Problems with phase change have in common a characteristic nonlinearity resulting from the moving boundary (solid-liquid interface), which complicates their analyses and which also renders each problem somewhat different. Furthermore, they have one or more singularities due to the appearance or disappearance of one phase or another. The nonlinearity and the singularity of these problems make them intrinsically difficult to solve;This study examines the two-dimensional freezing problem in pipe flows. Analytical analysis and numerical methods are used to determine solutions of the equations which govern the process. The governing equations and boundary conditions for the pipe flow are obtained from the fundamental equations of conservation of mass, momentum, and energy. Two types of initial conditions are considered: one which is a hydrodynamically and thermally fully developed flow and the other a thermally developing flow. The governing equations are formulated using the stream function and vorticity and then are nondimensionalized using proper scales for the solid and liquid phases. The moving-boundary problem in physical coordinates is transformed into a fixed-boundary problem in dimensionless coordinates. With the boundary of the domain fixed, numerical methods are aptly applied to determine solutions of the dependent variables. The numerical method employed in this study is a finite-difference method. It uses a fully implicit, Gauss-Seidel iterative scheme;Results obtained using the numerical model for the freezing of a flowing liquid in a pipe are presented and discussed. The freezing time required to reach an asymptotic steady state depends mainly on Stefan number (Ste) and superheat number (Su), whereas the thickness of the steady-state ice layer depends only on Su. The freezing time, t[subscript]95, is found to be (approximately) inversely proportional to Ste. The steady-state ice thickness increases as Su decreases, and the amount of transient ice growth from an initial state is larger for smaller Su. The effects of internal freezing in a pipe flow on heat transfer and pressure drop are obtained from this steady-state ice layer profile. When there is internal freezing, heat transfer is enhanced and pressure drop is increased for a given mass flow rate. The fluid flow is significantly accelerated as the flow passage decreases due to ice growth, but the velocity profile remains very similar to a parabolic distribution. The ice-band structure, which was one of the foremost interests in this study, was not observed in the present results. Nevertheless, the results help to provide a general understanding of the freezing problem in pipe flows.