Acknowledgements

We gratefully acknowledge financial support from the Knut and Alice Wallenberg Foundation (Project number KAW 2012.0078) and Vetenskapsrådet (Swedish Research Council) (Project number 2013-5171). The simulations were performed using resources provided by the Swedish National Infrastructure for Computing (SNIC) at PDC, HPC2N, and NSC.) We also acknowledge PRACE for awarding us access to Hazelhen based in Germany at Rechenzentrum Stuttgart.

where the products involving the Levi-Civita tensor ε are defined in the following way: ε : A ¼ εαβγAγβ and A � ε � B ¼ AαβεβγδBδ. The quantities η, ~η1, and ~η<sup>3</sup> are shear viscosities, ~γ<sup>1</sup> is the twist viscosity, η<sup>V</sup> is the volume viscosity, ~η<sup>2</sup> is the cross coupling coefficient relating the difference between the rotation and the director angular velocity, and the symmetric traceless pressure. According to the Onsager reciprocity relations (ORR), this coefficient is equal to ~γ2=2, the cross coupling coefficient relating the traceless strain rate and the antisymmetric pressure. The trace of the strain rate and the symmetric traceless pressure are related by the cross coupling coefficient, ζ, which, according to the ORR, is equal to the cross coupling coefficient κ between the traceless strain rate and the difference between the trace of the pressure tensor and the equilibrium

Variational Principle for Nonequilibrium Steady States Tested by Molecular Dynamics…

DOI: http://dx.doi.org/10.5772/intechopen.80977

Application of a planar Couette velocity gradient, <sup>∇</sup><sup>u</sup> <sup>¼</sup> <sup>γ</sup> <sup>e</sup>zex, where <sup>γ</sup> <sup>¼</sup> <sup>∂</sup>zux is the shear rate and fixation of the director in the zx-plane at an angle θ relative to the stream lines, see Figure 2, by application of an electric or a magnetic field gives the following relations between the pressure tensor components and the strain rate in a director based coordinate system ð Þ e1; e2; e<sup>3</sup> where the director points in the e3-

3

<sup>3</sup> <sup>þ</sup> <sup>2</sup> ~η3 3

> γ <sup>2</sup> � <sup>~</sup>γ<sup>2</sup> γ 2

where ^λ<sup>2</sup> is the external torque density acting on the system. From these equa-

In a planar elongational flow [8, 26–28], where the elongation direction is parallel to the x-axis, the contraction direction is parallel to the negative z-axis, and the velocity field is equal to u ¼ γð Þ xe<sup>x</sup> � ze<sup>z</sup> , so that the velocity gradient becomes ∇u ¼ γð Þ exe<sup>x</sup> � eze<sup>z</sup> . Then the linear relations between the velocity gradient and the pressure become the following in a director-based coordinate system ð Þ e1; e2; e<sup>3</sup> where the director points in the e1-direction, and θ is the angle between the director

3

6 <sup>γ</sup> cos 2<sup>θ</sup> <sup>þ</sup> <sup>~</sup>η<sup>2</sup>

¼ �~γ<sup>1</sup>

tions, it is apparent that the various elements of the pressure tensor are linear functions of sin 2θ and cos 2θ, so the various viscosity coefficients can be evaluated by fixing the director at a few different angles relative to the stream lines and

<sup>γ</sup> sin 2θ, (A.3a)

<sup>3</sup> <sup>~</sup>η<sup>1</sup> <sup>þ</sup> <sup>~</sup>η3Þ<sup>γ</sup> sin 2θ, (A.3b)

<sup>γ</sup> sin 2θ, (A.3c)

<sup>γ</sup> cos 2θ, (A.4a)

<sup>3</sup> <sup>~</sup>η<sup>1</sup> <sup>þ</sup> <sup>~</sup>η3Þ<sup>γ</sup> cos 2θ, (A.4b)

, (A.3d)

cos 2θ, (A.3e)

γ 2

<sup>p</sup><sup>11</sup> <sup>¼</sup> <sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>3</sup>

<sup>p</sup><sup>22</sup> <sup>¼</sup> <sup>1</sup>

<sup>p</sup><sup>33</sup> ¼ � <sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>1</sup>

<sup>p</sup><sup>31</sup> <sup>¼</sup> <sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>1</sup>

2 pa 2 <sup>¼</sup> ^λ<sup>2</sup>

calculating the averages of the pressure tensor elements.

and the elongation direction or x-axis, e<sup>2</sup> ¼ e<sup>y</sup> and e<sup>3</sup> ¼ e<sup>1</sup> � e2,

<sup>p</sup><sup>22</sup> ¼ � <sup>2</sup>

<sup>p</sup><sup>11</sup> ¼ �<sup>2</sup> <sup>η</sup> <sup>þ</sup> <sup>~</sup>η<sup>3</sup>

pressure.

direction:

and

139
