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

Abstract

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 C 1 -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.

An extended stochastic diffusion model for ternary mixtures

“…The demanding continuity requirements for spatial discretization will be accomplished by a B-spline based finite element method according to our earlier works, cf. Anders and Weinberg (2011b), Anders et al (2012) and Anders and Weinberg (2011c). To illustrate the flexibility and versatility of our approach this section is also devoted to representative two-dimensional simulations of several ternary morphologies.…”

Isogeometric analysis and hierarchical refinement for higher-order phase-field models

“…C 1 continuity. 2 To account for this type of problems, we introduce at next hierarchical refinement strategies.…”

“…In combination with finite element based solution strategies they enable a fascinating variety of multi-field and multi-physics simulations, cf. [1][2][3][4][5] and many others. In order to demonstrate the capabilities of the presented mathematical framework an in-depth investigation of a series of higher-order problems is presented.…”

A Chemo-Mechanical Model of Diffusion in Reactive Systems

“…To get insight into the mathematical details of these discretization schemes the authors refer to their original papers on the treatment of similar Cahn-Hilliard type phase-field problems, cf. [16][17][18][19].…”

“…The elastic energy (10) employs concentration dependent material parameters. Thus the concentration independent elastic moduli are set within (17) and (18) to be K A = G A = 10 GPa and K B = G B = 9 GPa, respectively. Please note that in this model the material softens with rising concentration c B ; an opposite model is possible as well.…”