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# CONSISTENT AND CONSERVATIVE PHASE-FIELD METHOD FOR MULTIPHASE FLOW PROBLEMS

This dissertation focuses on a consistent and conservative Phase-Field method for multiphase flow problems, and it includes both model and scheme development. The first general question addressed in the present study is the multiphase volume distribution problem. A consistent and conservative volume distribution algorithm is developed to solve the problem, which eliminates the production of local voids, overfilling, or fictitious phases, but follows the mass conservation of each phase. One of its applications is to determine the Lagrange multipliers that enforce the mass conservation in the Phase-Field equation, and a reduction consistent conservative Allen-Cahn Phase-Field equation is developed. Another application is to remedy the mass change due to implementing the contact angle boundary condition in the Phase-Field equations whose highest spatial derivatives are second-order. As a result, using a 2nd-order Phase-Field equation to study moving contact line problems becomes possible.

The second general question addressed in the present study is the coupling between a given physically admissible Phase-Field equation to the hydrodynamics. To answer this general question, the present study proposes the

*consistency of mass conservation*and the*consistency of mass and momentum transport*, and they are first implemented to the Phase-Field equation written in a conservative form. The momentum equation resulting from these two consistency conditions is Galilean invariant and compatible with the kinetic energy conservation, regardless of the details of the Phase-Field equation. It is further illustrated that the 2nd law of thermodynamics and*consistency of reduction*of the entire multiphase system only rely on the properties of the Phase-Field equation. All the consistency conditions are physically supported by the control volume analysis and mixture theory. If the Phase-Field equation has terms that are not in a conservative form, those terms are treated by the proposed consistent formulation. As a result, the proposed consistency conditions can always be implemented. This is critical for large-density-ratio problems.The consistent and conservative numerical framework is developed to preserve the physical properties of the multiphase model. Several new techniques are developed, including the gradient-based phase selection procedure, the momentum conservative method for the surface force, the boundedness mapping resulting from the volume distribution algorithm, the "DGT" operator for the viscous force, and the correspondences of numerical operators in the discrete Phase-Field and momentum equations. With these novel techniques, numerical analyses ensure that the mass of each phase and momentum of the multiphase mixture are conserved, the order parameters are bounded in their physical interval, the summation of the volume fractions of the phases is unity, and all the consistency conditions are satisfied, on the fully discrete level and for an arbitrary number of phases. Violation of the consistency conditions results in inconsistent errors proportional to the density contrasts of the phases. All the numerical analyses are carefully validated, and various challenging multiphase flows are simulated. The results are in good agreement with the exact/asymptotic solutions and with the existing numerical/experimental data.

The multiphase flow problems are extended to including mass (or heat) transfer in moving phases and solidification/melting driven by inhomogeneous temperature. These are accomplished by implementing an additional consistency condition, i.e.,

*consistency of volume fraction conservation*, and the diffuse domain approach. Various problems are solved robustly and accurately despite the wide range of material properties in those problems.## History

## Degree Type

- Doctor of Philosophy

## Department

- Mechanical Engineering

## Campus location

- West Lafayette