Two-dimensional flow and linear stability properties of smectic A liquid crystals

We examine some leading-order flow and stability properties of smectic A (SmA) liquid crystals (LCs) in two spatial dimensions by analysing a fully nonlinear continuum theory of these materials. We derive a system of equations for the dynamic variables describing the flow velocity and orientation of the material under suitable assumptions upon these quantities. This system can provide insight into the leading-order behaviour under quite general circumstances, and we provide an example of utilising this system to determine the flow induced by a constant pressure gradient applied normally to the smectic layers. We then consider the effect of oscillatory perturbations on a relaxed, stationary sample of SmA, and provide criteria under which one would expect to see the onset of instability in the form of inequalities between the material parameters and perturbative wave number. We find that instability occurs for physically realisable values of these quantities, and, in particular, that certain viscosities characterising the SmA phase can act as 'destabilising agents' such that one could, for a given sample with known parameter values, manipulate the behaviour of that sample.