Summary: The stratified flow of gas-liquid mixtures is important in many industries, in particular oil and gas, where the formation of liquid slugs is usually undesirable. In this paper, we analyse the hydraulic model for turbulent, two-layer, gas-liquid flow in a horizontal channel developed by D. J. Needham et al. [IMA J. Appl. Math. 73, No. 1, 274–322 (2008; Zbl 1154.76055)] who considered only the thin layer limit, for liquid layers of arbitrary thickness. This allows us to consider the effect of the prescribed inlet velocity ratio and the interaction of the gas layer with the upper wall of the channel on the formation of slug formation.
We show that there is a critical Froude number above which small-amplitude disturbances to the uniform flow are unstable and grow to form non-linear roll waves. We investigate the existence of periodic travelling wave solutions and find that these are strongly influenced by the existence of two equilibrium solutions that are not present when the liquid layer is thin. The system of ordinary differential equations that describe travelling wave solutions has a complicated phase portrait which we study in some typical cases. We find that periodic solutions that have the channel close to filled with liquid can exist, and we identify them as slugs. We also study an initial value problem driven by noise at the inlet of the channel and find that structures locally similar to these periodic travelling waves are generated. We provide numerical evidence for the formation of slugs in cases where the initial liquid to homogeneous velocity ratio is greater than 0.5.