in realistic systems. However, under certain conditions most of this information can be shown to be negligible. In particular, for Newtonian fluids near equilibrium and far from boundaries, the viscosity depends only space- and time-dependent macroscopic fields defining local equilibrium.
Nevertheless, viscosity may still carry a non-negligible dependence on several system properties, such as temperature, pressure, and the amplitude and frequency of any external forcing. Therefore, precision measurements of viscosity are only defined
with respect to a specific fluid state. To standardize comparisons among experiments and theoretical models, viscosity data is sometimes extrapolated to ideal limiting cases, such as the zero shear limit, or the zero density limit.

Momentum transport

Transport theory provides an alternative interpretation of viscosity in terms of momentum transport: viscosity is the material property which characterizes momentum transport within a fluid, just as thermal conductivity characterizes heat transport, and diffusivity characterizes mass transport. This perspective is implicit in Newton's law of viscosity,, because the shear stress has units equivalent to a momentum flux, i.e., momentum per unit time per unit area. Thus, can be interpreted as specifying the flow of momentum in the direction from one fluid layer to the next. Per Newton's law of viscosity, this momentum flow occurs across a velocity gradient, and the magnitude of the corresponding momentum flux is determined by the viscosity.
The analogy with heat and mass transfer can be made explicit. Just as heat flows from high temperature to low temperature and mass flows from high density to low density, momentum flows from high velocity to low velocity. These behaviors are all described by compact expressions, called constitutive relations, whose one-dimensional forms are given here:
where is the density, and are the mass and heat fluxes, and and are the mass diffusivity and thermal conductivity. The fact that mass, momentum, and energy transport are among the most relevant processes in continuum mechanics is not a coincidence: these are among the few physical quantities that are conserved at the microscopic level in interparticle collisions. Thus, rather than being dictated by the fast and complex microscopic interaction timescale, their dynamics occurs on macroscopic timescales, as described by the various equations of transport theory and hydrodynamics.