In this project, we consider the liquid-vapor flow as a homogenized mixture of the two-phases. The resulting models  pose a major challenge to mathematics, since there are a number of important open questions to be studied. The primary goal is to improve and validate numerical schemes for such models. Numerical solutions are needed in many diverse engineering applications involving phenomena such as liquid sprays or bubbly flows. In order to improve the quality of numerical results we need to address some mathematical issues concerning the modeling and resulting well-posedness of the equations. Also we will have to develope a deeper understanding of the theory and numerical methods for hyperbolic systems of equations containing non-conservative derivatives. Another challenge is phase extinction, which is related to vacuum states in gas dynamics. Further, it will be necessary to encorporate phase transitions into the models and numerical computations.

For a large class of practical applications there is a need to calculate two-phase flows with many dispersed particles, e.g. \ bubbles in liquids or droplets in a gas. The engineering applications include malfunctions in chemical reactors, spray cooling, spray painting or transport of bubbly liquids. Normally, we are not interested in a detailed description of particle interaction; instead we want to describe the flow as a whole. This is exactly where the homogenized approaches come into play. Further, they may also be used in cases where the phases are well separated, because they can deal with topological problems arising when the different regions of one phase merge or are created.

The current state of art of both mathematical models for the homogenized two-phase flows and the numerical methods for their solution is insufficient. An important issue concerning the systems of governing equations for these models is that they are instrinsically non-conservative. This comes as the result of the averaging-procedures on the constitutive equations for each phase. It appears that the mathematical structure of these non-conservative systems is much more complicated than for consevation laws. Also, there is a lack of theory for numerical methods for such systems.

On the other hand, the development of efficient numerical methods for the solution of two-phase flows is of great importance. Since the equations are intrinsically non-conservative, one has to provide non-conservative methods for their solution. Here we hope to get insight from considering the mathematically related Euler equations in a duct of variable cross-section. This relatively simple system exhibits not only remarkable similarities to the much more complex systems of governing equations of two-phase flow, but can be obtained from this system be simplefying assumptions.

Therefore, there are four main goals of the project:

- Gain more insight into the mathematical structure of the two-phase flow equations under consideration and the modeling of phase

   transition.

- Develop better algorithms for the numerical approximation of solutions to these two-phase flow equations, especially concerning non-

  conservative terms and phase extinction.

- Mathematical foundation of the numerical algorithms and their properties by numerical analysis

- Validation through comparison with other computations and experiments.