Consider these structured materials to consist of a Newtonian solvent that has suspended in it small rigid rod-shaped particles.

Constitutive equations: Here we consider the constitutive equation for a material with microstructure, such as liquid crystals which are now used in many different everyday components (e.g. digital displays, computer screens, etc.). Consider these structured materials to consist of a Newtonian solvent that has suspended in it small rigid rod-shaped particles. These particles are orientable and we denote the average orientation vector N (with N · N = 1), which is also called the director field (see figure); in general the director field is unknown and has to be solved for simultaneously with the velocity field, which is a complicated problem! Here we wish only to think about the constitutive equation relating stress to the rate of strain. We wish to treat this new “fluid” (thinking about the solvent and particles as an effective medium) as an incompressible continuum (velocity field u) since the suspended particles are very small, but we also wish to account in the constitutive equation for the orientability of the microstructure. Hence, we expect N to enter the form of the viscous stress tensor t, where recall that the total stress tensor is T = -pI + t .

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