Stability and bifurcation Problems for Equilibrium States of a Liquid Bridge

The results of an investigation on the stability of a doubly connected axisymmetric equilibrium free surface pinned to the edges of two coaxial disks are presented. The general boundary of the region where the interface is stable is constructed in the plane of the parameters determining the slenderness of a liquid bridge and its relative volume. Surface forces and arbitrary (not only axisymmetric) perturbations are taken into account. The general boundary of the stability region was calculated completely in the past only for a weightless fixed-contactline liquid bridge between equal disks. The influence of axially directed gravity, isorotation and disks inequality on the evolution of this boundary has been analyzed successively. As a result, the families of the stability boundaries have been obtained for fixed-contact-line liquid bridges between equal disks in a wide range of Bond numbers, for isorotating weightless bridges between equal disks, in a wide range of Weber numbers and for weightless fixed-contact-line liquid bridges when the disk radii ratio is varied. Basing on the solution of the bifurcation problem for the critical equilibrium states, the conclusion on the results of stability losing has been made for starting system of fixed-contact line weightless bridge between equal-disks. The stability of the melt during crystal growth using the floating zone technique can be considered as a special case of the presented results. Finally as an example, for a crystal growth system using the Stepanov's method the effect of free surface unconnectivity on the stability has been investigated under zero gravity conditions

Stability of Liquid Bridges between Unequal Disks under Zero-Gravity Conditions

The stability, of axisymmetric equilibrium shapes of a liquid bridge between two coaxial disks of different radii under zero-gravity conditions is investigated. The stability regions have been evaluated for different values of the ratio of the disk radii in terms of the dimensionless parameters which characterize the length and the volume of the bridge. It has been found that disk radii unequality radically changes the upper boundary of the stability region. The analysis of the shape of marginally stable equilibrium surfaces has been carried out. Relationships between the critical values of the parameters have been deduced for some particular cases, which are of special interest for the materials purification processes and growing of single crystals by the floating zone method: for typical values of the growing angle for semiconductor materials and for liquid volumes close to that of the cylinder having a radius equal to the mean radius of the disk

Stability of liquid bridges between twisted elliptical disks

The influence in the stability of long liquid bridges supported between two elliptical-shaped disks of their main axis relative orientation is investigated. A numerical continuation method capable of finding equilibrium shapes, both stable and unstable, is used to calculate a series of equilibrium shapes supported by disks of increasing eccentricity for different relative orientation of the disks axis. The stable or unstable character of each of the shapes is calculated to determine the position of the stability limit and its characte

Stability of an Isorotating Liquid Bridges between Equal Disks under Zero-gravity Conditions

The stability of the relative equilibrium of an isorotating axisymmetric liquid bridge between two equal‐radius coaxial disks under zero‐gravity conditions has been investigated in detail. The free surface is assumed to be pinned to the edges of the disks and in equilibrium and only perturbations compatible with this pinning are considered. In the plane of the dimensionless variables characterizing the liquid bridge length and the liquid bridge volume, the stability regions for a set of values of the Weber number have been calculated. The stability region structure and the nature of critical perturbations change when the Weber number, W, passes through the values W0 (2.05<W0<2.06) and W1 (2.44<W1<2.45). It has been found that, for W<W0, the stability region is connected, and the neutral stability may take place with respect to nonaxisymmetric perturbations as well as to axisymmetric ones. In the latter case, it has been established whether the critical axisymmetric perturbations are reflectively symmetric or reflectively antisymmetric about the equatorial plane. When the increasing Weber number passes through the value W0, the stability region breaks into two disconnected parts. The first exists for all Weber numbers larger than W0. For the states belonging to the boundary of this part, only nonaxisymmetric perturbations are critical. The second part exists only for Weber numbers between W0 and W1. Its boundary is determined by the states that may be neutrally stable to nonaxisymmetric perturbations or to axisymmetric ones. The characteristics of the shape of the neutrally stable surfaces have been calculated for a wide range of the Weber numbe

Stability of Liquid Bridges between Equal Disks in an Axial Gravity Field

The stability of axisymmetric liquid bridges spanning two equal-diameter solid disks subjected to an axial gravity field of arbitrary intensity is analyzed for all possible liquid volumes. The boundary of the stability region for axisymmetric shapes (considering both axisymmetric and nonaxisymmetric perturbations) have been calculated. It is found that, for sufficiently small Bond numbers, three different unstable modes can appear. If the volume of liquid is decreased from that of an initially stable axisymmetric configuration the bridge either develops an axisymmetric instability (breaking in two drops as already known) or detaches its interface from the disk edges (if the length is smaller than a critical value depending on contact angle), whereas if the volume is increased the unstable mode consists of a nonaxisymmetric deformation. This kind of nonaxisymmetric deformation can also appear by decreasing the volume if the Bond number is large enough. A comparison with other previous partial theoretical analyses is presented, as well as with available experimental results

The Stability of Two Connected Pendant Drops

The stability of an equilibrium system of two drops suspended from circular holes is examined. The drop surfaces are disconnected surfaces of a connected liquid body. For holes of equal radii and identical pendant drops axisymmetric perturbations are always the most dangerous. The stability region for two identical drops differs considerably from that for a single drop. Loss of stability leads to a transition from a critical system of identical drops to a stable system of axisymmetric non-identical. This system of non-identical drops reaches its own stability limit (to isochoric or non-isochoric paturbations). For non-identical drops, loss of stability results in dripping or streaming from the holes. Critical volumes for non-identical drops have been calculated as functions of the Bond number, B. For unequal hole radii, stability regions have been constructed for a set of hole radius, K. The dependence of critical volumes on K and B is analyzed

A Review on the Stability of Liquid Bridges

Errors in theAlthough early studies dealing with the stability of liquid bridges were published long time ago, these studies were mainly concerned with the stability of axisymmetric liquid bridges between parallel, coaxial, equal-in-diameter solid disks, with regard to axisymmetric perturbations. Results including effects such as solid rotation of the liquid column, supporting disks of different diameters and an axial acceleration acting parallel to the liquid column can be found in several works published in the early eighties, although most of these analysis were restricted to liquid bridge configurations having a volume of liquid equal or close enough to that of a cylinder of the same radius. Leaving apart some asymptotic studies, the analysis of non-axisymmetric effects on the stability of liquid bridges (lateral acceleration, eccentricity of the supporting disks) and other not so-classical effects (electric field) has been initiated much more recently, the results concerning these aspect of liquid bridge stability being yet scarce

Bifurcation of The Equilibrium States of A Weightless Liquid Bridge

The bifurcation of the solutions of the nonlinear equilibrium problem of a weightless liquid bridge with a free surface pinned to the edges of two coaxial equidimensional circular disks is examined. The bifurcation is studied in the neighborhood of the stability boundary for axisymmetric equilibrium states with emphasis on the boundary segment corresponding to nonaxisymmetric critical perturbations. The first approximations for the shapes of the bifurcated equilibrium surfaces are obtained. The stability of the bifurcated states is then determined from the bifurcation structure. Along the maximum volume stability limit, depending on values of the system parameters, loss of stability with respect to nonaxisymmetric perturbations results in either a jump or a continuous transition to stable nonaxisymmetric shapes. The value of the slenderness at which a change in the type of transition occurs is found to be Λs=0.4946. Experimental investigation based on a neutral buoyancy technique agrees with this prediction. It shows that, for Λ\u3cΛs, the jump is finite and that a critical bridge undergoes a finite deformation to a stable nonaxisymmetric state

Stability of liquid bridges between an elliptical and a circular supporting disk

A numerical method has been developed to determine the stability limits for liquid bridges held between noncircular supporting disks and the application to a configuration with a circular and an elliptical disk subjected to axial acceleration has been made. The numerical method led to results very different from the available analytical solution which has been revisited and a better approximation has been obtained. It has been found that just retaining one more term in the asymptotic analysis the solution reproduces the real behavior of the configuration and the numerical results

Bifurcation of The Equilibrium States of A Weightless Liquid Bridge

The bifurcation of the solutions of the nonlinear equilibrium problem of a weightless liquid bridge with a free surface pinned to the edges of two coaxial equidimensional circular disks is examined. The bifurcation is studied in the neighborhood of the stability boundary for axisymmetric equilibrium states with emphasis on the boundary segment corresponding to nonaxisymmetric critical perturbations. The first approximations for the shapes of the bifurcated equilibrium surfaces are obtained. The stability of the bifurcated states is then determined from the bifurcation structure. Along the maximum volume stability limit, depending on values of the system parameters, loss of stability with respect to nonaxisymmetric perturbations results in either a jump or a continuous transition to stable nonaxisymmetric shapes. The value of the slenderness at which a change in the type of transition occurs is found to be Λs=0.4946. Experimental investigation based on a neutral buoyancy technique agrees with this prediction. It shows that, for Λ\u3cΛs, the jump is finite and that a critical bridge undergoes a finite deformation to a stable nonaxisymmetric state