The temperature dependence of liquid viscosity is usually expressed by one of the following models:

Exponential model

$\backslash mu(T)=\backslash mu\_0\; \backslash exp(-bT)$

where T is temperature and $\backslash mu\_0$ and $b$ are coefficients. See first-order fluid and second-order fluid. This is an empirical model that usually works for a limited range of temperatures.

Arrhenius model

The model is based on the assumption that the fluid flow obeys the Arrhenius equation for molecular kinetics:

$\backslash mu(T)=\backslash mu\_0\; \backslash exp(\; \backslash frac\; \{E\}\{RT\}\; )$

where T is temperature, $\backslash mu\_0$ is a coefficient, E is the activation energy and R is the universal gas constant. A first-order fluid is another name for a power-law fluid with exponential dependence of viscosity on temperature.

WLF model

The Williams-Landel-Ferry model or WLF for short is usually used for polymer melt's or other fluids that have a glass transition temperature.

The model is:

$\backslash mu(T)=\backslash mu\_0\; \backslash exp\; \backslash left(\; \backslash frac\; \{-C\_1\; (T-T\_r)\}\; \{C\_2+\; T\; -T\_r\}\; \backslash right)$

where T-temperature, $C\_1$, $C\_2$, $T\_r$ and $\backslash mu\_0$ are empiric parameters (only three of them are independent from each other).

If one selects the parameter $T\_r$ based on the glass transition temperature, then the parameters $C\_1$, $C\_2$ become very similar for the wide class of polymers. Typically, if $T\_r$ is set to match the glass transition temperature $T\_g$, we get

$C\_1\; \backslash approx$17.44

and

$C\_2\; \backslash approx$51.6 K.

Van Krevelen recommends to choose

$T\_r=T\_g+43$ K, then

$C\_1\; \backslash approx$8.86

and

$C\_2\; \backslash approx$101.6 K.

Using such universal parameters allows one to guess the temperature dependence of a polymer by knowing the viscosity at a single temperature.

In reality the universal parameters are not that universal, and it is much better to fit the WLF parameters from the experimental data.

Non-Newtonian fluids