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 T η {\displaystyle \eta } P 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:
ρ [ ∂ u ∂ t + u ⋅ ∇ u ] = − ∇ P + ∇ [ ζ ( ∇ ⋅ u ) ] + ∇ ⋅ [ η ( ∇ u + ( ∇ u ) T − 2 3 ( ∇ ⋅ u ) I ) ] + ρ g {\displaystyle \rho \left[{\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right]=-\nabla P+\nabla [\zeta (\nabla \cdot \mathbf {u} )]+\nabla \cdot \left[\eta \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{T}-{\frac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right)\right]+\rho \mathbf {g} }
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