We present a systematic nonlinear theory of pattern selection for parametric surface waves (Faraday waves), not restricted to fluids of low viscosity. Typeset using REVT E X 1

We present a systematic nonlinear theory of pattern selection for parametric surface waves (Faraday waves), not restricted to fluids of low viscosity. Typeset using REVT E X 1

A mesoscopic model of a diblock copolymer is used to study the stability of a lamellar structure under a uniform shear flow. We first obtain the nonlinear lamellar solutions under both steady and oscillatory shear flows. Regions of existence of these solutions are determined as a function of the parameters of the model and of the flow. Finally, we address the stability of the lamellar solution against long wavelength perturbations.

A mesoscopic model of a diblock copolymer is used to study the stability of a uniform lamellar phase under a reciprocating shear flow. Approximate viscosity contrast between the microphases is allowed through a linear dependence of the (Newtonian) shear viscosity on monomer composition. We first show that viscosity contrast does not affect the composition of the base lamellar phase in an unbounded geometry, and that it only couples weakly to long wavelength perturbations. A perturbative analysis is then presented to address the stability of uniform lamellar structures under long wavelength perturbations by self-consistently solving for the composition and velocity fields of the perturbations. Stability boundaries are obtained as functions of the physical parameters of the polymer, the parameters of the flow and the initial orientation of the lamellae. We find that all orientations are linearly stable within specific ranges of parameters, but that the perpendicular orientation is generally stable within a larger range than the parallel orientation. Secondary instabilities are both of the Eckhaus type (longitudinal) and zig-zag type (transverse). The former is not expected to lead to re-orientation of the lamella, whereas in the second case the critical wavenumber is typically found to be along the perpendicular orientation.

Nonlinear wave dynamics in parametrically driven surface waves are studied in numerical simulations of the two-dimensional Navier-Stokes equation, with an emphasis on the evolution and interaction between different wave number modes. The dynamics are found to be closely correlated with the single-mode nonlinear saturated wave amplitudes. Modulating behavior of primary wave modes in a particular parameter range and in time scales much longer than the underlining wave periods is observed.