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.     

Flows Resulting from the Incidence of a Discontinuous Wave on a Bottom Step

The solvability of the problem of the flows resulting from the incidence of a discontinuous wave on a bottom step is studied using a single-layer shallow water model. Solutions in which the total energy of the flow is conserved at the step and those in which it is lost at the step are considered. Regions of double and triple hystereses in the obtained self-similar solutions are found. An analogy is drawn with the problem of single-layer flow over a bottom obstacle. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 8–22, March–April, 2006.     

Solvabilitity of a model problem of heat and mass transfer in thawing snow

A model problem of the motion of water and air in thawing snow is examined using the Masket-Leverett equations of two-phase filtration. The theorem of existence of a self-similar solution is proved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 13–23, July–August, 2008.     

Bifurcation of self-similar solutions describing a thermocapillary flow of a fluid in a thin layer

Thermocapillary flows of a fluid in a lamina with a rigid lower wall and a free upper surface, along which the temperature gradient is given in the radial direction, are investigated for large Marangoni numbers. Self-similar solutions which describe the axisymmetric flow regimes of a fluid without the circumferential velocity component are constructed numerically and asymptotically for a system of Prandtl equations. It is shown that a pair of new self-similar flow regimes of a fluid with rotation branches off from the regimes obtained. The new regimes ere calculated numerically and asymptotically. Rostov State University, Rostov-on-Don 344090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 137–142, May–June, 1999.     

Explosive discharge of a gas-saturated liquid from channels and tanks

A mathematical model for the discharge of a gas-saturated liquid from cylindrical channels is developed. Two limiting cases of linear and quadratic, relations between the flow friction force and the flow velocity are considered. It is established that the process of evacuation, from a semi-infinite channel consists of two stages. In the initial stage, the flow drag can be ignored, and the process of discharge is described by a Riemann wave solution. For the next stage, in which inertia is insignificant, nonlinear equations are obtained and self-similar solutions are constructed for them. The problem of flow through a slot in a tank of finite volume is solved. It is shown that the discharge proceeds either in a gas-dynamic choking regime or in a subsonic regime, depending on the conditions inside the tank and at the outlet. Examples of numerical calculations are given. Institute of Mechanics, Ufa Scientific Center, Russian Academy of Sciences, Ufa 450000 Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 64–73, January–February, 1999.     

Self-similar solution of the one-dimensional problem of thermocapillary motion of an emulsion

It is proved that the problem of one-dimensional motion of an emulsion under the action of thermocapillary forces has a self-similar solution in a semi-infinite interval. The behavior of the solution is illustrated by numerical examples for aluminum-lead emulsions, in which the carrier phase is lead or aluminum. The solution is compared with the solution of the self-similar problem linearized in the low impurity concentration. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 61–70, September–October, 2007.     

An investigation into the spreading of a thin liquid drop under gravity on a slowly rotating disk

The axisymmetric spreading of a thin liquid drop under the influence of gravity and rotation is investigated. The effects of the Coriolis force and surface tension are ignored. The Lie group method is used to analyse the non-linear diffusion-convection equation modelling the spreading of the liquid drop under gravity and rotation. A stationary group invariant solution is obtained. The case when rotation is small is considered next. A straightforward perturbation approach is used to determine the effects of the small rotation on the solution given for spreading under gravity only. Over a short period of time no real difference is observed between the approximate solution and the solution for spreading under gravity only. After a long period of time, the approximate solution tends toward a dewetting solution. We find that the approximate solution is valid only in the interval t∈[0,t∗), where t∗ is the time when dewetting takes place. An approximation to t∗ is obtained.     

Physicomathematical modeling of the processes of capillary impregnation of porous materials

The process of capillary impregnation of porous materials is studied numerically. A physicomathematical model of liquid diffusion in a porous sample is proposed. The model involves an analytical presentation of the diffusion coefficient, which describes available experimental data. A method of solving one-dimensional unsteady problems of impregnation is developed and tested on a self-similar solution of the corresponding boundary-value problem of impregnation. If the impregnation process is sufficiently long, the motion of the liquid in the sample is described by a stable self-similar solution. A classification of moisture diffusion on the basis of the initial humidity on the sample boundary is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 42–51, January–February, 2009.     

A theoretical analysis of forced convection in a porous-saturated circular tube: Brinkman–Forchheimer model

Fully developed forced convection inside a circular tube filled with saturated porous medium and with uniform heat flux at the wall is investigated on the basis of a Brinkman–Forchheimer model. The matched asymptotic expansion method is applied at small Darcy numbers. For large Darcy numbers, the solution for the Brinkman–Forchheimer momentum equation is found in terms of an asymptotic expansion. Once the velocity distribution is determined, the energy equation is solved using the same asymptotic technique. The results for the two limiting cases of clear fluid and Darcy flow conditions show good agreement with those available in the literature.     

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.     

Effect of the adsorption-desorption process intensity on solutal convection near a drop in a horizontal channel

The interaction between gravity convection and Marangoni convection in a horizontal rectangular channel filled with a liquid containing a surfactant and a drop of another liquid is numerically investigated. For large Schmidt numbers the occurring oscillatory regime of solutal convection is analyzed. In the model with a surface phase the effect of the adsorption and desorption processes on the convective flow structure is determined. The corresponding initial and boundary value problem is solved using a difference method.     

Effect of acoustic disturbances on a free convective flow

A gas-dynamic flow in an axisymmetric convective jet is studied experimentally. It is demonstrated that the jet flow with Grashof numbers Gr = (0.4–2.0) · 10⁶ is self-similar. Acoustic oscillations directed perpendicular to the axis of symmetry transform the profiles of the gas-flow parameters; two temperature maximums located outside the axis can appear. The results obtained indicate that flow instability is generated in high-gradient regions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 27–33, September–October, 2006.     

Hypersonic viscous shock layer in a swirling gas flow on a permeable surface

A hypersonic swirling flow of viscous compressible gas past rotating axisymmetric blunt bodies is considered, its velocity vector being parallel to the axis of rotation of the body. The body surface is assumed permeable, while, in the general case, the gas is not injected (drawn off) along the normal to the body surface. An analytic solution of the problem, valid at small Reynolds numbers, is found in the first approximation of the integral method of successive approximations. On the basis of the results of the numerical solution, obtained in a wide range of variation of the determining parameters of the problem, we investigate the influence of the swirling of the free-stream flow, the angular velocity of rotation of the body, the Reynolds number and the injection (suction) parameter on the structure of the compressed layer, and the coefficients of friction and heat transfer on the body surface. The influence of the swirling of the flow on the nature of the asymptotic behavior of the viscous shock layer equations at large Reynolds numbers is studied. It is shown that the presence of a nonzero peripheral component for the velocity vector of the gas in the shock layer can lead to a qualitative change in the nature of the flow. Deceased Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 27–37, November–December, 1986. The authors thank G. G. Chernyi for his useful discussion of the results of the work.     

Asymptotic behavior of the far region of turbulent wake vortices

An approximate mathematical model, formulation of the problem, and its approximate solution are proposed for the far region of a turbulent vortex wake past a moving body, where the departure of the horizontal velocity component from the uniform flow is slight. It is assumed that the single important parameter that defines the main flow characteristics in this region is the vortex momentum per unit length produced in the fluid by the lift equal to the weigth of the moving body uncompensated by the buoyancy force. Thus, the flow is self-similar, and the self-similarity law determines the intensity, shape, and location of vortex lines as functions of the downstream distance with accuracy up to a constant factor, which cannot be determined theoretically and should be obtained by comparison of theory with experiment. A boundary-value problem is formulated to determine the flow structure of vortex lines (vorticity distribution). A solution of the problem is obtained numerically in the limit of “vanishing turbulent viscosity.” The variation in the maximum velocity of a vortex line with distance, determined by self-similarity, is in agreement with available experimental data. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 13–23, March–April, 1999.     

Subsonic compressible cavitation flow past a plate at small cavitation numbers

A solution of the problem of symmetric inviscid compressible cavitation flow past a plate at small cavitation numbers is presented. Kazan’. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–104, July–August, 1998. This research was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-00111).     

On Singularity Formation of a Nonlinear Nonlocal System

We investigate the singularity formation of a nonlinear nonlocal system. This nonlocal system is a simplified one-dimensional system of the 3D model that was recently proposed by Hou and Lei (Comm Pure Appl Math 62(4):501–564, 2009) for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. In the nonlocal system we consider in this paper, we replace the Riesz operator in the 3D model by the Hilbert transform. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the nonlocal system for a large class of smooth initial data with finite energy. We also prove global regularity for a class of smooth initial data. Numerical results will be presented to demonstrate the asymptotically self-similar blow-up of the solution. The blowup rate of the self-similar singularity of the nonlocal system is similar to that of the 3D model.     

Dynamic contact angle for etting of a surface by a van der waals fluid

We consider the creeping motion of a thin layer of a nonvolatile viscous fluid spreading due to capillary forces over a rigid surface covered by a thin homogeneous film (microfilm). The influence of van der Waals forces on the asymptotic slope of the free boundary of the layer is studied in the region of large thickness, where capillary forces dominate. A solution of the problem of the slope angle is obtained for the entire possible range of the microfilm thickness. In the limit of small thickness of the microfilm, this solution is in agreement with the well-known solution of the problem of the dynamics of wetting of a dry surface in the presence of a precursory film and van der Waals forces. The role of the condition at the end of the precursory film is studied. Institute of Mechanics of Multiphase Systems, Siberian Division, Russian Academy of Sciences. Tyumen' 625000. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika. Vol. 41, No. 4, pp. 101–105, July–August. 2000.     

Turbulent time and length scale measurements in high-velocity open channel flows

In high-velocity open channel flows, the measurements of air–water flow properties are complicated by the strong interactions between the flow turbulence and the entrained air. In the present study, an advanced signal processing of traditional single- and dual-tip conductivity probe signals is developed to provide further details on the air–water turbulent level, time and length scales. The technique is applied to turbulent open channel flows on a stepped chute conducted in a large-size facility with flow Reynolds numbers ranging from 3.8E+5 to 7.1E+5. The air water flow properties presented some basic characteristics that were qualitatively and quantitatively similar to previous skimming flow studies. Some self-similar relationships were observed systematically at both macroscopic and microscopic levels. These included the distributions of void fraction, bubble count rate, interfacial velocity and turbulence level at a macroscopic scale, and the auto- and cross-correlation functions at the microscopic level. New correlation analyses yielded a characterisation of the large eddies advecting the bubbles. Basic results included the integral turbulent length and time scales. The turbulent length scales characterised some measure of the size of large vortical structures advecting air bubbles in the skimming flows, and the data were closely related to the characteristic air–water depth Y ₉₀. In the spray region, present results highlighted the existence of an upper spray region for C > 0.95–0.97 in which the distributions of droplet chord sizes and integral advection scales presented some marked differences with the rest of the flow.     

An Approximate Lie Group Investigation into the Spreading of a Liquid Drop on a Slowly Dropping Flat Plane II: Small Surface Tension Effects

The spreading of a thin liquid drop under gravity and small surfacetension on a slowly dropping flat plane is investigated. The initialslope of the flat plane is assumed to be small. By considering astraightforward forward perturbation, the fourth-order nonlinear partialdifferential equation modelling the spreading of the liquid drop reducesto a second-order nonlinear partial differential equation. Thisresulting equation is solved using the classical Lie group method. Thegroup invariant solution is found to model the long time behaviour ofthe liquid drop.     

Stability of Poiseuille flow in the presence of a longitudinal magnetic field

The stability of the plane flow of an electrically conducting fluid with respect to small perturbations was studied at large Reynolds numbers in the presence of a longitudinal magnetic field. The dependence of the critical Reynolds number on the electrical conductivity is investigated. At large Reynolds numbers, a new branch of instability and a sudden change in the critical Reynolds numbers is found. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 45–53, May–June, 2008.