# Sutherland’s Law

When working on practical engineering applications, you often encounter problems where there are significant changes in temperature locally in the flow field. For instance, if you are doing a heat transfer calculation, you usually want to use the film properties (aka the properties adjacent to the wall). Doing this in any practical sense requires a model for viscosity and thermal conductivity.

For pure, non-reacting gas, the viscosity is only dependent on temperature (Anderson 2006). Therefore, viscosity can be modelled using temperature alone. One of these models, and one that is built in with software like OpenFOAM, is the Sutherland’s law.

It is given by:

$\mu=\mu_o\frac{T_o + C}{T+C}\left(\frac{T}{T_o}\right)^{3/2}$

It is also often simplified (as it is in OpenFOAM) to:

$\mu=\frac{C_1 T^{3/2}}{T+C}=\frac{A_s T^{3/2}}{T+T_s}$

The right hand side gives the OpenFOAM notation. For air, $\mu_o=1.716\times 10^{-5}$, $T_o=273.15$ $C=T_s=110.4$ $C_1=A_s=1.458\times10^{-6}$ .

The coefficients for several other gases can be found in Frank White’s Viscous Fluid Flow (2006).

## When is it applicable?

This is an important question. If you are doing engineering or research, you should know the limits of these equations. According to Anderson (2006), “Sutherland’s law is accurate for air over a range of several thousand degrees and is certainly appropriate for hypersonic viscous-flow calculations”. In the crucial undergraduate textbook by Frank White, states that it is “adequate over a wide range of temperatures”.

For air, another source (Rathakrishnan 2013) states that the relationship is valid from 0.01 to 100 atm, and between 0 and 3000K. And Frank White’s Viscous Fluid Flow mentions 2% error between 170K and 1900K for air.

## What if you can’t find the coefficients?

If you can’t find a good reference listing the coefficients of the gas you are looking at, one option is to head over to the NIST Webbook, download 2 viscosity values at two different temperatures and solve for the 2 coefficients. Or curve-fit in Matlab using a series of data from NIST or some other source. Basically, something along those lines. This is explained in the undergraduate book by Munson (2014).

FYI that is a good idea anyway, if you want to quantify the error in your work or simulations.

## References

Rathakrishnan, E. (2013). Theoretical aerodynamics. John Wiley & Sons.

Anderson Jr, J. D. (2006). Hypersonic and high-temperature gas dynamics. American Institute of Aeronautics and Astronautics.

Munson, B. R., Okiishi, T. H., Rothmayer, A. P., & Huebsch, W. W. (2014). Fundamentals of fluid mechanics. John Wiley & Sons.

White, F. M. (2009). Fluid mechanics. Boston, Mass: WCB/McGraw-Hill.

White, F. M., & Corfield, I. (2006). Viscous fluid flow (Vol. 3, pp. 433-434). New York: McGraw-Hill.

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