Turing instability conditions in confined systems with an effective position-dependent diffusion coefficient.

We study a reaction-diffusion system within a long channel in the regime in which the projected Fick-Jacobs-Zwanzig operator for confined diffusion can be used. We found that under this approximation, Turing instability conditions can be modified due to the channel geometry. The dispersion relation, range of unstable modes where pattern formation occurs, and spatial structure of the patterns itself change as functions of the geometric parameters of the channel. This occurs for the three channels analyzed, for which the values of the projected operators can be found analytically. For the reaction term, we use the well-known Schnakenberg kinetics.

Membrane stress and torque induced by Frank's nematic textures: A geometric perspective using surface-based constraints.

An elastic membrane with embedded nematic molecules is considered as a model of anisotropic fluid membrane with internal ordering. By considering the geometric coupling between director field and membrane curvature, the nematic texture is shown to induce anisotropic stresses additional to Canham-Helfrich elasticity. Building upon differential geometry, analytical expressions are found for the membrane stress and torque induced by splaying, twisting, and bending of the nematic director as described by the Frank energy of liquid crystals. The forces induced by prototypical nematic textures are visualized on the sphere and on cylindrical surfaces.