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A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation

The mixing and de‐mixing properties of highly viscous fluids can be described as a diffusive system driven by an external velocity field. In order to analyze the micro‐morphological evolution with a diffusion theory of heterogeneous mixtures we apply a Cahn‐Hilliard phase‐field model, which is here extended by a convective term. As the model is based on an energetic variational formulation, we state the underlying energy minimizing principles. For consistent finite element analysis we provide a piecewise smooth and globally C1‐continuous approximation by means of rational B‐spline basis functions. Numerical examples exhibit the physical properties of the model. Simulations of phase decomposition and coarsening controlled by diffusion and by convection show the robustness and the versatility of our approach.