Multiphase flow is a familiar phenomenon from daily life and occupies an important role in physics, engineering, and medicine. The understanding of multiphase flows relies largely on the theory of interfaces, which is not well understood in many cases. To date, the Navier-Stokes-Korteweg equations and the Cahn-Hilliard equation have represented two major branches of phase-field modeling. The Navier-Stokes-Korteweg equations describe a single component fluid material with multiple states of matter, e.g., water and water vapor; the Cahn-Hilliard type models describe multi-component materials with immiscible interfaces, e.g., air and water. In this dissertation, a unified multiphase fluid modeling framework is developed based on rigorous mathematical and thermodynamic principles. This framework does not assume any ad hoc modeling procedures and is capable of formulating meaningful new models with an arbitrary number of different types of interfaces. In addition to the modeling, novel numerical technologies are developed in this dissertation focusing on the Navier-Stokes-Korteweg equations. First, the notion of entropy variables is properly generalized to the functional setting, which results in an entropy-dissipative semi-discrete formulation. Second, a family of quadrature rules is developed and applied to generate fully discrete schemes. The resulting schemes are featured with two main properties: they are provably dissipative in entropy and second-order accurate in time. In the presence of complex geometries and high-order differential terms, isogeometric analysis is invoked to provide accurate representations of computational geometries and robust numerical tools. A novel periodic transformation operator technology is also developed within the isogeometric context. It significantly simplifies the procedure of the strong imposition of periodic boundary conditions. These attributes make the proposed technologies an ideal candidate for credible numerical simulation of multiphase flows. A general-purpose parallel computing software, named PERIGEE, is developed in this work to provide an implementation framework for the above numerical methods. A comprehensive set of numerical examples has been studied to corroborate the aforementioned theories. Additionally, a variety of application examples have been investigated, culminating with the boiling simulation. Importantly, the boiling model overcomes several challenges for traditional boiling models, owing to its thermodynamically consistent nature. The numerical results indicate the promising potential of the proposed methodology for a wide range of multiphase flow problems. / text

The current theoretical understanding of thermodynamics of clathrate hydrates is based on the van der Waals and Plattew (vdWP) theory developed using statistical thermodynamics approach. vdWP theory has been widely used to predict the phase equilibrium of clathrate hydrates over the decades. However, earlier studies have shown that this success could be due to the presence of a large number of parameters. In this thesis, a systematic and a rigorous analysis of vdWP theory is per-formed with the help of Monte Carlo molecular simulations for methane hydrate. The analysis revealed that long range guest-water interactions and guest-guest interactions are important, Monte Carlo integration to is superior to the spherical shell approximation for the Langmuir constant calculation and even after inclusion of all the interactions and using Monte Carlo integration for Langmuir constant, the vdWP theory still fails to regress parameters correctly. This failure of vdWP theory is attributed to the rigid water lattice approximation. To address the rigid water lattice approximation, a new method is proposed. In the proposed method, the Langmuir constant is computed in flexible water lattice, by considering the movement of water molecules. The occupancy values predicted using the proposed method are in excellent agreement with the values obtained from Monte Carlo molecular simulations for variety of hydrates, methane, ethane, carbon dioxide and tetrahydrofuran(THF) hydrates . In addition to small guest molecules like methane, ethane etc. which are mod- heled as rigid, the method is extended for large guest molecules like propane and isobutane, using configurationally bias Monte Carlo method. The phase equilib-rium and occupancy along the phase equilibrium predictions from vdWP theory are compared with the exact phase equilibrium computed from Monte Carlo molecular simulations. This comparison is done for a wide variety of hydrate systems, single hydrates , binary hydrates and quaternary hydrate. In all the cases, the vdWP theory with the flexible water lattice showed significant improvement over the rigid lattice model with significantly less absolute relative deviations in pressure. Guest-cavity interactions for hydrates are calculated using abinitio calculations. In general, these guest-cavity interaction from first principle calculations are used to develop classical force field parameters in alternative to Lorentz-Berthelot rule. In the study, comparison of guest-cavity interactions from MP2 and CCSD(T) methods revealed that less expensive MP2 method, which is generally used, is insouciant to capture the dispersion interactions accurately. These guest-cavity interactions using CCSD(T) method extrapolated to complete basis set are used to model the interaction parameters between cyclopropane and water. The potential parameters obtained from ab-initio calculations are used in the calculation of Langmuir constant using vdWP theory. Langmuir constant calculated using vdWP theory with flexible water lattice gave close agreement with the values obtained from experimental occupancy data. In addition, simulation methodology to calculate ternary hydrate phase equilibrium is extended for binary hydrates. Simulations have been successful in the prediction of sIsII and sII-sI structural transitions as observed in experiments. Predicted methane-ethane binary hydrate is also compared with the available experimental phase equilibrium data. The phase equilibrium obtained from simulations showed very good qualitative agreement with the experimental data.