Stability of wave regimes in film flow along a vertical surface

The stability of nonlinear traveling waves with respect to all possible two-dimensional infinitesimal perturbations is numerically investigated. The stability zones are determined for two families. It is shown that regimes of the second family, which in the limit go over into positive solitons, are the more stable. Novosibirsk. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 126–131, September–October, 1988.     

Spatial flow regimes around a curved surface

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1, pp. 87–93, January–February, 1989.     

Grazing impact of continuous droplet streams with a superhydrophobic surface

When high-velocity droplets make grazing impact with a superhydrophobic surface, the droplets undergo significant deformation before recoiling and rebounding from the surface. Two distinct operating regimes describe the response of the reflected droplet stream after impact. In the first regime, the droplets remain discrete and uniform after the impact, but exhibit rotation and significant oscillations. This regime dominates if each droplet can clear the impact region before the next droplet arrives. In the second regime, droplets cannot avoid coalescing into a puddle at the surface. A secondary jet is ejected from the puddle which breaks up into a random droplet stream after traveling a short distance due to the lack of a forced unstable perturbation. The droplet-to-droplet spacing in the incoming stream determines which regime rules, with the critical value correlated by a Weber number. In the first regime, a detailed accounting of the kinetic and potential energies reveals that neither droplet oscillation nor rotation can fully account for the loss of translational kinetic energy, indicating significant internal circulation must occur in the droplets at impact. An application of droplet rebound from a superhydrophobic surface is proposed.     

Stokes flow over a cavity on a superhydrophobic surface containing a gas bubble

Within the approximation of Stokes hydrodynamics, several problems of a steady-state flow over a two-dimensional cavity containing a gas bubble are solved using the method of boundary integral equations. In contrast to previous publications, the method developed makes it possible to study the situation in which the cavity is only partially filled with gas, and the edges of a curved phase interface do not coincide with the cavity corners. Using periodic boundary conditions for the velocity, the flows with pure-shear and parabolic velocity profiles, and also the flow over a group of cavities were considered. The aim of the study was to calculate the effective (average) slip velocity over a microcavity, as applied to flows near textured superhydrophobic surfaces. A parametric numerical study of the effective velocity slip as a function of the radius of curvature of the interface and the position of the interface relative to the cavity boundaries was performed. The accuracy of the method is validated by the calculations of a number of limiting flows over a cavity, for which a quantitative agreement with the results known in the literature is demonstrated.     

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.     

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.     

Flow regimes in a single dimple on the channel surface

The boundaries of the domains of existence of flow regimes past single dimples made as spherical segments on a flat plate are determined with the use of available experimental results. Regimes of a diffuser-confuser flow, a horseshoe vortex, and a tornado-like vortex in the dimple are considered. Neither a horseshoe vortex nor a tornado-like vortex is observed in dimples with the relative depth smaller than 0.1. Transformations from the diffuser–confuser flow regime to the horseshoe vortex regime and from the horseshoe vortex flow to the tornado-like vortex flow are found to depend not only on the Reynolds number, but also on the relative depth of the spherical segment. Dependences for determining the boundaries of the regime existence domains are proposed, and parameters at which the experimental results can be generalized are given.     

Convection regimes and heat transfer characteristics along a continuously moving heated vertical plate

The steady laminar flow and thermal characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot. The velocity and temperature variations, obtained by a finite volume method, are used to map out the entire forced, mixed and natural convection regimes. The effects of the Prandtl number (Pr) and the buoyancy force parameter (B) on the friction and heat transfer coefficients are investigated. Comparisons with experimental measurements and solutions by others in the pure forced and pure natural convection regions are made. In the mixed convection region, the results are compared with available finite-difference solutions of the boundary layer equations showing excellent agreement. The region close to the extrusion slot is characterized as a non-similar forced-convection dominated region in which Nu_xRe_x^−1/2 drops sharply with increasing Richardson number (Ri_x). This is followed by a self-similar forced-convection dominated region in which Nu_xRe_x^−1/2 levels off with increasing Ri_x until the buoyancy effect sets in. The existence and extent of the latter region depend upon the value of B. A non-similar mixed convection region where increasing buoyancy effect enhances the heat transfer rate follows. Finally, this region is followed downstream by a self-similar natural-convection dominated region in which Nu_xRe_x^−1/2 approaches the pure natural convection asymptote at large Ri_x. Critical values of Ri_x to distinguish the various convection regimes are determined for different Pr and B.     

New solutions for surface tension driven spreading of a thin film

The standard fourth-order non-linear PDE modelling the flow of thin fluid film subject to surface tension is studied. The Lie group method is used to reduce the model equation from a fourth-order PDE to a fourth-order ODE. Analytical solutions are obtained for certain cases. Where analytical progress cannot be made, we determine numerical solutions.     

The spreading of nonlinear elastic surface waves

A theory for the lateral spreading of a beam of nonlinear surface acoustic waves across the surface of an arbitrary, homogeneous, elastic half-space is developed. The resulting evolution equation generalizes that obtained for uni-directional waves by replacing an ordinary derivative by a diffusion operator of Schrödinger type. The coefficients arising in the evolution equation are related to partial derivatives of the dispersion relation for linearized surface waves on the half space. Details are given for isotropic materials and for two special cases of beams travelling along axes of high elastic symmetry.     

正交异性复合材料界面上反平面动态自相似扩展裂纹问题的解

采用复变函数论的方法，对复合材料界面上的裂纹扩展问题进行研究。并根据任意的自相似指数的断裂动力学问题，进行自相似求解，导出解析解的一般表示。应用该法可以迅速地将所论问题转化为Riemann-Hil-bert问题，并可以相当简单地得到问题的闭合解。文中分别对裂纹中心受阶跃载荷，裂纹面受到瞬时脉冲载荷作用下的界面裂纹扩展问题进行求解。得到了裂纹的位移。尖端的应力和动态应力强度因子的解析解。应用该解并通过叠加原理。就可以很容易的求得任意复杂问题的解。     

Motion of a three-phase contact line along a wet surface

The motion of the contact line in gas-liquid-solid systems is theoretically investigated for small values of the capillary number and Reynolds number. The possible existence on the solid substrate of a residual microscopic film formed by adsorbed liquid molecules is taken into account and the spreading characteristics of the liquid on dry and wet substrates are compared. It is shown that, in accordance with the experimental data, in the model employed the motion of the liquid during wetting is rolling motion, and that the increase in the dynamic contact angle is slower for a wet than for a dry substrate. The maximum dynamic contact angle is much less than 180°. The flow structure in the neighborhood of the moving contact line is analyzed and it is shown that under certain conditions regions with closed streamlines may be formed. The reason for this is the self-induced Marangoni effect — the reaction of the surface tension gradient on the liquid-solid boundary caused by the liquid flow on the flow that caused it.Based on a paper read at the Seventh Congress on Theoretical and Applied Mechanics, Moscow, August 1991. Presented by R. I. Nigmatulin.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 81–89, November–December, 1992.     

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.     

Sliding of a filament along the surface of a horizontal shelf

We consider the problem of the steady motion of a linear-elastic and envelope surface of a fixed horizontal shelf of a flexible filament of semiinfinite length. We investigate the effect of longitudinal waves reflected from the point of encounter and point of descent of the filament on the parameters of motion of the filament.Yu. Akhunbabaev Tashkent Institute of Textiles and light Industry, Uzbekistan. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 90–96, February, 1994.     

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.     

Simulations of the spreading of a vesicle on a substrate surface mediated by receptor-ligand binding

A continuum model was introduced for the adhesion of vesicles to substrate surfaces. In the model, the vesicle membrane was assumed to be a closed shell with hyperelasticity. The vesicle cavity is filled with a liquid of fixed volume. The receptors on the membrane are mobile and initially uniformly distributed while the ligands on the substrate surface are fixed and also uniformly distributed. The formation of localized regions of tight binding between receptors and ligands, results in vesicle adhesion to the substrate surface. An adhesive model was introduced to describe the adhesive interaction between the receptors and the ligands. The growth of the adhesion area occurs via recruiting receptors from the non-adhered region through diffusion. Finite-element methods were used to solve the governing equations for the deformation of the vesicle and the receptor diffusion on the membrane surface. Effects of the membrane stiffness, the cohesive parameters and the receptor density on the adhesion kinetics of the vesicle were studied. In addition, the instability of the advancing front of the adhesion was also analyzed.