The unstable one-dimensional incompressible two-fluid model including a hydrostatic force is reduced to a two equation model in terms of the liquid volume fraction and the liquid velocity. For small density ratios the model may be simplified to a formulation that is equivalent tothe Shallow Water Theory (SWT) equations [Whitham, 1975] with a source term corresponding to the two-fluid model constitutive relations for wall and interfacial shear and to a void gradient term that contains the Kelvin-Helmholtz mechanism.

Linear stability of the SWT equations shows that the model is made well-posed stable by the hydrostatic force. However, unlike the SWT equations, the two equation two-fluid model is only conditionally stable. As the gas velocity increases the model becomes unstable once the kinematic instability occurs, i.e., the Viscous Kelvin-Helmholtz (VKH) instability. When the gas velocity is increased further the model becomes dynamically unstable, i.e., it reaches the Inviscid Kelvin-Helmholtz (IKH) instability limit. Beyond the IKH limit the model becomes ill-posed and requires higher order modelling, e.g. surface tension. Simple analytic expressions for the two instabilities are obtained because of the simplified mathematics of the two equation model. Furthermore, the wave “sheltering” effect, which allows more accurate predictions of the flow regime transition, may be easily incorporated into the analysis. The theory is validated with the new HAWAC flow regime map [Vallee et al., 2010].

The two equation two-fluid model is consistent with all previous results of two-fluid model linear stability for stratified flow and, since it is a special case of SWT, it is amenable to non-linear stability analysis and a very broad body of mathematics literature on non-linear kinematic waves, e.g., Whitham (1974) is now directly applicable when the model is not IKH unstable.

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