AbstractWe consider a thin liquid film flowing down an inclined plate in the presence of a counter-current turbulent gas. By making appropriate assumptions, Tseluiko & Kalliadasis (J. Fluid Mech., vol. 673, 2011, pp. 19–59) developed low-dimensional non-local models for the liquid problem, namely a long-wave (LW) model and a weighted integral-boundary-layer (WIBL) model, which incorporate the effect of the turbulent gas. By utilising these models, along with the Orr–Sommerfeld problem formulated using the full governing equations for the liquid phase and associated boundary conditions, we explore the linear stability of the gas–liquid system. In addition, we devise a generalised methodology to investigate absolute and convective instabilities in the non-local equations describing the gas–liquid flow. We observe that at low gas flow rates, the system is convectively unstable with the localised disturbances being convected downwards. As the gas flow rate is increased, the instability becomes absolute and localised disturbances spread across the whole domain. As the gas flow rate is further increased, the system again becomes convectively unstable with the localised disturbances propagating upwards. We find that the upper limit of the absolute instability region is close to the ‘flooding’ point associated with the appearance of large-amplitude standing waves, as obtained in Tseluiko & Kalliadasis (J. Fluid Mech., vol. 673, 2011, pp. 19–59), and our analysis can therefore be used to predict the onset of flooding. We also find that an increase in the angle of inclination of the channel requires an increased gas flow rate for the onset of absolute instability. We generally find good agreement between the results obtained using the full equations and the reduced models. Moreover, we find that the WIBL model generally provides better agreement with the results for the full equations than the LW model. Such an analysis is important for an understanding of the ranges of validity of the reduced model equations. In addition, a comparison of our theoretical predictions with the experiments of Zapke & Kröger (Intl J. Multiphase Flow, vol. 26, 2000, pp. 1439–1455) shows a fairly good agreement. We supplement our stability analysis with time-dependent computations of the linearised WIBL model. To provide some insight into the mechanisms of instability, we perform an energy budget analysis.
Kinetic Aspects of Phosphorus Removal from Electromagnetically Levitated 600 MPa Steel Droplets
