Exact analytical solution is obtained for the interface shape between two immiscible fluids and for the capillary pressure in the case of unidirectional axial laminar pipe flow. The solution is determined by three dimensionless parameters: the holdup, fluid/wall wettability angle and the Eotvos number, Eo = Delta rho g cos alpha R-2/sigma(12). The ranges of parameter values, for which the model of flat interface (Eo-->infinity), or models of constant interfacial curvature can be applied, are explored. Finally, the implications to modeling of stratified flow characteristics and stability are discussed.