This article discusses several models of this dependence, ranging from rigorous first-principles calculations for monatomic gases, to empirical correlations for liquids.
Understanding the temperature dependence of viscosity is important for many applications, for instance engineering lubricants that perform well under varying temperature conditions (such as in a car engine), since the performance of a lubricant depends in part on its viscosity.
Engineering problems of this type fall under the purview of tribology.
This transfer of momentum can be thought of as a frictional force between layers of flow.
Increasing temperature results in a decrease in viscosity because a larger temperature means particles have greater thermal energy and are more easily able to overcome the attractive forces binding them together.
An everyday example of this viscosity decrease is cooking oil moving more fluidly in a hot frying pan than in a cold one.
The kinetic theory of gases allows accurate calculation of the temperature-variation of gaseous viscosity.
In particular, given a model for intermolecular interactions, one can calculate with high precision the viscosity of monatomic and other simple gases (for more complex gases, such as those composed of polar molecules, additional assumptions must be introduced which reduce the accuracy of the theory).
The predictions of the first three models (hard-sphere, power-law, and Sutherland) can be simply expressed in terms of elementary functions.
If one models gas molecules as elastic hard spheres (with mass
), then elementary kinetic theory predicts that viscosity increases with the square root of absolute temperature
trend is not accurate; the viscosity of real gases increases more rapidly than this.
[3] This is not a realistic model for real-world gases (except possibly at high temperature), but provides a simple illustration of how changing intermolecular interactions affects the predicted temperature dependence of viscosity.
, which results in faster increase of viscosity compared with the hard-sphere model.
, called the Sutherland constant, can be expressed in terms of the parameters of the intermolecular attractive force.
obtained from fitting to experimental data are shown in the table below for several gases.
In general, it has been argued that the Sutherland model is actually a poor model of intermolecular interactions, and is useful only as a simple interpolation formula for a restricted set of gases over a restricted range of temperatures.
Under fairly general conditions on the molecular model, the kinetic theory prediction for
is called the collision integral and is a function of temperature as well as the parameters of the intermolecular interaction.
[5] It is completely determined by the kinetic theory, being expressed in terms of integrals over collisional trajectories of pairs of molecules.
is a complicated function of both temperature and the molecular parameters; the power-law and Sutherland models are unusual in that
Nevertheless, it is frequently used for non-spherically symmetric molecules provided these do not possess a large dipole moment.
estimated from experimental data are shown in the table below for several common gases.
In contrast with gases, there is no systematic microscopic theory for liquid viscosity.
[9] However, there are several empirical models which extrapolate a temperature dependence based on available experimental viscosities.
A simple and widespread empirical correlation for liquid viscosity is a two-parameter exponential: This equation was first proposed in 1913, and is commonly known as the Andrade equation (named after British physicist Edward Andrade).
Comprehensive tables of these parameters for hundreds of liquids can be found in the literature.
[12] One can also find tabulated exponentials with additional parameters, for example and Representative values are given in the tables below.
, often a polynomial fit to experimental data, has been added to the Walther formula.
The Seeton model is based on curve fitting the viscosity dependence of many liquids (refrigerants, hydrocarbons and lubricants) versus temperature and applies over a large temperature and viscosity range: where