Viscosity depends strongly on temperature. In liquids it usually decreases with increasing temperature, whereas, in most gases, viscosity increases with increasing temperature. 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. Here dynamic viscosity is denoted by μ {\displaystyle \mu } and kinematic viscosity by ν {\displaystyle \nu } . The formulas given are valid only for an absolute temperature scale; therefore, unless stated otherwise temperatures are in kelvins.
Physical causes Viscosity in gases arises from molecules traversing layers of flow and transferring momentum between layers. This transfer of momentum can be thought of as a frictional force between layers of flow. Since the momentum transfer is caused by free motion of gas molecules between collisions, increasing thermal agitation of the molecules results in a larger viscosity. Hence, gaseous viscosity increases with temperature. In liquids, viscous forces are caused by molecules exerting attractive forces on each other across 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.
Gases
The kinetic theory of gases allows accurate calculation of the temperature-variation of gaseous viscosity. The theoretical basis of the kinetic theory is given by the Boltzmann equation and Chapman–Enskog theory, which allow accurate statistical modeling of molecular trajectories. 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 viscosity predictions for four molecular models are discussed below. The predictions of the first three models (hard-sphere, power-law, and Sutherland) can be simply expressed in terms of elementary functions. The Lennard–Jones model predicts a more complicated T {\displaystyle T} -dependence, but is more accurate than the other three models and is widely used in engineering practice.
Hard-sphere kinetic theory If one models gas molecules as elastic hard spheres (with mass m {\displaystyle m} and diameter σ {\displaystyle \sigma } ), then elementary kinetic theory predicts that viscosity increases with the square root of absolute temperature T {\displaystyle T} :
μ = 1.016 ⋅ 5 16 σ 2 ( k B m T π ) 1 / 2 {\displaystyle \mu =1.016\cdot {\frac {5}{16\sigma ^{2}}}\left({\frac {k_{\rm {B}}mT}{\pi }}\right)^{1/2}}
where k B {\displaystyle k_{\text{B}}} is the Boltzmann constant. While correctly predicting the increase of gaseous viscosity with temperature, the T 1 / 2 {\displaystyle T^{1/2}} trend is not accurate; the viscosity of real gases increases more rapidly than this. Capturing the actual T {\displaystyle T} dependence requires more realistic models of molecular interactions, in particular the inclusion of attractive interactions which are present in all real gases.
Power-law force A modest improvement over the hard-sphere model is a repulsive inverse power-law force, where the force between two molecules separated by distance r {\displaystyle r} is proportional to 1 / r α {\displaystyle 1/r^{\alpha }} , where α {\displaystyle \alpha } is an empirical parameter. 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. In this case, kinetic theory predicts an increase in temperature as T s {\displaystyle T^{s}} , where s = ( 1 / 2 ) + 2 / ( α − 1 ) {\displaystyle s=(1/2)+2/(\alpha -1)} . More precisely, if μ ′ {\displaystyle \mu '} is the known viscosity at temperature T ′ {\displaystyle T'} , then
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