Kinematics and Material Frame-Indifference

The Eulerian description of the instantaneous motion of a fluid with microstructure employs two independent vector fields. The first is the usual velocity v(x, t) and the second is an axial vector w(x, t) which, in the case of polar fluids, represents the angular velocity of the polar fluid particle at position x at time t. In the context of liquid crystals, w is interpreted as being the local angular velocity of the liquid crystal material element, that is, it represents the local angular velocity of the director n. In ordinary continuum theory the only independent field is the velocity v of the fluid because the angular velocity in such theories equals one half of the curl of the velocity. We denote this particular angular velocity by w defined 

The quantity uj is called the relative angular velocity and is introduced to measure the difference between the local angular velocity w of the liquid crystal director and the regional angular velocity w of the fluid in the neighbourhood of the director. 

It will always be assumed that the director n satisfies the constraint 

Since w represents the angular velocity of the director, we have that n, being a unit vector, also satisfies 

This derivative can also be expressed using the usual index notation and summation convention as employed in earlier Chapters, giving 

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