The Grad closure

Grad, in the late 1940s, introduced an assumption on the fourth moment [99]:

$$\mathcal{R}_{iklh}=\frac{1}{mn}(\mathcal{P}_{ik}\mathcal{P}_{lh}+... ...hcal{T}_{lh}-\mathcal{T}_{il}\mathcal{T}_{kh}-\mathcal{T}_{ih}\mathcal{T}_{kl})$$ (2.118)

The simple physical meaning of Grad's relation is that it is satisfied by relatively smooth distribution functions and it does not allow for 'spiky' distribution functions. This is the only requirement that the distribution function $P$ must fulfill, no specific assumptions on the form of this distribution are needed. Using this assumption, the Maxwell equations for the first moments (2.122) become a closed set: it is a set of $20$ equations for $20$ unknown scalar functions. The unknown functions are $n$, $\mathbf{u}$ ($3$ components), $\mathcal{P}$ ($6$ components), $\mathcal{Q}$ ($10$ components). The equations are the first three groups of the set (2.122), which are $1+3+6=10$ scalar equations, plus the equation for the heat flow tensor that must be rewritten using the Grad relation:

$$\begin{multline} \frac{\partial \mathcal{Q}_{klh}}{\partial t}+ \left(\mathcal{Q... ...rtial u_l}{\delta t} -\mathcal{P}_{kl}\frac{\delta u_h}{\delta t} \end{multline}$$