A. Appendix 1: Relation between the pressure tensor, velocity gradient, and viscosity coefficients

The relation between the velocity gradient, ∇u, and the pressure tensor, P, is more complicated in an axially symmetric system such as nematic liquid crystal than in an isotropic fluid due to the lower symmetry. In order to derive the linear phenomenological relations between the velocity gradient and the pressure, it is appropriate to begin by identifying the thermodynamic forces and fluxes in the expression for the irreversible entropy production [3, 4, 23, 36]:

$$\sigma = -\frac{1}{T} \left\{ \overline{\mathbf{P}} : \overline{\mathbf{Vu}} + 2\mathbf{P}^d \cdot (\mathbb{W}\nabla \times \mathbf{u} - \mathfrak{Q}) + \left( \frac{1}{3} \text{Tr}(\mathbf{P}) - p\_{eq} \right) \nabla \cdot \mathbf{u} \right\},\tag{A.1}$$

where T is the absolute temperature, and u is the streaming velocity. The various parts of the second rank tensor are denoted in the following manner: the symmetric traceless part is given by <sup>A</sup> <sup>¼</sup> <sup>½</sup> <sup>A</sup> <sup>þ</sup> <sup>A</sup><sup>T</sup> � � � ð Þ <sup>1</sup>=<sup>3</sup> Trð Þ <sup>A</sup> <sup>1</sup> and the pseudovector dual of the antisymmetric part is denoted by <sup>A</sup><sup>a</sup> ¼ �½ε:A ¼ �½εαβγAγβ, where <sup>ε</sup> is the Levi-Civita tensor. Three pairs of thermodynamic forces and fluxes can be identified by inspection of the irreversible entropy production, namely, the symmetric traceless pressure tensor and the traceless strain rate, P and ∇u, the antisymmetric pressure and the difference between the rotation and the director angular velocity, <sup>P</sup><sup>a</sup> and ½<sup>∇</sup> � <sup>u</sup> � <sup>Ω</sup>, and the difference between the trace of the pressure tensor and the equilibrium pressure of a quiescent liquid crystal, and the trace of the strain rate, 1ð Þ =3 Trð Þ� P peq and ∇�u. Note that the strain rate is defined as ½ ∇u þ ð Þ ∇u <sup>T</sup> h i, and it is always symmetric. In a uniaxially symmetric nematic liquid crystal, the relations between the pressure and the velocity gradient can be deduced by symmetry arguments, and they can be expressed in a few different equivalent ways [23, 36]. It has been found that a notation due to Hess [36] is the most convenient one for deducing Green-Kubo relations and NEMD-algorithms:

$$\begin{split} \langle \overline{\mathbf{P}} \rangle = -2\eta \overline{\nabla \mathbf{u}} - \tilde{\eta}\_1 \overline{\overline{\mathbf{m}} \mathbf{\overline{u}} \cdot \overline{\nabla \mathbf{u}}} - 2\tilde{\eta}\_3 \overline{\mathbf{m}} \, \overline{\mathbf{m}} \mathbf{\overline{m}} \overline{\nabla \mathbf{u}} + 2\tilde{\eta}\_2 \overline{\overline{\mathbf{m}} \mathbf{\overline{u}} \cdot \mathbf{e} \cdot (\mathbb{M} \nabla \times \mathbf{u} - \boldsymbol{\Omega})} - \zeta \overline{\mathbf{m}} \nabla \cdot \mathbf{u}, \end{split} \tag{A.2a}$$

$$
\langle \mathbf{P}^{\mathfrak{a}} \rangle = -\frac{\tilde{\mathcal{V}}\_1}{2} (\mathsf{W} \nabla \times \mathbf{u} - \mathfrak{Q}) \ -\frac{\tilde{\mathcal{V}}\_2}{2} \mathbf{e} : (\overline{\mathbf{m}} \mathbf{\overline{n}} \cdot \overline{\nabla \mathbf{u}}), \tag{A.2b}
$$

and

$$\frac{1}{3}\langle \text{Tr}(\mathbf{P})\rangle - p\_{eq} = -\eta\_V \nabla \cdot \mathbf{u} - \kappa \overline{\mathbf{m}} \overline{\mathbf{n}} \overline{\nabla \mathbf{u}},\tag{A.2c}$$
