Stability diagrams for disconnected capillary surfaces

Abstract

Disconnected free surfaces ͑or interfaces͒ of a connected liquid volume ͑or liquid volumes͒ occur when the boundary of the liquid volume consists of two or more separate surface components ⌫ i (iϭ1,...,m) that correspond to liquid-gas ͑or liquid-liquid͒ interfaces. We consider disconnected surfaces for which each component ⌫ i is axisymmetric and crosses its own symmetry axis. In most cases, the stability problem for an entire disconnected equilibrium capillary surface subject to perturbations that conserve the total liquid volume reduces to the same set of problems obtained when separately considering the stability of each ⌫ i to perturbations that satisfy a fixed pressure constraint. For fixed pressure perturbations, the stability of a given axisymmetric ⌫ i can be found through comparison of actual and critical values of a particular boundary parameter. For zero gravity, these critical values are found analytically. For non-zero gravity, an analytical representation of the critical values is not generally possible. In such cases, a determination of stability can be accomplished by representing all possible equilibrium surface profiles on a dimensionless ''heightradius'' diagram. This diagram is contoured with critical values of the boundary parameter. The stability diagram can, in most cases, be used to determine the stability of a disconnected surface ͑subject to perturbations that conserve the total volume͒ that is composed of components that are represented by given equilibrium profiles on the diagram. To illustrate this approach, solutions of stability problems for systems consisting of a set of sessile or pendant drops in contact with smooth planar walls or with the edges of equidimensional perforated holes in a horizontal plate are presented.

Capillary surfaces in and around exotic cylinders with application to stability analysis

“…The line only intersects with the solution curves with and there is only one intersection point for each of the solution curves. In the region , the contour line intersects with the abscissa at which satisfies the relation (Slobozhanin & Alexander 2003) We can see that as . In the region , the contour lines will intersect with all solution curves, and there is only one intersection point for each of the solution curves.…”

“…Stability to pressure disturbances for axisymmetric menisci: ( a ) the solution curves fixed at the origin for (see also figure 3 in Slobozhanin & Alexander (2003)) and ( b , c ) the solution curves with located at the water line for ( b ) and ( c ) . Panel ( b ) is essentially the same as panel ( a ) except for the different choices of the ordinates.…”

“…The study is then extended to stability analysis for exotic cylinders. Using the method of Slobozhanin & Alexander (2003), the stability of the meniscus is determined by comparing the critical value and the boundary parameter of the exotic cylinder, where the Sturm–Liouville problem is solved for . The spectral parameter power series (SPPS) method (Kravchenko & Porter 2010) is applied to the problems to obtain the contour lines of for two families of menisci.…”

Liquid-bridge mediated droplet switch: A tristable capillary system

“…This motivates our focus on contact-line imperfections. Both imperfections have been studied previously in the context of single bridges, 22 coupled droplets, 6,15 and contact-line distortion from circular for coupled droplets. 6 The interfaces cannot be perfectly pinned because the end rings have finite thickness and, therefore, achieving perfectly indistinguishable states ͓for example, ͑c − ͒ and ͑c + ͒ in Fig.…”

“…12 Another approach, based on work by Orel, 14 is employed by Slobozhanin and Alexander. 15 They use an eigenvalue problem to characterize system stability. The resulting criteria, for disturbances that preserve total system volume, can be paraphrased: ͑A͒ if all individual components are stable to fixed-pressure disturbances, the system is stable; ͑B͒ if at least two components are unstable to fixed-pressure disturbances, the system is unstable; ͑C͒ if at least one component is unstable to fixedvolume disturbances, the system is unstable.…”