Shear flow near a boundary

Consider flow near a stationary solid boundary at y = 0 as shown in Fig. 5.5. Following Leslie [163] and the exposition by Muller [207], we expect solutions to equations (5.121) and (5.122) to satisfy the boundary conditions 

Integrating (5.126) between y and infinity and employing the boundary condition (5.123)3 results in the relation 

This provides y as a function of 0, that is, the solution 0 is implicitly given as a function of y in (5.129) clearly, from the form of (5.129), (/ increases monotonically with y. Once the solution (j has been determined from equation (5.129), an integration of equation (5.121) combined with the boundary condition (5.123)i concludes the solution to the problem by determining the velocity 

As commented upon by Muller [207], the constant c can be determined by the shear stress required to keep the lower plate in place, since, by (4.56), (5.108) and the given dependence of n and v upon y. 

The solutions for j y) and v y) are given via equations (5.129) and (5.130). In general, the evaluations of the resultant integrals have to be carried out numerically for given values of the elastic constants and viscosities once c is known. 

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Section 5.5 describes two simple shear flows shear flow near a boundary and shear flow between two parallel plates. This leads on to a discussion of scaling properties, and it is pointed out in Section 5.5.5 that the apparent viscosity, defined by equation (5.146) below, scales differently from that of an isotropic fluid. Stability and instabiUty of oscillatory shear flow are discussed in Section 5.6 this Section contains what is perhaps the most advanced analysis in this Chapter. 

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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