Stability capillary instability

Linear stability theories have also been applied to analyses of liquid sheet breakup processes. The capillary instability of thin liquid sheets was first studied by Squire[258] who showed that instability and breakup of a liquid sheet are caused by the growth of sinuous waves, i.e., sideways deflections of the sheet centerline. For a low viscosity liquid sheet, Fraser et al.[73] derived an expression for the wavelength of the dominant unstable wave. A similar formulation was derived by Li[539] who considered both sinuous and varicose instabilities. Clark and DombrowskF540 and Reitz and Diwakar13161 formulated equations for liquid sheet breakup length. 

We begin with capillary instability of a liquid thread. This is a problem that was discussed qualitatively already in Chap. 2. It is a problem with a physically clear mechanism for instability and thus provides a good framework for introducing the basic ideas of linear stability theory. This problem is one of several examples in which the viscosity of the fluid plays no role in determining stability, but only influences the rate of growth or decay of the infinitesimal disturbances that are analyzed in a linear theory. 

A key idea from the preceding analysis, as well as the analysis of capillary instability in section A, is that viscous effects often cannot change the conditions for instability of a rest state, but only moderate the rate of growth or decay of disturbances. In such cases, analysis of the stability (or instability) of the inviscid limit can be extremely useftd in identifying the conditions for instability. In view of the fact that the inviscid analysis is very much simpler, this is an important observation. 

Problem 12-2. Capillary Instability of a Varicose Sheet. We have seen that a cylinder of fluid surrounded by air is unstable because of growing capillary waves. An equivalent problem is the stability of capillary waves on a round laminar jet when the velocity profile across the jet is uniform. We now wish to consider whether the same type of instability is relevant to a fluid sheet of finite width <7 and infinite lateral extent (the sheet has an interface above and below). Equivalently, we could also ask whether a planar (2D) laminar jet is unstable to capillary wave growth. Assume that the sheet is subject to an infinitesimal ID wavelike disturbance that is symmetric about the center plane and corresponds to an initial sheet thickness h = ho( + e sin foe). Prove whether the sheet is linearly stable or unstable to this type of disturbance. 

Problem 12-4. The Capillary Instability of an Annular Fluid Film. Two problems that are closely related to the classic problem of capillary instability of a iquid thread in air are the stability of an annular fluid film that may either coat the surface of a wire or coat the inside surface of a circular tube. We assume in both cases that the unperturbed thickness of the fluid film is a constant, independent of position. There are many applications for both of these problems. For example, the former is related to the stability of a fluid film in a wire... 

Abstract This chapter deals with capillary instability of straight free liquid jets moving in air. It begins with linear stability theory for small perturbations of Newtonian liquid jets and discusses the unstable modes, characteristic growth rates, temporal and spatial instabilities and their underlying physical mechanisms. The linear theory also provides an estimate of the main droplet size emerging from capillary breakup. Formation of satellite modes is treated in the framework of either asymptotic methods or direct numerical simulations. Then, such additional effects like thermocapiUarity, or swirl are taken into account. In addition, quasi-one-dimensional approach for description of capillary breakup is introduced and illustrated in detail for Newtonian and rheologically complex liquid jets (pseudoplastic, dilatant, and viscoelastic polymeric liquids). 

Keywords Capillary instability of liquid jets Curvature Elongational rheology Free liquid jets Linear stability theory Nonlinear theory Quasi-one-dimensional equations Reynolds number Rheologically complex liquids (pseudoplastic, dilatant, and viscoelastic polymeric liquids) Satellite drops Small perturbations Spatial instability Surface tension Swirl Temporal instability Thermocapillarity Viscosity... 

Equation 5.127 can be applied to some of the situations we have already analyzed, such as gravitationally produced instability of superposed fluids and capillary instability of a fluid cylinder. It has also been applied to more complex simations, such as stability of a pendant drop (Huh, 1969 Pitts, 1974). We remark that the expression in braces in Equation 5.127 is the local force acting in the normal direction on the deformed interface to restore it to its initial configuration. Hence application of the force method amoimts to a requirement that this expression be positive for stability. 

Use the force method to investigate the stability of a cylinder of A in B when both fluids rotate at an angular velocity co around the axis of the cylinder. Assume that interfacial tension exerts a local restoring force equal to -yd(2H), where H is the local mean curvature. If pg > Pa, how large must co be to overcome the basic capillary instability described in Section 4. 

Assemblies of liquid films have also their stability conditions, and we saw ( 210) how Mr. Lamarle subjected this topic to a precise theory. I will finish this chapter with a summary of a work of Mr. Duprez " on a phenomenon in which the capillary pressures combine with the action of gravity to produce curious effects of stability and instability. 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

Shale stability is an important problem faced during drilling. Stability problems are attributed most often to the swelling of shales. It has been shown that several mechanisms can be involved [680,681]. These can be pore pressure diffusion, plasticity, anisotropy, capillary effects, osmosis, and physicochemical alterations. Three processes contributing to the instability of shales have to be considered [127] ... 

Benjamin (B5) has given a detailed treatment of the onset of two-dimensional instability in film flow, taking capillary effects into account. The expression for neutral stability found in this work can be given as... 

The theoretical framework, within which the existence of surface instabilities created by capillary waves can be predicted is the linear stability analysis [23, 24]. This model assumes a spectrum of capillary waves with wave vectors q and time constant r (Fig. 1.8a). 

The breakup or bursting of liquid droplets suspended in liquids undergoing shear flow has been studied and observed by many researchers beginning with the classic work of G. I. Taylor in the 1930s. For low viscosity drops, two mechanisms of breakup were identified at critical capillary number values. In the first one, the pointed droplet ends release a stream of smaller droplets termed tip streaming whereas, in the second mechanism the drop breaks into two main fragments and one or more satellite droplets. Strictly inviscid droplets such as gas bubbles were found to be stable at all conditions. It must be recalled, however, that gas bubbles are compressible and soluble, and this may play a role in the relief of hydrodynamic instabilities. The relative stability of gas bubbles in shear flow was confirmed experimentally by Canedo et al. (36). They could stretch a bubble all around the cylinder in a Couette flow apparatus without any signs of breakup. Of course, in a real devolatilizer, the flow is not a steady simple shear flow and bubble breakup is more likely to take place. 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

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