On the thermodynamics of the Swift–Hohenberg theory

We present the microbalance including the microforces, the first- and second-order microstresses for the Swift–Hohenberg equation concomitantly with their constitutive equations, which are consistent with the free-energy imbalance. We provide an explicit form for the microstress structure for a free-energy functional endowed with second-order spatial derivatives. Additionally, we generalize the Swift–Hohenberg theory via a proper constitutive process. Finally, we present one highly resolved three-dimensional numerical simulation to demonstrate the particular form of the resulting microstresses and their interactions in the evolution of the Swift–Hohenberg equation

Energy exchange analysis in droplet dynamics via the Navier-Stokes-Cahn-Hilliard model

We develop the energy budget equation of the coupled Navier-Stokes-Cahn-Hilliard (NSCH) system. We use the NSCH equations to model the dynamics of liquid droplets in a liquid continuum. Buoyancy effects are accounted for through the Boussinesq assumption. We physically interpret each quantity involved in the energy exchange to gain further insight into the model. Highly resolved simulations involving density-driven flows and the merging of droplets allow us to analyse these energy budgets. In particular, we focus on the energy exchanges when droplets merge, and describe flow features relevant to this phenomenon. By comparing our numerical simulations to analytical predictions and experimental results available in the literature, we conclude that modelling droplet dynamics within the framework of NSCH equations is a sensible approach worthy of further research