Momentum transport in gases is mediated by discrete molecular collisions, and in liquids by attractive forces that bind molecules close together. Because of this, the dynamic viscosities of liquids are typically much larger than those of gases. In addition, viscosity tends to increase with temperature in gases and decrease with temperature in liquids. Above the liquid-gas
critical point, the liquid and gas phases are replaced by a single
supercritical phase. In this regime, the mechanisms of momentum transport interpolate between liquid-like and gas-like behavior. For example, along a supercritical
isobar (constant-pressure surface), the kinematic viscosity decreases at low temperature and increases at high temperature, with a minimum in between. A rough estimate for the value at the minimum is : \nu_{\text{min}} = \frac{1}{4 \pi} \frac{\hbar}{\sqrt{m_\text{e} m}} where \hbar is the
Planck constant, m_\text{e} is the
electron mass, and m is the molecular mass. In general, however, the viscosity of a system depends in detail on how the molecules constituting the system interact, and there are no simple but correct formulas for it. The simplest exact expressions are the
Green–Kubo relations for the linear shear viscosity or the
transient time correlation function expressions derived by Evans and Morriss in 1988. Although these expressions are each exact, calculating the viscosity of a dense fluid using these relations currently requires the use of
molecular dynamics computer simulations. Somewhat more progress can be made for a dilute gas, as elementary assumptions about how gas molecules move and interact lead to a basic understanding of the molecular origins of viscosity. More sophisticated treatments can be constructed by systematically coarse-graining the
equations of motion of the gas molecules. An example of such a treatment is
Chapman–Enskog theory, which derives expressions for the viscosity of a dilute gas from the
Boltzmann equation.
Pure gases Elementary calculation of viscosity for a dilute gas Consider a dilute gas moving parallel to the x-axis with velocity u(y) that depends only on the y coordinate. To simplify the discussion, the gas is assumed to have uniform temperature and density. Under these assumptions, the x velocity of a molecule passing through y = 0 is equal to whatever velocity that molecule had when its mean free path \lambda began. Because \lambda is typically small compared with macroscopic scales, the average x velocity of such a molecule has the form u(0) \pm \alpha \lambda \frac{d u}{d y}(0), where \alpha is a numerical constant on the order of 1. (Some authors estimate \alpha = 2/3; on the other hand, a more careful calculation for rigid elastic spheres gives \alpha \simeq 0.998.) Next, because half the molecules on either side are moving towards y=0, and doing so on average with half the
average molecular speed (8 k_\text{B} T/\pi m)^{1/2}, the momentum flux from either side is \frac{1}{4} \rho \cdot \sqrt{\frac{8 k_\text{B} T}{\pi m}} \cdot \left(u(0) \pm \alpha \lambda \frac{d u}{d y}(0)\right). The net momentum flux at y=0 is the difference of the two: -\frac{1}{2} \rho \cdot \sqrt{\frac{8 k_\text{B} T}{\pi m}} \cdot \alpha \lambda \frac{d u}{d y}(0). According to the definition of viscosity, this momentum flux should be equal to -\mu \frac{d u}{d y}(0), which leads to \mu = \alpha \rho \lambda \sqrt{\frac{2 k_\text{B} T}{\pi m}}. Viscosity in gases arises principally from the
molecular diffusion that transports momentum between layers of flow. An elementary calculation for a dilute gas at temperature T and density \rho gives :\mu = \alpha\rho\lambda\sqrt{\frac{2k_\text{B} T}{\pi m}}, where k_\text{B} is the
Boltzmann constant, m the molecular mass, and \alpha a numerical constant on the order of 1. The quantity \lambda, the
mean free path, measures the average distance a molecule travels between collisions. Even without
a priori knowledge of \alpha, this expression has nontrivial implications. In particular, since \lambda is typically inversely proportional to density and increases with temperature, \mu itself should increase with temperature and be independent of density at fixed temperature. In fact, both of these predictions persist in more sophisticated treatments, and accurately describe experimental observations. By contrast, liquid viscosity typically decreases with temperature. For rigid elastic spheres of diameter \sigma, \lambda can be computed, giving : \mu = \frac{\alpha}{\pi^{3/2}} \frac{\sqrt{k_\text{B} m T}}{\sigma^2}. In this case \lambda is independent of temperature, so \mu \propto T^{1/2}. For more complicated molecular models, however, \lambda depends on temperature in a non-trivial way, and simple kinetic arguments as used here are inadequate. More fundamentally, the notion of a mean free path becomes imprecise for particles that interact over a finite range, which limits the usefulness of the concept for describing real-world gases.
Chapman–Enskog theory A technique developed by
Sydney Chapman and
David Enskog in the early 1900s allows a more refined calculation of \mu. It is based on the
Boltzmann equation, which provides a statistical description of a dilute gas in terms of intermolecular interactions. The technique allows accurate calculation of \mu for molecular models that are more realistic than rigid elastic spheres, such as those incorporating intermolecular attractions. Doing so is necessary to reproduce the correct temperature dependence of \mu, which experiments show increases more rapidly than the T^{1/2} trend predicted for rigid elastic spheres. Indeed, the Chapman–Enskog analysis shows that the predicted temperature dependence can be tuned by varying the parameters in various molecular models. A simple example is the Sutherland model, which describes rigid elastic spheres with
weak mutual attraction. In such a case, the attractive force can be treated
perturbatively, which leads to a simple expression for \mu: \mu = \frac{5}{16 \sigma^2} \left(\frac{k_\text{B} m T}{\pi}\right)^{\!\!1/2} \ \left(1 + \frac{S}{T} \right)^{\!\!-1}, where S is independent of temperature, being determined only by the parameters of the intermolecular attraction. To connect with experiment, it is convenient to rewrite as \mu = \mu_0 \left(\frac{T}{T_0}\right)^{\!\!3/2}\ \frac{T_0 + S}{T + S}, where \mu_0 is the viscosity at temperature T_0. This expression is usually named Sutherland's formula. If \mu is known from experiments at T = T_0 and at least one other temperature, then S can be calculated. Expressions for \mu obtained in this way are qualitatively accurate for a number of simple gases. Slightly more sophisticated models, such as the
Lennard-Jones potential, or the more flexible
Mie potential, may provide better agreement with experiments, but only at the cost of a more opaque dependence on temperature. A further advantage of these more complex interaction potentials is that they can be used to develop accurate models for a wide variety of properties using the same potential parameters. In situations where little experimental data is available, this makes it possible to obtain model parameters from fitting to properties such as pure-fluid
vapour-liquid equilibria, before using the parameters thus obtained to predict the viscosities of interest with reasonable accuracy. In some systems, the assumption of
spherical symmetry must be abandoned, as is the case for vapors with highly
polar molecules like
H2O. In these cases, the Chapman–Enskog analysis is significantly more complicated.
Bulk viscosity In the kinetic-molecular picture, a non-zero bulk viscosity arises in gases whenever there are non-negligible relaxational timescales governing the exchange of energy between the translational energy of molecules and their internal energy, e.g.
rotational and
vibrational. As such, the bulk viscosity is 0 for a monatomic ideal gas, in which the internal energy of molecules is negligible, but is nonzero for a gas like
carbon dioxide, whose molecules possess both rotational and vibrational energy.
Pure liquids In contrast with gases, there is no simple yet accurate picture for the molecular origins of viscosity in liquids. At the simplest level of description, the relative motion of adjacent layers in a liquid is opposed primarily by attractive molecular forces acting across the layer boundary. In this picture, one (correctly) expects viscosity to decrease with increasing temperature. This is because increasing temperature increases the random thermal motion of the molecules, which makes it easier for them to overcome their attractive interactions. Building on this visualization, a simple theory can be constructed in analogy with the discrete structure of a solid: groups of molecules in a liquid are visualized as forming "cages" which surround and enclose single molecules. These cages can be occupied or unoccupied, and stronger molecular attraction corresponds to stronger cages. Due to random thermal motion, a molecule "hops" between cages at a rate which varies inversely with the strength of molecular attractions. In
equilibrium these "hops" are not biased in any direction. On the other hand, in order for two adjacent layers to move relative to each other, the "hops" must be biased in the direction of the relative motion. The force required to sustain this directed motion can be estimated for a given shear rate, leading to {{NumBlk|:|\mu \approx \frac{N_\text{A} h}{V} \operatorname{exp}\left(3.8 \frac{T_\text{b}}{T}\right),|}} where N_\text{A} is the
Avogadro constant, h is the
Planck constant, V is the volume of a
mole of liquid, and T_\text{b} is the
normal boiling point. This result has the same form as the well-known empirical relation {{NumBlk|:|\mu = A e^{B/T},|}} where A and B are constants fit from data. On the other hand, several authors express caution with respect to this model. Errors as large as 30% can be encountered using equation (), compared with fitting equation () to experimental data. More fundamentally, the physical assumptions underlying equation () have been criticized. It has also been argued that the exponential dependence in equation () does not necessarily describe experimental observations more accurately than simpler, non-exponential expressions. In light of these shortcomings, the development of a less ad hoc model is a matter of practical interest. Foregoing simplicity in favor of precision, it is possible to write rigorous expressions for viscosity starting from the fundamental equations of motion for molecules. A classic example of this approach is Irving–Kirkwood theory. On the other hand, such expressions are given as averages over multiparticle
correlation functions and are therefore difficult to apply in practice. In general, empirically derived expressions (based on existing viscosity measurements) appear to be the only consistently reliable means of calculating viscosity in liquids. Local atomic structure changes observed in undercooled liquids on cooling below the equilibrium melting temperature either in terms of radial distribution function
g(
r) or structure factor
S(
Q) are found to be directly responsible for the liquid fragility: deviation of the temperature dependence of viscosity of the undercooled liquid from the Arrhenius equation (2) through modification of the activation energy for viscous flow. At the same time equilibrium liquids follow the Arrhenius equation.
Mixtures and blends Gaseous mixtures The same molecular-kinetic picture of a single component gas can also be applied to a gaseous mixture. For instance, in the
Chapman–Enskog approach the viscosity \mu_{\text{mix}} of a binary mixture of gases can be written in terms of the individual component viscosities \mu_{1,2}, their respective volume fractions, and the intermolecular interactions. As for the single-component gas, the dependence of \mu_{\text{mix}} on the parameters of the intermolecular interactions enters through various collisional integrals which may not be expressible in
closed form. To obtain usable expressions for \mu_{\text{mix}} which reasonably match experimental data, the collisional integrals may be computed numerically or from correlations. This is a common approach in the development of
reference equations for gas-phase viscosities. An example of such a procedure is the Sutherland approach for the single-component gas, discussed above. For gas mixtures consisting of simple molecules,
Revised Enskog Theory has been shown to accurately represent both the density- and temperature dependence of the viscosity over a wide range of conditions. can be motivated from various theoretical models of amorphous materials at the atomic level. A two-exponential equation for the viscosity can be derived within the Dyre shoving model of supercooled liquids, where the Arrhenius energy barrier is identified with the high-frequency
shear modulus times a characteristic shoving volume. Upon specifying the temperature dependence of the shear modulus via thermal expansion and via the repulsive part of the intermolecular potential, another two-exponential equation is retrieved: : \mu = \exp{\left\{ \frac{V_c C_G}{k_{B}T} \exp{\left[(2+\lambda)\alpha_T T_g \left(1-\frac{T}{T_g}\right)\right]}\right\}} where C_{G} denotes the high-frequency
shear modulus of the material evaluated at a temperature equal to the
glass transition temperature T_{g} , V_{c} is the so-called shoving volume, i.e. it is the characteristic volume of the group of atoms involved in the shoving event by which an atom/molecule escapes from the cage of nearest-neighbours, typically on the order of the volume occupied by few atoms. Furthermore, \alpha_{T} is the
thermal expansion coefficient of the material, \lambda is a parameter which measures the steepness of the power-law rise of the ascending flank of the first peak of the
radial distribution function, and is quantitatively related to the repulsive part of the
interatomic potential. Finally, k_{B} denotes the
Boltzmann constant.
Eddy viscosity In the study of
turbulence in
fluids, a common practical strategy is to ignore the small-scale
vortices (or
eddies) in the motion and to calculate a large-scale motion with an
effective viscosity, called the "eddy viscosity", which characterizes the transport and dissipation of
energy in the smaller-scale flow (see
large eddy simulation). In contrast to the viscosity of the fluid itself, which must be positive by the
second law of thermodynamics, the eddy viscosity can be negative. ==Prediction==