This paper investigates the spatial instability of a double-layer viscous liquid sheet moving in a stationary gas medium. A linear stability analysis is conducted and two situations are considered, an inviscid-gas situation and a viscous-gas situation. In the inviscid-gas situation, the basic state of the entire gas phase is stationary and the analytical dispersion relation is derived. Similar to single-layer sheets, the instability of double-layer sheets presents two unstable modes, the sinuous and the varicose modes. However, the result of the base-case double-layer sheet indicates that the cutoff wavenumber of the dispersion curve is larger than that of a single-layer sheet. A decomposition of the growth rate is performed and the result shows that for small wavenumbers, the surface tension of all three interfaces and the aerodynamic forces of both the lower and upper gases contribute significantly to the unstable growth rate. In contrast, for large wavenumbers the major contribution to the unstable growth rate is only the surface tension of the upper interface and the aerodynamic force of the upper gas. In the viscous-gas situation, although the majority of the gas phase is stationary, gas boundary layers exist at the vicinity of the moving liquid sheet, and the stability problem is solved by a spectral collocation method. Compared with the inviscid-gas solution, the growth rate at large wavenumber is significantly suppressed. The decomposition of growth rate indicates that all the aerodynamic and surface tension terms behave consistently throughout the entire unstable wavenumber range. The effects of various parameters are discussed. In addition, the effect of gas viscosity and the gas velocity profile is investigated separately, and the results indicate that both factors affect the maximum growth rate and the dominant wavenumber, although the effect of the gas velocity profile is stronger than that of the gas viscosity.

This paper investigates the axisymmetric instability of a viscoelastic compound jet, for which the constitutive relation is described by the Oldroyd B model. It is found that a viscoelastic compound jet is more unstable than a Newtonian compound jet, regardless of whether the viscoelastic compound jet is inner-Newtonian-outer-viscoelastic, inner-viscoelastic-outer-Newtonian, or fully viscoelastic. It is also found that an increase in the stress relaxation time of the inner or outer fluid renders the jet more unstable, while an increase in the time constant ratio makes the jet less unstable. An analysis of the energy budget of the destabilization process is performed, in which a formulation using the relative rate of change of energy is adopted. The formulation is observed to provide a quantitative analysis of the contribution of each physical factor (e.g., release of surface energy and viscous dissipation) to the temporal growth rate. The energy analysis reveals the mechanisms of various trends in the temporal growth rate, including not only how the growth rate changes with the parameters, but also how the growth rate changes with the wavenumber. The phenomenon of the dispersion relation presenting two local maxima, which occurred in previous research, is explained by the present energy analysis.

The instability of gas-surrounded Rayleigh viscous jets is investigated analytically and numerically in this paper. Theoretical analysis is based on a second-order perturbation expansion for capillary jets with surface disturbances, while the axisymmetric two-dimensional, two-phase simulation is conducted by applying the Gerris code for jets subjected to velocity disturbances. The relation between the initial surface and velocity disturbance amplitude was obtained according to the derivation of Moallemi et al. [“Breakup of capillary jets with different disturbances,” Phys. Fluids 28, 012101 (2016)], and the breakup lengths resulting from these two disturbances agree well. Analytical and numerical breakup profiles also coincide satisfactorily, except in the vicinity of the breakup point, which shrinks forcefully. The effects of various parameters (i.e., oscillation frequency, Reynolds number, Weber number, and gas-to-liquid density ratio) have also been examined by comparing spatial growth rate, second-order disturbance amplitude, breakup length, and the breakup profiles at low frequency, where obvious satellite droplets form, versus different parameters. In addition, the competition between Rayleigh instability and Kelvin-Helmholtz instability has been examined using an energy approach.

This paper investigates the temporal instability of an eccentric compound liquid thread. Results of linear stability are obtained for a typical case in the context of compound threads in microencapsulation. It is found that the disturbance growth rate of an eccentric compound liquid thread is close to that of the corresponding concentric one, in terms of both the maximum growth rate and the dominant wavenumber. Furthermore, linear stability results over a wide parameter range are obtained and the conclusion is basically unchanged. Energy balance of the destabilization process is analyzed to explain the mechanism of instability, and it is found that although the disturbance growth rate of an eccentric compound thread is close to that of the corresponding concentric thread, their energy balances are distinctively different. The disturbance interface shape and disturbance velocity distributions are plotted. It is found that the behavior of the disturbance velocity in the cross section plane is different from that of the axial disturbance velocity. The disturbance velocity distributions in the cross section plane explain the trend in the disturbance interface shape. A fully nonlinear simulation of the destabilization process is performed by the Gerris flow solver and the results agree well with those obtained by linear stability analysis.

A second-order perturbation analysis has been performed on the nonlinear temporal instability of para-sinuous disturbances on annular viscous sheets moving in an inviscid stationary gas medium. The mathematical expressions of second-order interface disturbances, velocity, and pressure have been derived. The nonlinear instability of annular viscous sheets has several characteristics which differ from that of planar viscous sheets: (1) both the first-order interface disturbances and the second-order interface disturbances contribute to breakup; (2) the zero-wavenumber component of interface disturbances in the second-order solution is nonzero; (3) the second-order interface disturbance is para-varicose in most cases, but para-sinuous for some cases. As with planar viscous sheets, it was found that viscosity plays a dual role in the nonlinear instability of annular viscous sheets. However, with the decrease in the ratio of inner radius to sheet thickness, the interval between the upper and the lower critical Reynolds numbers shrinks, and when the ratio of inner radius to sheet thickness is less than a certain value, the dual effect of viscosity vanishes.