Spatial instability of double-layer viscoelastic liquid sheets in a viscous gas medium

Abstract

The spatial instability of double-layer viscoelastic liquid sheets in a viscous gas medium is investigated. A linear stability analysis is used to simplify the governing equations and boundary conditions combined with the Oldroyd-B model for describing the viscoelastic liquid. The growth rate and the cut-off wavenumber have been studied by analyzing the dispersion curve which is based on the spectral collocation method. Moreover, we use the decomposition of growth rate to interpret some special cases. We conclude that the thickness ratio, the liquid density ratio, the liquid viscosity ratio, and the gas-to-liquid viscosity ratio stabilize the liquid sheets, while the gas-to-liquid density ratio, the Reynolds number, and the Weber number destabilize the liquid sheets. The dispersion curves are nearly-identical when the surface tension ratio is smaller than one, while the surface tension ratio is a stabilizing factor when the ratio is larger than one, but the second maximum and the counterintuitive destabilization exist in the large wavenumber region. As specific parameters concerning the viscoelasticity, the elasticity number transforms from the destabilizing factor into the stabilizing factor with the increased Reynolds number in the region nearby the cut-off wavenumber, because the relationship between the change rates of viscous dissipation terms and elasticity terms in the decomposition of the growth rate is varying. Besides, it is discovered that the trend of the change of the growth rate under the varying stress relaxation time ratio is non-monotonic and double-layer liquid sheets are most stable when the stress relaxation time ratio is near 3 (not 1), because the difference between the change rate of the viscous dissipation and the deformation retardation dissipation determines the most stable region (not the elasticity difference), which can also be interpreted with the variation of the energy terms.