  A numerical framework for rigorous linear stability analysis of two-phase
stratified flows of two immiscible fluids in horizontal circular pipes is
presented. For the first time, three-dimensional disturbances, including those
at the interface between two fluids, are considered. The proposed numerical
framework is based on a finite-volume method and allows solving the problem
numerically in bipolar cylindrical coordinates. In these coordinates, both the
pipe wall and the unperturbed interface (of a constant curvature, e.g., plane
interface, as considered in this work) coincide with the coordinate surfaces.
Thereby, the no-slip as well as the interfacial boundary conditions can be
imposed easily. It also enables investigation of the local behavior of the flow
field and shear stresses in the vicinity of the triple points, where the
interface contacts the pipe wall. The results obtained in the bipolar
coordinates are verified by an independent numerical solution based on the
problem formulation in Cartesian coordinates, where the pipe wall is treated by
the immersed boundary method. Two representative examples of gas-liquid and
liquid-liquid flows are included to demonstrate the applicability of the
proposed numerical technique for analyzing the flow stability.
