  The one-dimensional (1D) two-fluid model (TFM) for stratified flow in
channels and pipes suffers from an ill-posedness issue: it is only
conditionally well-posed. This results in severe linear instability for
perturbations of vanishing wavelength, and non-convergence of numerical
solutions. This issue is typically only examined from the perspective of linear
stability analysis.
  In order to analyze this long-standing problem from a nonlinear perspective,
we show the novel result that the TFM (in its incompressible, isothermal form)
satisfies an energy conservation equation, which arises naturally from the mass
and momentum conservation equations that constitute the TFM. This result
extends upon earlier work on the shallow water equations (SWE), with the
important difference that we include non-conservative pressure terms in the
analysis, and that we propose a formulation that holds for ducts with an
arbitrary cross-sectional shape, with the 2D channel and circular pipe
geometries as special cases.
  The second result of this work is a new finite volume scheme for the TFM that
satisfies a discrete form of the continuous energy equation. This
discretization is derived in a manner that runs parallel to the continuous
analysis. Due to the non-conservative pressure terms it is essential to employ
a staggered grid, and this distinguishes the discretization from existing
energy-conserving discretizations of the SWE. Numerical simulations confirm
that the discrete energy is conserved.
