Spatial instability of viscous double-layer liquid sheets

Abstract

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.

On the nature of instabilities in externally perturbed liquid sheets

“…2011; Ye et al. 2016). To predict the spatial growth rate, we used the MATLAB tool ‘Hydrostab’ recently developed by Ye et al.…”

“…In the present analysis, we have employed the spectral collocation method, which has been adapted by many researches for the growth rate prediction for a viscous liquid sheet and viscous gas (Söderberg & Alfredsson 1998;Söderberg 2003;Tammisola et al 2011;Ye et al 2016). To predict the spatial growth rate, we used the MATLAB tool 'Hydrostab' recently developed by Ye et al (2016). Details of this method with a sample code can be found in Ye et al (2016).…”

“…Here, we have provided the governing equations, the boundary conditions and the gas velocity profile used in the current analysis. For consistency, these equations are presented in a similar way as given in Ye et al (2016). We consider the liquid sheet as shown in figure 2 with viscous liquid and viscous surrounding gas.…”

“…To predict the spatial growth rate, we used the MATLAB tool ‘Hydrostab’ recently developed by Ye et al. (2016). Details of this method with a sample code can be found in Ye et al.…”

“…Details of this method with a sample code can be found in Ye et al. (2016). The liquid sheet thickness in the viscous analysis is taken to be constant and calculated from the expression given by Miller (1960).…”