Linear instability, nonlinear instability and ligament dynamics in three-dimensional laminar two-layer liquid-liquid flows

We consider the linear and nonlinear stability of two-phase density-matched but viscosity-contrasted fluids subject to laminar Poiseuille flow in a channel, paying particular attention to the formation of three-dimensional waves. A combination of Orr-Sommerfeld-Squire analysis (both modal and non-modal) with direct numerical simulation of the three-dimensional two-phase Navier-Stokes equations is used. For the parameter regimes under consideration, under linear theory, the most unstable waves are two-dimensional. Nevertheless, we demonstrate several mechanisms whereby three-dimensional waves enter the system, and dominate at late time. There exists a direct route, whereby three-dimensional waves are amplified by the standard linear mechanism; for certain parameter classes, such waves grow at a rate less than but comparable to that of the most dangerous two-dimensional mode. Additionally, there is a weakly nonlinear route, whereby a purely spanwise wave grows according to transient linear theory and subsequently couples to a streamwise mode in weakly nonlinear fashion. Consideration is also given to the ultimate state of these waves: persistent three-dimensional nonlinear waves are stretched and distorted by the base flow, thereby producing regimes of ligaments, 'sheets' or 'interfacial turbulence'. Depending on the parameter regime, these regimes are observed either in isolation, or acting together.