governing the evolution of $\Pi^{\mu\nu}$ must be given. Usually, this is done within the gradient expansion for hydrodynamic variables [@deGroot; @ferz; @kremer; @revhydro1; @revhydro2; @hyd1; @hyd2]. Here, instead, $\Pi^{\mu\nu}$ is an algebraic function of $\lambda^{(1)ij}$ and $\zeta^{a(1)}$, so we must obtain the evolution equation for $\lambda^{(1)ij}$. We will get the latter from the kinetic equation.

It will prove convenient to express our results in terms of a new relaxation time $\tau_\pi$ (related by a constant to the previously introduced $\tau$) = \^[-1]{} and the shear viscosity = \^[-1]{} \^2 which naturally arise in the context of second order fluid dynamics [@revhydro1; @revhydro2; @hyd1; @hyd2]. For simplicity, we also introduce a new variable \^[ij]{}T \^[(1)ij]{}

We shall deal with the conservation equations first, and then go over to discuss the evolution equation for $\gamma^{ij}$.

### Conservation equations

Inserting the expression for $\mathbf{f}$ given by Eq. (\[closfin\]) into Eqs. (\[5\]) and (\[8\]), and using that $T^{\mu\nu}_{m,\nu}=-T^{\mu\nu}_{YM,\nu}=
\mathrm{tr}\mathbf{J}_{\lambda}\mathbf{F}^{\mu\lambda}$ in the latter, we get equations of motion for the velocity $u^\mu$, the energy density $\rho$ and $\zeta^{a(1)}$.

For the matter energy-momentum tensor $T^{\mu\nu} = T^{\mu\nu}_0 + \Pi^{\mu\nu}_1+\Pi^{\mu\nu}_2$, where T\^\_0 = (u\^u\^+\^) \[T0\] is the perfect-fluid energy-momentum tensor, \^\_1 = \^ \[pi1\] and

\^\_2 &= 