Linear temporal instability of a two-layer axisymmetric cylindrical liquid sheet

The instability of an axial uniform flow of a two-layer axisymmetric cylindrical sheet composed of two inviscid fluids is examined analytically and numerically on the basis of thin-layer approximation. As the eigenmodes for disturbances of large wavelength, we obtain an internal mode in which the displacement of interface between these fluids is larger than the displacements of two free surfaces of the sheet, as well as the modes similar to varicose and sinuous modes in a single-layer case. All the modes are unstable only for disturbances of large wavelength, and the growth rate of sinuous-like mode is much larger than that of two other modes. Furthermore, unlike a single-layer case, the wavenumber of disturbances of maximum growth rate is non-zero if the density ratio is sufficiently far from one. In such cases, the ratio of amplitudes of these disturbances is much different from that in a typical sinuous mode. The dependences of maximum growth rate on the ratios of surface tension coefficients, fluid densities, and sheet thicknesses are also examined.