The shear viscosity (or "viscosity" for short) of a fluid is a material property that describes the friction between internal neighboring fluid surfaces (or "sheets") flowing with different fluid velocities. This friction is the effect of linear momentum exchange, caused by molecules with sufficient energy to move or "jump" between these fluid sheets due to fluctuations in their motion.
The viscosity is not a material constant, but a material property that depends on temperature, pressure, fluid mixture composition, and local velocity variations. This functional relationship is described by a mathematical viscosity model called a constitutive equation, which is usually far more complex than the defining equation of shear viscosity. One such complicating feature is the relation between the viscosity model for a pure fluid and the model for a fluid mixture which is called mixing rules. When scientists and engineers use new arguments or theories to develop a new viscosity model, instead of improving the reigning model, it may lead to the first model in a new class of models.
This article will display one or two representative models for different classes of viscosity models. These classes are:
Elementary kinetic theory and simple empirical models - viscosity for dilute gas with nearly spherical molecules Power series - simplest approach after dilute gas Equation of state analogy between PVT and TP Corresponding state model - scaling a variable with its value at the critical point Friction force theory - internal sliding surface analogy to a sliding box on an inclined surface Multi- and one-parameter version of friction force theory Transition state analogy - molecular energy needed to squeeze into a vacancy analogous to molecules locking into each other in a chemical reaction Free volume theory - molecular energy needed to jump into a vacant position in the neighboring surface Significant structure theory - based on Eyring's concept of liquid as a blend of solid-like and gas-like behavior / features
Selected contributions from these development directions are displayed in the following sections. This means that some known contributions of research and development directions are not included. For example, the group contribution method applied to a shear viscosity model is not displayed. Even though it is an important method, it is thought to be a method for parameterization of a selected viscosity model, rather than a viscosity model in itself.
The microscopic or molecular origin of fluids means that transport coefficients like viscosity can be calculated by time correlations which are valid for both gases and liquids; however, these calculations are computer-intensive. Another approach utilises the Boltzmann equation, which describes the statistical behaviour of a thermodynamic system not in a state of equilibrium. It can be used to determine how physical quantities (such as heat energy and momentum) change when a fluid is in transport. This approach also involves computer-intensive simulations.
From Boltzmann's equation, one may also analytically derive analytical mathematical models for properties characteristic to fluids, such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). (See also the convection–diffusion equation.) The complexity of the mathematics for polar and non-spherical molecules makes it very difficult to get practical models for viscosity. The purely theoretical approach will therefore be left out in the rest of this article, except for some discussions related to dilute gas and significant structure theory.
Use, definition and dependence The classic Navier-Stokes equation is the balance equation for momentum density for an isotropic, compressional and viscous fluid. It is used in fluid mechanics in general and fluid dynamics in particular:
is the mass density, and is the fluid velocity. On the right hand side is the divergence of the total stress tensor , which consists of a pressure tensor and a dissipative (or viscous or deviatoric) stress tensor . The dissipative stress consists of a compression stress tensor (the 2nd term) and a shear stress tensor (the 3rd term). The rightmost term is the gravitational force, which constitutes the body force contribution.
For fluids, the spatial or Eularian form of the governing equations is preferred to the material or Lagrangian form, and the concept of velocity gradient is preferred to the equivalent concept of strain rate tensor. Stokes assumptions for a wide class of fluids therefore says that for an isotropic fluid, the compression and shear stresses are proportional to their velocity gradients, and respectively, and named this class of fluids for Newtonian fluids. The classic defining equation for volume viscosity and shear viscosity are respectively:
The classic compression velocity "gradient" is a diagonal tensor that describes a compressing (or expanding) flow or attenuating sound waves:
The classic Cauchy shear velocity gradient is a symmetric and traceless tensor that describes a pure shear flow (where pure means "excluding normal outflow", which in mathematical terms means a traceless matrix) around, for example, a wing, propeller, ship hull, or in, for example, a river, pipe or vein with or without bends and boundary skin:
where the symmetric gradient matrix with non-zero trace is
How much the volume viscosity contributes to the flow characteristics - for example, in a choked flow such as convergent-divergent nozzle or valve flow - is not well known, but the shear viscosity is by far the most utilized viscosity coefficient.
The volume viscosity will now be ignored, and the rest of this article will focus on the shear viscosity.
Another application of shear viscosity models is Darcy's law for multiphase flow.
where is water, oil or gas, and and are absolute and relative permeability, respectively. These 3 vector equations model flow of water, oil, and natural gas in subsurface oil and gas reservoirs in porous rocks. Although the pressures changes are big, the fluid phases will flow slowly through the reservoir due to the flow restriction caused by the porous rock.
The above definition is based on a shear-driven fluid motion that, in its most general form, is modelled by a shear stress tensor and a velocity gradient tensor. However, the fluid dynamics of a shear flow is well-illustrated by the simple Couette flow. In this experimental layout, the shear stress and the shear velocity gradient (where now ) takes the simple form:
Inserting these simplifications gives a defining equation that can be used to interpret experimental measurements:
where is the area of the moving plate and the stagnant plate, and is the spatial coordinate normal to the plates. In this experimental setup, value for the force is first selected, and then a maximum velocity is measured. Both values are then substituted into the equation to calculate viscosity, giving a single value for the viscosity of the selected fluid. If another value of the force is selected, another maximum velocity will be measured. This will result in another viscosity value if the fluid is a non-Newtonian fluid such as paint, but will give the same viscosity value for a Newtonian fluid such as water, petroleum oil or gas. If another parameter like temperature, , is changed, and the experiment is repeated with the same force, a new value for viscosity will be calculated for both non-Newtonian and Newtonian fluids.
The majority of material properties vary as a function of temperature, including viscosity. The viscosity is also a function of pressure and, of course, the material itself. For a fluid mixture, this means that the shear viscosity will also vary according to the fluid composition. Mapping the viscosity as a function of all these variables requires a large sequence of experiments that generates an even larger set of numbers called measured data, observed data or observations. Prior to, or at the same time as, the experiment, a material property model (or short material model) is proposed to describe or explain the observations. This mathematical model is called the constitutive equation for shear viscosity. It is usually an explicit function that contains some empirical parameters that are adjusted in order to match the observations as accurately as the mathematical function is capable of.
For a Newtonian fluid, the constitutive equation for shear viscosity is generally a function of temperature, pressure and fluid composition:
where is the liquid phase composition with molfraction for fluid component , and and are the gas phase and total fluid compositions, respectively.
For a non-Newtonian fluid (in the sense of a generalized Newtonian fluid), the constitutive equation for shear viscosity is also a function of the shear velocity gradient:
The existence of the velocity gradient in the functional relationship for non-Newtonian fluids implies that viscosity is generally not an equation of state, so the term "constitutional equation" will generally be used for viscosity equations or functions. The free variables in the two equations above also indicate that specific constitutive equations for shear viscosity will differ from the simple defining equation for shear viscosity. The rest of this article will show that this is true. Non-Newtonian fluids will therefore be ignored for the rest of this article to focus on Newtonian fluids.
Dilute gas limit and scaled variables
Elementary kinetic theory
In textbooks on elementary kinetic theory one can find results for dilute gas modeling that have widespread use. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallel plates are separated by a gas layer. This non-equilibrium flow is superimposed on a Maxwell–Boltzmann equilibrium distribution of molecular motions.
Let be the collision cross section of one molecule colliding with another. The number density is defined as the number of molecules per extensive volume . The collision cross section per volume (or collision cross section density) is , and it is related to the mean free path by:
Combining the kinetic equations for molecular motion with the defining equation of shear viscosity gives the equation for shear viscosity for dilute gases:
where
where is the Boltzmann constant, is the Avogadro constant, is the gas constant, is the molar mass and is the molecular mass. The equation above presupposes that the gas density is low (i.e. that the pressure is low), hence the subscript zero in the variable . This implies that the kinetic translational energy dominates over rotational and vibrational molecule energies. The viscosity equation displayed above further presupposes that there is only one type of gas molecule, and that the gas molecules are perfect elastic hard core particles of spherical shape. This assumption of particles being like billiard balls with radius implies that the collision cross section of one molecule can be estimated by:
However, molecules in reality are not hard particles. For a reasonably spherical molecule, the interaction potential is closer to the Lennard-Jones potential, or even closer to the Morse potential. Both have a negative component that attracts the other molecule from distances much greater than the hard core radius, and thus models the Van der Waals forces. The positive component models the repulsive forces as the electron clouds of the two molecules overlap. The radius for zero interaction potential is therefore appropriate for estimating (or defining) the collision cross section in kinetic gas theory, and the r-parameter (conf. ) is therefore referred to as the kinetic radius. The d-parameter (where ) is called the kinetic diameter.