Characteristic stability parameter of the axisymmetric equilibrium surface of a capillary liquid

In [1] the question of stability of the equilibrium state of a capillary liquid in weak force fields was reduced to determination of conditions such that the smallest eigenvalue λ* of a certain boundary problem would be positive. In [2] it was shown that λ* is a monotonic function of the parameter χ, dependent on the form of the vessel. The basic properties of the function λ*(χ) were also described. In the present study, these properties are used to study the general problem of stability of an axisymmetric liquid surface. A method for calculation of the critical values of the parameter χ and construction of the maximum stability region is given. Special attention is given to the cases of complete weightlessness, and action of gravitational and centrifugal forces. Critical values of the parameter χ are presented for these cases either graphically or analytically, which, given the shape of the vessel, permits evaluation of the stability of any of the family of axisymmetric equilibrium surfaces. We note that in the case of action by gravitational forces χ values for certain equilibrium surfaces were obtained by Barnyak.     

On the stability margin of equilibrium states of a capillary liquid in a slit

The stability margin is determined for symmetric equilibrium shapes of the free surface of a liquid suspended in a slit and subject to gravity and surface tension. The calculations are made in the range of variation of the parameters, the wetting angle and the Bond number, adjoining the boundary of the stability region.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 88–93, July–August, 1979.     

Determination of the equilibrium state of a capillary liquid in a vessel

This article describes a method for determination of the form of the equilibrium surface of a liquid in a given vessel of arbitrary axiosymmetric form. Capillary, gravitational, and centrifugal forces act on the liquid. Liquid volume and wetting angle are given. Curves are constructed for the case of negative overloads by a homogeneous gravitational field, which are used to find the equilibrium states. An example which illustrates the application of the method is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–7, July–August, 1973.     

Numerical study of equilibrium capillary surfaces under low gravitational conditions

Summary The shape of the equilibrium capillary surface of a liquid in a vessel of rotational symmetry is investigated numerically by transforming the governing equation (Young-Laplace-Gauss-equation) into a system of four first order differential equations. In addition a differential equation of first order concerning the liquid volume is taken into consideration. The problem under study is solved as an initial value problem. Essential for the numerical procedure is the horizontal section method which works for all tank geometries with monotone continuous curvature, i.e. the technically important cylindrical, conical and spheroidal tank configurations.

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Bestimmung der kapillaren gleichgewichtsoberflächen unter restschwere

Übersicht Die Gestalt der kapillaren Gleichgewichtsoberfläche einer Flüssigkeit in einem drehsymmetrischen Behälter wird numerisch bestimmt durch Transformation der Young-Laplace-Gauß-Gleichung in ein System von vier Differentialgleichungen erster Ordnung. Hinzu kommt eine weitere gewöhnliche Differentialgleichung erster Ordnung für das Flüssigkeitsvolumen. Das resultierende Problem wird als Anfangswertproblem gelöst. Als wesentlich für die numerische Lösung erweist sich das Horizontalschnittverfahren, das bei allen monoton stetig gekrümmten Tankberandungen angewandt werden kann. Das sind vor allem die technisch wichtigen Tankgeometrien Kreiszylinder, Kegel und Kugel.

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Dedicated to Professor Dr. W. Schnell on the occasion of his sixty-fifth anniversary     

Hydrostatics in weak force fields. On annular figures of equilibrium of a rotating liquid and their stability

We consider the problem of annular equilibrium figures of a rotating weightless liquid, having surface tension, and their stability. This question has been studied by Charreaux (for a discussion of his results see [1]), who examined the evolution of the forms of annular equilibrium figures and showed that there exists a family of stable equilibrium shapes. However, these studies of Charreaux are incomplete, and the conclusion on the existence of stable forms is valid only for axisymmetric disturbances.In the following we examine the properties of annular equilibrium figures of a rotating liquid. From the results of numerical integration on a digital computer we construct a family of equilibrium forms and present data which permit finding the corresponding equilibrium form from the ensemble of physical parameters which define the equilibrium state. In studying the stability we use the technique of [2, 3].The results of the numerical calculation and the asymptotic representat ons show that stable annular equilibrium figures of a rotating liquid do not exist.The author wishes to thank M. A. Belyaev for compiling the program for the numerical calculation, and also N. D. Kopachevskii, A. D. Myshkis, and A. D. Tyuptsov for discussions of the results and helpful remarks.     

Bifurcation and stability analysis for liquid surfaces

A more comprehensive discussion on the bifurcation problems for the shape of liquid surfaces is made in this paper. The necessary conditions for bifurcation are given, and the bifurcating solutions near bifurcation points can be obtained by perturbation technique. Finally the stability of the bifurcating states is analyzed by means of the principle of minimum potential energy.     

Stability of the equilibrium state of a capillary liquid with disconnected free surface

A study is made of the stability of the equilibrium state of a liquid with disconnected free surface in open systems. The solution of two three-dimensional problems illustrates the differences from the case of a connected surface. A method is proposed for investigating the stability in the presence of axial symmetry for each of the connected components of the equilibrium surface. This method is used in a study of stability of equilibrium under conditions of weightlessness and when a gravity field is acting.     

On equilibrium configurations and their stability for a system of two coupled pendulums

A mechanical system consisting of two identical mathematical pendulums connected by a linear spring is considered under the assumption that the pendulum suspension points lie on a horizontal straight line and the system is in a homogeneous gravitational field. The equilibrium configurations of this mechanical system and their stability are studied. The results are represented in the form of bifurcation diagrams.     

Stable equilibrium of the surface of a capillary liquid in contact with the edge of a solid

The equilibrium conditions which must be satisfied at the free surface of a capillary liquid and at the line of its contact with the smooth surface of a solid are well known [1]. These conditions are equivalent to the condition of steady-state conditions for the potential energy [2, 3], and the question of the stability of the equilibrium states reduces to a study of the sign of its second variation [2–4]. The case is discussed below where the surface of the solid has a break at the line of contact with the free surface. Stable equilibrium states are identified with the points of a local minimum of the potential energy. The necessary and sufficient conditions for a minimum are obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 3–6, January–February, 1974.The authors thank A. D. Myshkis for his evaluation of the results obtained.     

Liquid equilibrium in a capillary of random form

A statistical analysis of depth distribution of droplets of a given volume is given for a capillary with radius varying randomly along the axis.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 149–152, November–December, 1974.     

A modified finite element method for determining equilibrium capillary surfaces of liquids with specified volumes

This paper describes a modified finite element method (MFEM) for determining the static equilibrium shape of the capillary surface of a liquid with a prescribed volume constrained by rigid boundaries with arbitrary shapes. It is assumed that the liquid is in static equilibrium under the influence of surface tension, adhesion, and gravity forces. This problem can be solved by employing the conventional FEM; however, a major difficulty arises due to the presence of the volume (integral) constraint and usually requires the use of the Lagrange multiplier method, the sequential unconstrained minimization technique, or the augmented Lagrange multiplier method. With the MFEM, the space variables defining the equilibrium surfaces (or curves) are expanded in terms of parametric interpolation functions, which are designed such that the boundary conditions and the integral constraint equation are automatically satisfied during each iteration of a direct numerical search process. Hence, there is no need to include Lagrange multipliers and/or penalty factors and the problem can be treated more simply as one involving unconstrained optimization. This investigation indicates that the MFEM is more efficient and reliable than the other methods. Results are presented for several case study problems involving liquid solder drops. Copyright © 2000 John Wiley & Sons, Ltd.     

Influence of surface tension on the stability of heterogeneous equilibrium of barotropic liquid phases

On the basis of the results of earlier work of the author [1] a study is made of the equilibrium and stability of a two-phase single-component heterogeneous liquid system with respect to perturbations of arbitrary shape. Allowance is made for the influence of surface tension, which plays a critical part in the formation of nucleating centers of a new phase [2]. Conditions of equilibrium are derived, and also a criterion of radial stability of a nucleating center of a new phase bounded by a closed spherical boundary. It is shown that radial stability of spherical nucleating centers also guarantees stability with respect to perturbations of arbitrary shape. The part played by the finite size of the system and the boundary conditions is elucidated. For this, two different cases are studied: a) a system under a constant external pressure, b) a system with fixed volume. In the first case, all equilibrium states are unstable. In the second, there are both unstable and stable configurations (depending on the corresponding values of two dimensionless parameters). The equation of the hyperbola of neutral stability is derived. The limits of a very small coefficient of surface tension and a very large size of the container are considered. The first situation corresponds to stable configurations, the second to unstable. For simplicity, the considered systems are assumed to be isothermal, and the equilibrium and stability are analyzed on the basis of the mechanical analog of Gibbs's principle, namely, the principle of a minimum of the mechanical potential energy of the barotropic heterogeneous liquid system. The case of nonisothermal perturbations leads to similar results, but the expressions for the corresponding dimensionless parameters are more cumbersome and less physically perspicuous.     

Stability of the equilibrium of a capillary fluid in a horizontal slit

The stability of axisymmetric shapes of equilibrium of a capillary fluid between two horizontal plates is investigated. It is shown that stability is lost with respect to axisymmetric perturbations. The boundaries of the stability region are calculated in the case when the wetting angles on the lower and upper plates are the same.     

On stability of the equilibrium of a cottrell crack

The Wiener–Hopf method is used to find the exact solution to the static symmetric plane problem of elasticity for a homogeneous isotropic plate with a finite-length crack emerging from the point of intersection of two semi-infinite straight slip (dislocation) lines. An expression for the crack-tip stress intensity factor is derived. Crack initiation is described by the Cottrell mechanism. The equilibrium of the crack is analyzed for stability     

Electroconvective stability of a weakly conducting liquid

In inhomogeneous electric fields, at sufficiently high field strengths, a weakly conducting liquid becomes unstable and is set in motion [1–4]. The cause of the loss of stability and the motion is the Coulomb force acting on the space charge formed by virtue of the inhomogeneity of the electrical conductivity of the liquid [4–13]. This inhomogeneity may be due to external heating [4–6], a local raising of the temperature by Joule heating [2, 7, 8], and nonlinearity of Ohm's law [9–13]. In the present paper, in the absence of a temperature gradient produced by an external source, a condition is found whose fulfillment ensures that the influence of Joule heating on the stability can be ignored. Under the assumption that this condition is satisfied, a criterion for stability of a weakly conducting liquid between spherical electrodes is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 137–142, July–August, 1979.