Numerical and analytical study of the convective Cahn-Hilliard equation

We consider the convective Cahn-Hilliard equation that is used as a model of coarsening dynamics in driven systems and that in two spatial dimensions (x; y) has the form ut + Duux + r2(u 􀀀 u3 + r2u) = 0: Here t denotes time, u = u(x; y; t) is the order parameter and D is the parameter measuring the strength of driving. We primarily consider the case of one spatial dimension, when there is no y-dependence. For the case of no driving, when D = 0, the standard Cahn-Hilliard equation is recovered, and it is known that solutions to this equation are characterised by an initial stage of phase separation into regions of one phase surrounded by the other phase (i.e., clusters or droplets/holes or islands are obtained) followed by the coarsening process, where the average size of the clusters grows in time and the number of the clusters decreases. Moreover, two main coarsening mechanisms have been identified in the literature, namely, coarsening due to volume and translational modes. On the other hand, for the case of strong driving, when D ! 1, the well-known Kuramoto-Sivashinsky equation is recovered, solutions of which are characterised by complicated chaotic oscillations in both space and time. The primary aim of the present thesis is to perform a detailed and systematic investigation of the transitions in the solutions of the convective Cahn-Hilliard equation for a wide range of parameter values as the driving-force parameter is increased, and, in particular, to understand in detail how the coarsening dynamics is affected by driving. We find that one of the coarsening modes is stabilised at relatively small values of D, and the type of the unstable coarsening mode may change as D increases. In addition, we find that there may be intervals in the driving-force parameter D where coarsening is completely stabilised. On the other hand, there may be intervals where twomode solutions are unstable and the solutions can evolve, for example, into one-droplet/hole solutions, symmetry-broken two-droplet/hole solutions or time-periodic solutions. We present detailed stability diagrams for 2-mode solutions in the parameter planes and corroborate our findings by time-dependent simulations. Finally, we present preliminary results for the case of the (convective) Cahn-Hilliard equation in two spatial dimensions.