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Spatial instability of viscous double-layer liquid sheets
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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.
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