# Where does this viscosity formula come from?

I was reading a scientific article and i came across this formula:

$$\frac{\tau_{xy}}{V_{/x}} = \eta_{0}+\eta_{2}V_{/x}^{2}+\eta_{4}V_{/x}^{4}$$

which refers to the calculation of the viscosity of oxygen; $\tau_{xy}$ is the stress tensor and $V_{/x}$ is the derivative of the velocity field of the flow. So here's the question: has anyone the idea of where this formula comes from? I believe it's some sort of series expansion related to the more traditional formula $\tau = \mu\frac{\partial u}{\partial y}$ but I don't know how.

• Do you have a reference to the article where you saw this? It does look like a model for nonlinear behavior, with only even terms. – Floris Jan 6 '15 at 23:43
• Unfortunately the article is not free; it is entitled "oxygen transport properties estimation by dsmc-ct simulations"; the problem is that I don't understand where the formula comes from... it says by the way that the if the limit of vanishing $V_{/x}$, $\tau_{xy}$ is proportional through the viscosity to the imposed shear. – Federico Gentile Jan 6 '15 at 23:55

An outline (abstract & introduction) to the paper can be viewed for free here.

In the paper, the authors explain that how they attempt to compute the transport properties of rarefied oxygen by 'modeling binary collisions through an accurate potential energy surface' and then assess the accuracy of the model using a range of simulation techniques and comparing the results with experimental data.

The simulation technique involves using 'massively parallel computers' and GPU acceleration to calculate shear viscosity and heat conductivity using 'direct simulation Monte Carlo' (DSMC) methods coupled with a 'classical trajectory' (CT) calculations.

The 'formula' for viscosity which you have provided in your question is nothing more than a polynomial expression giving the ratio of oxygen viscosity to the shear rate as a function of powers of the shear rate. In other words, they are doing a regression to 'fit' the discrete data from the simulation to a 'continuous curve' using a polynomial expression, so that interpolations can be made and the results of different simulations compared to see how closely they correlate.

The precise form of the polynomial has no physical basis, but is purely a mathematical technique for obtaining interpolated data from 'discrete' data by trying to find a 'smooth' curve which 'fits' the discrete data.

The form of the polynomial is generally governed by the requirements for 'smoothness' imposed by the (partial) differential equations which govern the model. In this case, the simulation is based on the non-homogenous Boltzmann equation. For more detail on numerical methods for solving the Boltzmann equation, refer to this paper.

• Ok perfect!!! thank you so much!!! that's exactly what I was looking for! – Federico Gentile Jan 7 '15 at 1:07