ITR: Analysis and Simulation of Interface Dynamics in Multiphase Fluids and Solids

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ABSTRACTThe investigator and his colleagues are concerned with the development of new computational techniques for simulating interface problems in fluids and solids that are founded on recent advances in the analysis of the underlying mathematical models. The primary objectives of the research are to: (1) develop novel computational algorithms for fluids with moving interfaces governed by surface tension, with rigorous error bounds and convergence proofs, based on mollifications and higher order regularizations of boundary terms proposed by the investigator and his colleagues; (2) extend their existing solid collision algorithms to include non-interpenetration invariants and inexpensive energy approximations, allowing the efficient simulation of thin-shell impact problems; (3) develop new phase-field schemes for multiphase fluids with surface tensions effects based on their recent analytical proofs, which can handle topological changes and provably converge; (4) prove the existence of unique solutions for fluids with complex nonlinear surface energies, including surfactant stress and elastic membrane energies; (5) use these new analytical results for complex surface energies to develop convergent algorithms for such problems, focusing on fluid flow in elastic structures, and combine these methods with the existing solids collision code; (6) use the new analytical results and computational tools to investigate the problem of the existence and possible uniqueness of solutions to solids systems involving collisions.The physical mechanisms of multiphase flows that the investigator and his colleagues propose to model include droplet formation and breakup, spray, coalescence, wave breaking, solid-body collisions, fragmentation, and fracture. These phenomena are of primary importance in many areas of science such as ocean-atmosphere dynamics, aircraft and hydrocraft design and analysis, and many biological applications such as arterial blood flows. There are a host of engineering applications such as the manufacturing of advanced materials from nanotechnology to polymeric fluids. Solid body collision, damage, and fracture are of paramount importance in host of areas ranging from anti-ballistic design of armored vehicles and buildings to the response of the earth's crust to the sliding of tectonic plates. The new analytical and computational tools that are to be developed will be applicable to these challenging physical problems. Having stable and convergent numerical codes will greatly aid the investigation of such complex systems.