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started his research by improving the reliability of the solution procedure for small interface thicknesses, by implementing a two-level error estimation, which solves for the behavior at different scales in the space of the diffuse-interface model. The adaptive approximation framework builds a series of improved approximations in each step by following the Solve → Estimate → Mark → Refine steps. The adaptive refinement procedure was complemented by an additional procedure where the interface thickness parameter was adjusted during the refinement process, in such a way that the diffuse interface is adequately solved at each refinement level, for the interface thickness parameter at that level. In addition, a new way to solve problems was shown with a specific method by dividing the work. Furthermore, it was investigated in a structured way how the derivative structures behave in a typical two-dimensional model problem. Numerical results are presented for a two-dimensional vibrating droplet levitating in an environment, and these are confirmed with the results of a corresponding sharp-interface model.

The rate of approach

Next, the researcher developed new formulas for small oscillations of a droplet in a liquid in two dimensions. Using these expressions, the optimal order of the scaling relation of model parameters was clarified in terms of the rate of approach from the diffuse-interface solution to the sharp-interface solution. Furthermore, it was considered how a non-optimal choice of parameters affects the rate of approach. For two different modes of oscillation, the sharp-interface limit of the diffuse-interface model was investigated.

Appropriate boundary conditions for wetting

Finally, Demont considered how diffuse-interface models can be used in wetting situations by adding appropriate boundary conditions for wetting. He shows that the results do not match when the interface thickness goes to zero in the case of no-slip boundary conditions, which corresponds to the problem at the contact line in the traditional sharp-interface model. Furthermore, for the no-slip case, he considered how the position of the boundary deviates from the simulations of the traditional sharp-interface model, when the mobility parameter is fixed. He shows that, to solve the problems of no-slip boundary conditions in wetting processes, the diffuse-interface model transitions to the traditional sharp-interface model once the interface is compressed when we introduce slip at the wetting boundaries.

This comprehensive study not only advances the computational framework for diffuse-interface models but also provides insights into their behavior in limiting scenarios, supporting their broader application in fluid dynamics research.

Title of PhD thesis: . Supervisors: Prof. Harald van Brummelen, and Assistant Prof. Stein Stoter.