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.     

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.     

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.     

Cylindrical equilibrium surfaces of a capillary liquid and their stability

We consider the plane problem of the equilibrium of a capillary surface. We study the stability of a two-dimensional surface with respect to plane and spatial disturbances. We give data which can be used for deciding the question of the stability of any symmetric equilibrium surface in a field of gravitational forces and in conditions of weightlessness. We solve the problems of the stability of a liquid in a rectangular and a sectorial channel and also the problem of the separation of a plane drop from a horizontal wall.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 57–68, January–February, 1976.The authors are grateful to A. D. Myshkis and A. D. Tyuptsov for their evaluation and their useful comments.     

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.     

Hydrodynamics of contact of a falling drop with a liquid free surface

A fine structure of the flows developing during primary contact of freely falling drops with a deep quiescent fluid is studied using the macrophotography and high-speed video filming methods. Water drops falling in water, alcohol, and oil, as well as drops of oil, petroleum, and aqueous solutions of salt or alcohol falling in water are investigated. The work is focused on the visualization of the finespray scattering from the primary contact area. The collisions of small droplets with the surface of the submerging drop are first recorded. The direction of the spray and streamer scattering is determined by the surface tension coefficients of the coalescing liquids. The conditions under which the spray droplets collide with the drop surface are determined.     

Symmetric shapes of contact of a jet with the surface of a heavy liquid

The plane linearized problem of oblique impingement of a weightless jet of an ideal incompressible fluid on the surface of a heavy fluid is considered. Flows are sought with symmetric forms of the contact region. Mathematically we arrive at the problem of the eigenvalues and eigenfunctions of an integral equation; solving this problem we obtain various contact forms. The fundamental result for the infinitesimally thin jet of finite intensity is derived by passing to the limit, yielding a result analagous with the forms of free vibrations of a string. Some results are presented for the problem under consideration in the nonlinear formulation.The two-dimensional problem on (vertical) impingement of a jet on a liquid was solved by Olmstead and Raynor [1]. Some results for oblique impingement of a sufficiently thin, slightly curved jet are presented by Frolov [2], Information on other studies, primarily experimental, is presented in [3].This problem is related to the model of a jet curtain of an air-cushion vehicle; in this regard we note the study of Stepanov [4] in which, in particular, a result is obtained for an infinitesimally thin jet curtain.     

Stability of the equilibrium of a liquid with multiply connected free surface in closed systems

A study is made of the equilibrium of a capillary liquid that partly fills a container and is in contact with several isolated gas cavities. The conditions of its stability are obtained with allowance for the change in the internal energy of the gas when the free surface is perturbed. The analysis of these conditions is accompanied by the solution of examples.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 4–9, May–June, 1984.     

关于平动相对平衡问题的液面不动点

为了方便叙述,我们定义平动相对平衡问题变形前后两液面交线上的点为不动点． 一般《工程流体力学》教材均采用等加速度运动罐车或其它容器来作为典型例题,以说明平动相对平衡问题的求解方法．其中不少教材［１～６］既不说明容器的具体形状,也不说明是变形前还是变形后的液面中心,或者简单地给出‘腋面中心点不变化”［１］的结论后,将坐标原点取在液面中心点;另有一些教材则将坐标原点取在液面上的某一点,从而得出压强分布规律和等压面方程．这就引出了以下两个问题; （１）液面中心点是否一定是不动点？如果不是,则由该点的边界…     

Instability of the equilibrium state of a liquid with a free surface in the presence of volume heat sources

The problem of the convection of a weakly compressible fluid is considered. In the free convection equations a heat source function is taken into account. The stability of the equilibrium state of a horizontal layer relative to small perturbations is studied using the linearization method. On the basis of numerical calculations it is shown that the mechanical equilibrium state of the fluid is unstable. The neutral curves are plotted and the critical Rayleigh numbers are found. In the calculations values of the physical parameters typical of Lake Baikal were used.     

Stress analysis of a layered elastic solid in contact with a rough surface exhibiting fractal behavior

A contact stress analysis is presented for a layered elastic half-space in contact with a rough surface exhibiting self-affine (fractal) behavior. Relationships for the mean contact pressure versus representative strain and the real half-contact width versus elastic properties of the layer and the substrate, asperity radius, layer thickness, and truncated half-contact width were derived from finite element simulations of a layered medium compressed elastically by a rigid cylindrical asperity. These relationships were incorporated in a numerical algorithm that was used to obtain the contact pressure distributions and stresses generated by the asperity contacts formed at the interface of the layered medium and the fractal surface. Analytical solutions illustrate the significance of the elastic material properties, layer thickness, and surface topography (roughness) on global parameters such as normal load and real contact area. Results for the contact pressure distribution and the surface and subsurface stresses provide insight into the initiation of yielding and the tendency for cracking in the layered medium. It is shown that cracking at the surface and the layer/substrate interface is more likely to occur in the case of a stiff layer, whereas surface cracking is more prominent for a relatively compliant layer.     

Dynamic responses of a solid wall in contact with a bubbly liquid excited by thermal shock loading

Several MW-class spallation neutron sources are being developed in the world. Specifically, intensive and high energy protons are injected into heavy liquid metals (mercury, lead or lead-bismuth eutectic) to induce the spallation reaction that produces neutrons. At the moment when the proton beams are injected, thermal shock occurs in the liquid metal, causing pressure waves to propagate in the liquid metal, collide against the container and damage it.It is proposed that microbubbles are injected into the liquid metal to mitigate the impulsive pressure waves by means of absorption and attenuation effects. These effects are dependent on the relationship between bubble size and the rate of pressure increase. In the present experiment, a very rapid rise in pressure in the order of MPa/μs, equivalent to the rise in pressure due to proton beam injection, was simulated by the electric discharge method in a water loop test to investigate the impulsive pressure mitigation effect of injected microbubbles. The solid wall response was measured using an accelerometer, and the dynamic responses of microbubbles were observed using an ultra-high-speed camera filming at 5 × 10⁵ frame/s. The sound velocity in bubbly water was estimated using a differential image technique. It was confirmed from the experimental results that microbubbles are effective in reducing impulsive pressure waves and to suppressing the impact vibration of the solid wall in contact with the liquid.     

Piercing a liquid surface with an elastic rod: Buckling under capillary forces

When a thin elastic structure comes in contact with a liquid interface, capillary forces can be large enough to induce elastic deformations. This effect becomes particularly relevant at small scales where capillary forces are predominant, for example in microsystems (micro-electro-mechanical systems or microfluidic devices) under humid environments. In order to explore the interaction between capillarity and elasticity, we have developed a macroscopic model system in which an initially immersed vertical elastic rod is raised through a horizontal liquid surface. We follow a combined approach of experiments, theory and numerical simulations to study this system. In spite of its apparent simplicity, our experiment reveals a complex phase diagram, involving large hysteretic behaviour. We employ Kirchhoff equations for thin elastic rods and use path-following methods from which we obtain a variety of equilibrium states and associated transitions that are in excellent qualitative and quantitative agreement with those observed experimentally.     

Dynamic edge angles of wetting upon spreading of a drop over a solid surface

The hydrodynamic free-boundary problem of the axisymmetric spreading of a viscous-fluid drop over the smooth surface of a solid under the action of capillary forces and under the conditions of weak gravitation is considered. For finite inclination angles of the free surface and small capillary numbers, the problem is reduced to the simpler hydrodynamic problem in a region with known boundary by the asymptotic method. An expression for the dynamic edge angle of the drop is obtained. It is shown that in addition to the local inclination angle of the boundary near the contact line of three phases, one drop has several dynamic edge angles. These angles are calculated for small Reynolds and Bond numbers. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences, Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 101–107, January–February, 1999.     

Adhesive contact of an elastic semi-infinite solid with a rigid rough surface: Strength of adhesion and contact instabilities

The effect of adhesion on the contact behavior of elastic rough surfaces is examined within the framework of the multi-asperity contact model of Greenwood and Williamson (1966), known as the GW model. Adhesive surface interaction is modeled by nonlinear springs with a force–displacement relation governed by the Lennard–Jones (LJ) potential. Constitutive models are presented for contact systems characterized by low and high Tabor parameters, exhibiting continuous (stable) and discontinuous (unstable) surface approach, respectively. Constitutive contact relations are obtained by integrating the force–distance relation derived from the LJ potential with a finite element analysis of single-asperity adhesive contact. These constitutive relations are then incorporated into the GW model, and the interfacial force and contact area of rough surfaces are numerically determined. The development of attractive and repulsive forces at the contact interface and the occurrence of instantaneous surface contact (jump-in instability) yield a three-stage evolution of the contact area. It is shown that the adhesion parameter introduced by Fuller and Tabor (1975) governs the strength of adhesion of contact systems with a high Tabor parameter, whereas the strength of adhesion of contact systems with a low Tabor parameter is characterized by a new adhesion parameter, defined as the ratio of the surface roughness to the equilibrium interatomic distance. Applicable ranges of aforementioned adhesion parameters are interpreted in terms of the effective surface separation, obtained as the sum of the effective distance range of the adhesion force and the elastic deformation induced by adhesion. Adhesive strength of rough surfaces in the entire range of the Tabor parameter is discussed in terms of a generalized adhesion parameter, defined as the ratio of the surface roughness to the effective surface separation.     

Supercomputer simulation of liquid drop formation on a solid surface

Using a molecular-dynamics-type approach, we show how to simulate the formation of a liquid drop on a solid surface. Application is made to the case in which the liquid is water and the solid is graphite. The dynamical equations are large systems of nonlinear, ordinary differential equations which must be solved numerically. CRAY X-MP/24 simulations and related contact angle calculations are described and discussed.     

Short capillary waves on the surface of a stretching cylindrical jet of a viscous liquid

This paper presents asymptotic formulas describing the evolution of short–wave perturbations on the surface of a cylindrical viscous liquid jet with the radius decreasing in time. The effects of Reynolds and Weber numbers and the initial wavenumber on the decay of the perturbations are analyzed.     

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.