Inclined two-layered stratified channel flows: Long wave stability analysis of multiple solution regions

• Stability of stratified inclined flows in the multiple solution regions is analyzed.
• Stability boundaries on flow-pattern maps for various systems are obtained.
• There are systems and conditions where the multiple solutions are stable.
• Expressions for the shear stresses in phase with the wave slope are derived.
• These should be considered even in the framework of long wave stability analysis.

In the present work, the linear stability of two-layered stratified channel flows to long wave disturbances is studied. In particular, the study addresses the stability of laminar inclined counter-current and concurrent flows in the regions of multiple solutions for the holdup and pressure drop. The analysis is carried out by solving the Orr-Sommerfeld equations for two-plate geometry, through a formal power series in the wave number. The results are summarized in the form of stability boundaries on flow rate maps, which enable a systematic study of the effect of the system physical parameters on the stratified-smooth/wavy transition in gas–liquid and liquid–liquid systems. It is demonstrated that for counter-current flow there is a region of low flow rates where the two solutions for the holdup are stable. Likewise, the results of concurrent gas–liquid upward flows reveal a region where all three solutions are stable. Moreover, it was found that the middle solution is always stable within the entire 3-s domain. Additionally, the analysis of the wave induced stresses in the axial direction reveals that the terms in phase with the wave slope should be considered in long wave stability analyses of stratified flows.