Governing Equations for Shear Flow

To illustrate flow behaviour that is often considered to be more typical than the Newtonian behaviour indicated by (5.88), it will be assumed that the director and the velocity take the forms 

Clearly, the constraints (4.118) are satisfied and, because of the dependence of n and V upon y only, the governing Ericksen-Leslie dynamic equations (4.119) and 

Using the form for i 2 given by equation (5.101), the result (5.108) may be formulated conveniently as [163] 

Equations (5.95) have now been reduced to (5.112), the pressure being available via (5.111) to allow this reduction. 

Our attention is now turned to the remaining equations (5.96). These equations, using (4.131), reduce to 

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In this Section we describe in detail the shear flow examples discussed by Leslie [163] for nematic liquid crystals. We first make some comments on Newtonian and non-Newtonian behaviour of fluids in Section 5.5.1 before going on to derive the general explicit governing equations for shear flow, equations (5.121) and (5.122), in Section 5.5.2. We then specialise in Sections 5.5.3 and 5.5.4 to the specific problems of shear flow near a boundary and shear flow between parallel plates. Section 5.5.5 discusses some scaling properties for nematics. 

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