Conservation laws and the entropy inequality

We begin by writing down the conservation or balance laws (Ericksen ). We shall employ the cartesian tensor notation, repeated tensor indices being subject to the usual summation convention. The comma denotes partial differentiation with respect to spatial coordinates and the superposed dot a material time derivative. For example, 

We shall consider the medium to be incompressible v = 0, where v is the linear velocity) and at constant temperature t = Tj = 0). We shall assume further that the director is of constant magnitude. This implies that the external forces and fields responsible for elastic deformation, viscous flow, etc., are very much weaker than the molecular interactions giving rise to the spontaneous alignment of the neighbouring molecules. It is indeed a valid assumption in all the static and dynamic phenomena discussed in this chapter. We may therefore conveniently choose n to be a dimensionless unit vector (n n — 1). 

Let the material volume be Abounded by a surface A. The conservation laws take the following form  

N may be interpreted as the angular velocity of the director relative to that 

In conjunction with the balance equations we make use of the inequality 

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