Gravity-induced spreading of a drop of a viscous fluid over a surface

The unsteady-state nonlinear problem of spreading of a drop of a viscous fluid on the horizontal surface of a solid under the action of gravity and capillary forces is considered for small Reynolds numbers. The method of asymptotic matching is applied to solve the axisymmetrical problem of spreading when the gravity exerts a significant effect on the dynamics of the drop. The flow structure in the drop is determined at large times in the neighborhood of a self-similar solution. The ranges of applicability of the quasiequilibrium model of drop spreading with a dynamic edge angle and a self-similar solution are found. It is shown that the transition from one flow model to another occurs at very large 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. 3, pp. 59–67, May–June, 1999.     

Impact of a drop on a solid surface

A numerical solution is obtained to the unsteady-state problem of a direct collision between a liquid drop of cylindrical form and a rigid surface. It is shown that unsteady-state interaction between shock waves inside the drop leads to the development of broad zones of cavitation, promoting the dispersion of the drop.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 151–155, September–October, 1977.The authors thank L. F. Shaikhatarova for making the calculations.     

Dynamics of the impact of a drop on a solid surface

A numerical method is developed for calculating the non-steady-state motions of a compressible liquid, with application to problems of the impact type. A homogeneous, completely conservative difference scheme of the first order of exactness is used. For isolation of the discontinuities and smoothing of the solution, use is made of the introduction of an artificial viscosity, based on an analysis of the provisional properties of the solution. An investigation is made of the scheme stability and the necessary stability criteria are obtained. The article gives the detailed results of a calculation of the impact interaction between a spherical drop and a rigid surface. It is shown that the maximal pressure arising with the impact of a drop of liquid on a solid surface exceeds by several times the pressurecalculated using the one-dimensional theory; under these circumstances, the rate of expansion of the drop along the surface exceeds the initial velocity of the collision by an order of magnitude.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 36–43, January–February, 1978.In conclusion, the authors thank G. G. Chernyi for his continuing interest and his valuable evaluation of the results of the work, and L. F. Shaikhatarov, who participated in making the calculations.     

The impact of a single drop on a wetted solid surface

 The impact of single drops on a thin liquid film was studied to understand the mechanism of secondary atomisation of sprays colliding on a wetted, cold, solid surface. To span a wide range of conditions various mixtures of water and glycerol were used. The use of Weber number, Ohnesorge number and non-dimensional film thickness to describe the peculiarities of the phenomenon allowed to carry out the experiments under appropriate similarity conditions. The impact of millimetric drops was analysed in detail by photographic means, using both still photography to study impact morphology, and laser sheet visualisation to investigate secondary droplet formation. Two mechanisms of splash were identified, depending essentially on the liquid viscosity (Ohnesorge number), a parameter which appears to play an important role also in defining the splash morphology. A photographic documentation is annexed. The characteristic times of the crown formation, the non-linear evolution of cusps (jet formation) and the surface roughness influence are further discussed. The experimental results allow to propose an empirical correlation for the splashing/deposition limit, for a wide range of conditions, and a comparison to available previous works is presented. The influence of the film thickness and liquid viscosity on the splash is confirmed and quantified. Received: 1 March 1996/Accepted: 12 November 1996     

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.     

The cooling of a lava flow spreading over a flat surface

The cooling of a lava flow modeled by a viscous incompressible fluid spreading over a flat surface is considered. In order to model the free surface, a known analytical solution is used in the thin-layer approximation. The thermal boundary layer thickness is determined and the evolution of thermal fields in the lava profile is studied.     

Full-field spreading velocity measurement inside droplets impinging on a dry solid surface

Liquid droplet impacts onto solid surfaces have attracted enormous amount of attention from wide range of research fields including experimental and numerical investigations. Unlike experimental efforts, numerical and analytical studies generated various sets of data. In this study, we investigated the spreading velocities inside the water droplets impinging onto a dry glass substrate using time-resolved PIV. The method, together with the high spatiotemporal resolution and the additional treatments improving the robustness, allowed us to resolve the radial velocity profiles efficiently in the spreading phase. Several impact velocity cases ranging from 0.40 to 0.96 m/s were studied. They correspond to low and moderate level Weber numbers (4.9–27.6). We observed that instantaneous radial velocity distributions exhibit linear and nonlinear modes. The nonlinearity is caused by the vortical flows formed at outer regions of the spreading liquid lamella. We demonstrated that even at low impact velocities the linear parts of the profiles obey a quasi-one-dimensional theory proposed in the literature. The comparison of obtained results with a literature-based numerical study, performed for high range of Weber numbers, confirmed the simultaneous existence of linear and nonlinear parts in the radial velocity profiles. In spite of the scale differences in terms of Weber number, the agreements in the tendencies of the profiles imply the validity of the mechanism considered in the numerical study even at low and moderate level range of Weber numbers.     

Effects of tangential speed on low-normal-speed liquid drop impact on a non-wettable solid surface

A stream of water drops colliding on a rotating cylindrical Teflon surface was observed. The collision resulted in partial rebound, deposition, and split deposition. The collision outcome was found to depend on the normal Weber number and the tangential Weber number, which represent the collision momentum in the normal and the tangential direction, respectively. The extent of influence of the centrifugal acceleration induced by the rotation was kept negligibly under 2% compared with the impact momentum change. Through careful measurements on the elongation of the liquid drop after impact, the impact regime boundaries are seen to be governed by the long and short axes of the elliptical-disk shape at the maximal spread. Partial rebound first changes to deposition when the long axis is 1.1 times the short axis; and then changes to split deposition when the long axis is 1.46 times the short axis. Furthermore, the dimensionless excess spread area was found to be directly proportional to the tangential Weber number.     

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.     

Modelling the wetting of a solid occlusion by a liquid film

We investigate in this work how the presence of an occlusion affects the dynamics of the wetting front of a liquid film draining down a vertical surface. This numerical study is developed in the context of the lubrication approximation. Through a parametric study, we show that depending on the asymptotic film thickness and the fluid properties, there exists a critical substrate contact angle below which separation of the contact line from the occlusion wall is observed which results in the appearance of a dry zone in the wake of the occlusion. In analogy with external aerodynamics, we also show that a sharp corner in the occlusion can induce this contact line separation. Our numerical results also highlight the importance of the occlusion wettability on the morphology of the wetting front suggesting a possible mechanism to control and mitigate the often undesirable fingering instability.     

Axisymmetric spreading of a heavy liquid over a plane

A study is made of the steady flow over a horizontal plane of a heavy inviscid incompressible liquid which flows through the side surface of a circular cylinder which rises above the plane to height h and has a base radius ofa. The motion of the liquid is assumed to be symmetric with respect to the axis of the cylinder; the pressure p is constant (equal to the atmospheric pressure) on the free surface of the liquid. Fora/h = 1, this problem can be regarded as a problem of perturbation of the flow from a flat source by a free surface. Investigation showed that this perturbation problem is essentially nonlinear, and a solution of it in the complete region occupied by the liquid can be obtained only in variables of the boundary layer type. The problem admits linearization under the additional assumption that the parameter = Q²/(8²ga³) is small; here, Q is the constant volume flow rate of the liquid per unit height of the cylinder, and g is the acceleration of free fall. For the case 1, 1 the problem is solved by the method of integral transformations. A noteworthy feature of the solution is the slow damping of the perturbations of the velocity with the depth (inversely proportional to the square of the distance from the free surface), in contrast to the similar problem of the wave motions of a heavy liquid, for which the velocity perturbations are damped exponentially.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 3–7, March–April, 1984.     

Dynamic behaviors of droplet impact and spreading: A universal relationship study of dimensionless wetting diameter and droplet height

A notable universal relationship has been proposed in the literature for the evolution of dimensionless droplet height and wetting diameter during the initial spreading stage of droplet impingement. In this study, this universal relationship was investigated by employing three sets of measurements. Sequential images were recorded, and the whole droplet profile ensembles were plotted to facilitate this study. These sets of experiments were designed by changing impact velocity, surface hydrophobicity, or solution property. The experimental results illustrate that the importance of parameters causing the data variation is in the order of surface hydrophobicity > initial impact velocity > surfactant on wetting diameter, and surface hydrophobicity ≈ initial impact velocity > surfactant on droplet height. No universal relationship was observed for dimensionless droplet height and wetting diameter.     

Molecular dynamics studies on spreading of nanofluids promoted by nanoparticle adsorption on solid surface

Spreading of nanofluids on solid substrate was studied via molecular dynamics simulations. Simulation models for two immiscible fluids (oil and water based nanofluids) confined in a slit between two planar solid walls were set up. The influence of the volume concentration of the nanoparticles on the three-phase contact line motion was investigated. We found that the larger volume concentration results in more visible nanoparticle adsorption on solid surface. This effect further induces an advancing displacement of the contact line compared with the meniscus profiles in low concentration case and that with the absence of nanoparticles. These findings are consistent with the previous experimental and theoretical results and provide the atomic-scale understanding on nanofluid spreading.     

Effect of chamber pressure on spreading and splashing of liquid drops upon impact on a dry smooth stationary surface

Liquid drop impacts on a smooth surface were studied at elevated chamber pressures to characterize the effect of gas pressure on drop spreading and splashing. Five common liquids were tested at impact speeds between 1.0 and 3.5 m/s and pressure up to 12 bars. Based on experiments at atmospheric pressure, a modification to the “free spreading” model (Scheller and Bousfield in AIChE Paper 41(6):1357–1367, 1995) has been proposed that improves the prediction accuracy of maximum spread factors from an error of 15–5%. At high chamber pressures, drop spreading and maximum spread factor were found to be independent of pressure. The splash ratio (Xu et al. in Phys Rev Lett 94:184505, 2005) showed a non-constant behavior, and a power-law model was demonstrated to predict the increase in splash ratio with decreasing impact speed in the low impact speed regime. Also, drop shape was found to affect splash promotion or suppression for an asymmetry greater than 7–8% of the equivalent drop diameter. The observations of the current work could be especially useful for the study of formation of deposits and wall combustion in engine cylinders.     

Numerical investigation of surface wave generation upon solid body immersion in water

The immersion of a cylindrical solid in an undisturbed fluid is numerically investigated. Wave formation accompanying this process and the dependence of the wave parameters on those of the body are studied. The shallow water model in a cylindrical coordinate system is used. A total variation diminishing (TVD) scheme is applied on a C-type difference grid, according to Arakawa’s classification.     

Dynamics of a viscous liquid wetting a solid via Van der Waals Forces

Deposited at All-Union Institute of Scientific and Technical Information (VINITI), No. 3197-V93, December 27, 1993.     

The distribution of particles in a shock-induced boundary layer of a dusty gas over a solid surface

The laminar boundary layer behind a constant-speed shock wave moving through a dusty gas along a solid surface is studied. The Saffman lift force acting on a spherical particle in a gas boundary layer is taken into account. A method for calculating the density profile of dispersed phase near the wall is proposed and some numerical results are given. It is shown that behind the shock wave, there exists a curved thin layer where the density of particles is many times higher than the original one. This dust collection effect may be of essential importance to the problem of dust explosion in industry.     

Surface shape of a magnetic liquid drop near the edge of a magnetic wedge

The two-dimensional problem of the shape of the free surface of a magnetic fluid in a gravity field, a uniform external magnetic field and the nonuniform field of a magnetized metal wedge is considered. The results of numerically calculating the shape of the free surface of a magnetic liquid drop retained on an inclined plane by the field of a magnetizing wedge are presented. The changes in the shape of the free surface of an infinite volume of magnetic liquid near the edge of a wedge with increase in the external field are investigated. It is shown that for a certain critical field some of the magnetic liquid separates and adheres to the edge of the wedge. Experimental data on the determination of the maximum cross-sectional area of a drop retained by the magnetic field of a wedge and the critical rise of the magnetic liquid relative to the level outside the field are presented. The experimental and theoretical results are in agreement.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 113–119, July–August, 1992.The authors wish to thank V. V. Gogosov for useful discussions and his interest in the work.