Instability of two-phase co-axial jets at small Reynolds number

A precise knowledge of co-axial flow dynamics and a better understanding of the mechanisms that act to destabilize the interface between the two fluids is of fundamental interest in many industrial applications like lubricated transport, injection devices, atomization and controlled microdroplet production. The flow is in general unstable, since at least two mechanisms act to destabilize the cylindrical interface: shear and capillary instabilities. While these two mechanisms are active in a jet issuing from a tap, their respective influence strongly depends on the Reynolds number, the Ohnesorge number, but also on the viscosity, density, and aspect ratios. In this thesis, the global stability characteristics of two-phase co-axial flow are determined. The stability analysis follows two successive steps. First the steady base flow is determined, via the resolution of the nonlinear Navier-Stokes equations together with the location of the free interface. Second, these equations are linearized around the base flow and the dominant eigenmodes determined. The novelty lies in the formulation of models that can describe both the qualitative and quantitative characteristics of the two-phase flow configuration and the adaptation of the tools of global stability analysis for this configuration. We find that the dripping to jetting regime transition depends on the Capillary number, the degree of the confinement and the viscosity ratio, and we show that, surprisingly, the nozzle geometry does not affect the stability properties of the flow. Finally, the influence of surface viscosity on these coaxial flows has been considered. The governing and constitutive equations describing the continuum mechanics of the surface in the axisymmetric case are derived. With the addition of surface viscosity at the interface, the base flow evolves over a lengthscale which is much larger than the entry length in the Stokes regimes and than the typical unstable wavelength. We show that while the flow becomes eventually more convectively unstable once it reaches the fully developed profile, the surface viscosity creates an absolute region at the inlet, that is expected to promote droplet formation.