SynopsisThe construction of the approximate solution within a rectangle of a singular elliptic problem is discussed. It is found that, provided the boundary data satisfy certain continuity conditions at the corners of the rectangle, ordinary boundary layers and parabolic boundary layers only are necessary to describe the solution. A correction term, however, has to be added to the solution if the continuity conditions on the boundary data are not satisfied.

This paper presents a new method for determining the critical threshold values at which both homogeneous and roll-type instabilities may appear when thin layers of nematic liquid crystal are subjected to Couette flow. The anisotropic axis of the material is initially uniformly aligned parallel to the axial direction and the effect of a stabilising, axially oriented, uniform magnetic field is allowed for in calculations. Carrying out a linear stability analysis of the continuum equations, we employ a Fourier series method to derive an expression for determining the critical shear rate at which instability occurs. Thresholds for both homogeneous and roll instabilities are obtained and their variation with rotation rate and applied magnetic field strength is examined for the material MBBA.

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