# Temperature dependence of liquid viscosity

The temperature dependence of liquid viscosity is the phenomenon by which liquid viscosity tends to decrease (or, alternatively, its fluidity tends to increase) as its temperature increases. This can be observed, for example, by watching how cooking oil appears to move more fluidly upon a frying pan after being heated by a stove.

## Physical causes

Kinematic viscosity can be estimated as a typical (thermal) velocity times the mean free path.[1] A molecular view of liquids can be used for a qualitative picture of decrease in the shear (or bulk) viscosity of a simple fluid with temperature. As the temperature increases, the thermal velocity increases. However, much more important is the rapid decrease of the mean free path with temperature. The reason for this is that temperature increase releases more and more molecules to move around and interact with any given molecule. The actual process can be quite complex and is typically represented by simplified mathematical or empirical models, some of which are discussed below.[2] The models are valid over limited temperature ranges and for selected materials.

## Models for shear viscosity

### Exponential Model

An exponential model for the temperature-dependence of shear viscosity (μ) was first proposed by Reynolds in 1886.[3]

${\displaystyle \mu (T)\,=\,\mu _{0}\exp(-bT)}$

where T is temperature and ${\displaystyle \mu _{0}}$ and ${\displaystyle b}$ 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:

${\displaystyle \mu (T)\,=\,\mu _{0}\exp \left({\frac {E}{RT}}\right)}$

where T is temperature, ${\displaystyle \mu _{0}}$ 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.

### Williams-Landel-Ferry model

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

The model is:

${\displaystyle \mu (T)\,=\,\mu _{0}\exp \left({\frac {-C_{1}(T-T_{r})}{C_{2}+T-T_{r}}}\right)}$

where T-temperature, ${\displaystyle C_{1}}$, ${\displaystyle C_{2}}$, ${\displaystyle T_{r}}$ and ${\displaystyle \mu _{0}}$ are empiric parameters (only three of them are independent from each other).

If one selects the parameter ${\displaystyle T_{r}}$ based on the glass transition temperature, then the parameters ${\displaystyle C_{1}}$, ${\displaystyle C_{2}}$ become very similar for the wide class of polymers. Typically, if ${\displaystyle T_{r}}$ is set to match the glass transition temperature ${\displaystyle T_{g}}$, we get

${\displaystyle C_{1}\approx }$17.44

and

${\displaystyle C_{2}\approx 51.6}$ K.

Van Krevelen recommends to choose

${\displaystyle T_{r}\,=\,T_{g}+43}$ K, then
${\displaystyle C_{1}\approx 8.86}$

and

${\displaystyle C_{2}\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.

### Masuko and Magill model

The model is usually used for polymer melts or other fluids that have a glass transition temperature as well as the WLF model. Ordinarily, The WLF model is limited to the temperature interval between Tg and Tg+ 100 K, But this model can be applied to more wide temperature range. [4]

The model is:

${\displaystyle \log(\eta /\eta _{g})=A\left[\exp \left\{{\frac {B(T_{g}-T)}{T}}\right\}-1\right]}$

The A and B are empirical parameters that does not depend on the materials. The average values are:

A = 14.25 to 16.24,

and

B = 5.34 to 7.60.

### Viscosity of water

Viscosity of water equation accurate to within 2.5% from 0 °C to 370 °C:

${\displaystyle \mu (T)=2.414\times 10^{-5}\times 10^{247.8/(T-140)}}$[5]

where T has units of Kelvin, and μ has units of N*s/m^2.

## Models for kinematic viscosity

The effect of temperature on the kinematic viscosity (ν) has also been described by a number of empirical equations.

### Walther formula

The Walther formula[2] is typically written in the form

${\displaystyle \log _{10}[\log _{10}(\nu +\lambda )]=A-B\,\log _{10}(T)}$

where λ is a shift constant, and A, B are empirical parameters.

### Wright model

The Wright model[2] has the form

${\displaystyle \log _{10}[\log _{10}[\nu +\lambda +f(\nu )]]=A-B\,\log _{10}(T)}$

where an addition function f(ν), often a polynomial fit to experimental data, has been added to the Walther formula.

### Seeton model

The Seeton model[2] 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:

${\displaystyle \ln \left({\ln \left({\nu +0.7+e^{-\nu }K_{0}\left({\nu +1.244067}\right)}\right)}\right)=A-B*\ln \left(T\right)}$

where T is absolute temperature in kelvins, ${\displaystyle \nu }$ is the kinematic viscosity in centistokes, ${\displaystyle K_{0}}$ is the zero order modified Bessel function of the second kind, and A and B are liquid specific values. This form should not be applied to ammonia or water viscosity over a large temperature range.

For liquid metal viscosity as a function of temperature, Seeton proposed:

${\displaystyle \ln \left({\ln \left({\nu +0.7+e^{-\nu }K_{0}\left({\nu +1.244067}\right)}\right)}\right)=A-{B \over T}}$

## Notes

1. ^ Falkovich, Gregory (2011). Fluid mechanics (a short course for physicists. Cambridge University Press :. ISBN 9781107005754.
2. ^ a b c d Seeton, Christopher J. (2006), "Viscosity-temperature correlation for liquids", Tribology Letters, doi:10.1007/s11249-006-9071-2
3. ^ Reynolds O. (1886). Phil Trans Royal Soc London, v. 177, p.157.
4. ^ Toru Masuko, Joseph H. Magill (1988), A comprehensive expression for temperature dependence of liquid viscosity., Journal of the Society of Rheology Japan, Vol.16
5. ^ Al-Shemmeri, Tarik (2012). Engineering Fluid Mechnanics. Ventus Publishing ApS. pp. 17–18. ISBN 978-87-403-0114-4.