Instability of a spatially developing liquid sheet

Abstract

The linear stability of an inviscid two-dimensional liquid sheet falling under gravity in a still gas is studied by analysing the asymptotic behaviour of a localized perturbation (wave-packet solution to the initial value problem). Unlike previous papers the effect of gravity is fully taken into account by introducing a slow length scale which allows the flow to be considered slightly non-parallel. A multiple-scale approach is developed and the dispersion relations for both the sinuous and varicose disturbances are obtained to the zeroth-order approximation. These exhibit a local character as they involve a local Weber number Weη. For sinuous disturbances a critical Weη equal to unity is found below which the sheet is locally absolutely unstable (with algebraic growth of disturbances) and above which it is locally convectively unstable. The transition from absolute to convective instability occurs at a critical location along the vertical direction where the flow Weber number equals the dimensionless sheet thickness. This critical distance, as measured from the nozzle exit section, increases with decreasing the flow Weber number, and hence, for instance, the liquid flow rate per unit length. If the region of absolute instability is relatively small it may be argued that the system behaves as a globally stable one. Beyond a critical size the flow receptivity is enhanced and self-sustained unstable global modes should arise. This agrees with the experimental evidence that the sheet breaks up as the flow rate is reduced. It is conjectured that liquid viscosity may act to remove the algebraic growth, but the time after which this occurs could be not sufficient to avoid possible nonlinear phenomena appearing and breaking up the sheet.

Stability of an unsupported multi-layer surfactant laden liquid curtain under gravity

“…It was shown by Crapper et al [12] that viscosity does not influence the initial stages of wave development, whilst Lin et al [13] showed that the critical Weber number does not depend on the liquid-gas density ratio or the Reynolds number. [14] provide an explanation of the importance of considering the absolute or convective character of disturbances, taking gravity into account, to confirm this and present theoretical results on the behaviour of standing waves.…”

“…The new stability criterion provides an interesting connection to the hypothesis made by De Luca and Meola [14], that in the case of a varying surface tension the constant surface tension in (1) should be replaced by the surface pressure. The two surface pressures, corresponding to the two free surfaces of the curtain, are defined as j = γ s, j − γ j = RT Γ in,1 Γ j , having used the equation of state (16).…”

Destabilization of a Viscous Film Flowing Down in the Form of a Vertical Cylindrical Curtain

“…Since early studies of Savart [1] and Boussinesq [2], numerous investigations have focused on the formation and stability of liquid curtains [3][4][5][6], which are of some importance for coating techniques, both from fundamental and practical points of view. In most experiments, sheets are obtained by a flow through a slot which induces a jet at its outlet, with slot geometry allowing various sheet shapes [3,[7][8][9].…”

Trends in Comprehensive Modeling of Spray Formation

“…The symmetric waves have a higher growth rate at higher density ratios. Luca and Costa [93] analysed the flow of plane inviscid sheets under the influence of gravity by the asymptotic behavior of a localized perturbation. For antisymmetric disturbances a critical Weber number is found below which the sheet is locally absolutely unstable and above which it is locally convectively unstable.…”