This thesis describes the mathematical modelling and analysis of the two phase flow of water and steam in a steam generating pipe contained within the evaporating component of a nuclear power plant. As the fluid is heated the continuous phase (water) is evaporated and steam is produced. The phases adopt different geometrical configurations called flow regimes. The flow regime adopted by the phases depends on the amount of each phase present and also external features such as pipe geometry. The main regions of interest are the subcooled, bubbty and, annular flow regimes. The subcooled flow regime consists of water flowing at temperature below its saturation temperature and can be modelled as a single phase flow. The bubbly is characterised by that the gas phase flows as discrete bubbles in the continuous phase. The bubbly flow regime shall be modelled using a system of averaged two-phase flow equations. Of particular interest is the annular flow regime. The annular flow regime is characterised by the fact that the majority of the liquid phase is present in a thin film along the the pipe wall with the remaining liquid phase being present as droplets in a central gas core. We develop a non-linear singular integro-differential equation to describe the interface between the liquid film and gas core. The annular flow regime is further complicated by an exchange of mass between the film and gas core due to entrainment, deposition of droplets, and evaporation. The loss of mass from the liquid film leads to the phenomenon of dryout where the liquid film ceases to exist.