# Conservation Equations forming a Determinate Set

I was reading "molecular gas dynamics" by Bird and came across the the statement that

For conservation equations to form a determinate set shear stress and heat flux must be expressed in terms of lower order macroscopic quantities.

My question is what does a determinate set mean and secondly why does shear stress and heat flux need to be expressed in lower order terms.

## 1 Answer

The "determinant set" requirement means that all of the variables are defined and related. In other words, it is a closed system of equations. This require models for the thermodynamic and transport properties. The heat flux and shear stress terms introduce new variables, specifically $\kappa$ and $\mu$ (the heat diffusion coefficient and the viscous diffusion coefficient). In order to solve the governing equations, we need some expressions for those coefficients. These expressions make the system determinant.

The lower order terms of macroscopic quantities means whatever expressions we generate for $\kappa$ and $\mu$ must contain only things already known in the equation set, otherwise we introduce another round of variables and require yet another round of models for those variables to close the set. These terms need to be low order because that's all we know about them. It's not "low order" in the exponent sense -- for instance, you could have $\mu$ as a 20-th order polynominal in temperature if you wanted.

It is low order in a statistical sense. If we needed our expression for $\mu$ to be based on a time averaged temperature, we can't solve the equations because we don't know a time averaged temperature until we solve the equations. It's circular. Time averaging is a first-order term. If it depended on the skewness or kurtosis or even higher order moments of variables, or high order moments of correlations of variables, then it's even harder to solve (although, "harder" isn't really true because it's already impossible...).

To summarize, to be determinant we must have equations relating all of the variables to all of the other variables. These expressions must depend only on known information, such as the instantaneous values of the variables and are thus low-order terms.