Absolute Instability of Variable-Viscosity Jets with Counter-current Shear Flow

Variable viscosity flows occur in a variety of industrial situations, such as static mixers and continuously stirred tanks. Recently, it has been shown that low viscosity jets emerging into a medium of higher viscosity are absolutely unstable, while high viscosity jets are convectively unstable. It has long been known that the introduction of counterflow can substantially alter the instability characteristics of constant-property flows. In this work, we investigate the instability of variable-viscosity round jets subjected to coflow and counterflow using spatio-temporal linear stability analysis of local profiles. Both axisymmetric and helical modes are examined over the parameter ranges Reynolds number 500~3000 and ambient-to-jet centerline viscosity ratio 0.5~50. We determine the critical parameter combinations that trigger absolute instability and characterize the dominant unstable modes. Our results reveal that counterflow and viscosity variation act jointly to significantly enhance instability, shifting the boundary between absolute and convective regimes and promoting the onset of global modes. For example, co-flow can suppress instability in low viscosity jets, while counterflow can promote absolute instability in high viscosity jets. These findings provide new insights into how counterflow and viscosity contrast jointly shape jet instability mechanisms, with implications for both fundamental fluid dynamics and industrial applications.