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Revisiting the linear stability analysis and absolute–convective transition of two fluid core annular flow

Numerous experimental, numerical and theoretical studies have shown that core annular flows can be unstable. This instability can be convective or absolute in different situations: miscible fluids with matched density but different viscosities, creeping flow of two immiscible fluids or buoyant flow along a fibre. The analysis of the linear stability of the flow equation of two fluids injected in a co-current and concentric manner into a cylindrical tube leads to a rather complex eigenvalue problem. Until now, all analytical solution to this problem has involved strong assumptions (e.g. lack of inertia) or approximations (e.g. developments at long or short wavelengths) even for axisymmetric disturbances. However, in this latter case, following C. Pekeris, who obtained, almost seventy years ago, an elegant explicit solution for the dispersion relationship of the flow of a single fluid, we derive an explicit solution for the more general case of two immiscible fluids of different viscosity, density and inertia separated by a straight interface. This formulation is well adapted to commercial software. First, we review the creeping flow limit (zero Reynolds number) of two immiscible fluids as it is used in microfluidics. Secondly, we consider the case of two fluids of different viscosities but of the same density in the absence of surface tension and also without diffusion (i.e. miscible fluids with infinite Schmidt number). In both cases, we study the transition from convective to absolute instability according to the different control parameters.