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.     

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.     

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.     

Behaviour of the Fokker–Planck–Boltzmann equation near a Maxwellian

We consider the initial value problem for the Fokker–Planck–Boltzmann equation namely, viewed as the Boltzmann equation with an additional diffusion term in velocity space to describe, for instance, the transport in thermal baths of binary elastic collisional particles. The strong solution for initial data near an absolute Maxwellian is proved to exist globally in time and tends asymptotically in the -norm to another time dependent self-similar Maxwellian in large time. The effect of the diffusion in phase space is investigated. It produces a diffusion process in velocity space and results in a heating process on the macroscopic fluid-dynamic observable, accelerating the convergence of solutions to the equilibrium of a self-similar Maxwellian at a faster time-decay rate than the Boltzmann equation. This phenomena is also observed for homogeneous Fokker–Planck–Boltzmann equations, where the time-decay rate in the -norm to the self-similar Maxwellian is proved to be faster than exponential. Moreover, the Fokker–Planck–Boltzmann equation is shown to converge (under an appropriate scaling) strongly to the Boltzmann equation in the process of the zero diffusion limit.     

Determining the limiting solutions of nonstationary axisymmetric Hele-Shaw problems

Numerical solution of the Hele-Shaw problem reduces to solution of three boundary-value problems of determining analytic functions of a complex variable in each time step: conformal mapping of the range of the parametric variable to the physical plane, the Dirichlet problems for determining the electric-field strength, and the Riemann-Hilbert problem for calculating partial time derivatives of the coordinates of points of the interelectrode space (the images of the points on the boundary of the parametric plane are fixed). Unlike in the two-dimensional problem, the electric-field strength is determined using integral transformations of an analytic function. Approximation by spline function is performed, and more accurate and steady (than the well-known ones) general solution algorithms for the nonstationary axisymmetric problems are described. Results of a numerical study of the formation of stationary and self-similar configurations are presented. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 87–99, July–August, 2009.     

Charged jet of an incompressible liquid in an electric field

The problem of the jet flow of an incompressible liquid with free boundaries in an electric field is solved in the approximation of a laminar boundary layer. An exact solution for a round jet is found in the class of self-similar solutions. In the case of a flat slit jet, a solution is constructed in the form of a series in powers of the coordinate transverse to the plane of symmetry. The dependence of the radius (half-width) on the longitudinal coordinate is given. Branch of the Karpov Physicochemistry Institute, State Science Center, Obninsk 249020. Karpov Physicochemistry Institute, State Science Center, Moscow 115523. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 12–16, July–August, 1998.     

Construction of piecewise-analytical gas flows joined through shock waves near the axis or center of symmetry

Analytical solutions of a quasilinear system of equations with partial derivatives are constructed in the case where the initial data for different functions are specified on different surfaces and the resultant problem has singularities of the form u/x and w/x. Conditions for existence and uniqueness of a solution in the form of formal power series for the problem posed and sufficient conditions for convergence of the series are indicated. A generalization of the problem considered is given. Results of the study are used to solve the problem of the focussing of a compression wave generated by a piston moving smoothly in a quiescent gas: a solution for t=0, including determination of the piston trajectory, and a solution for t<0, including unequivocal construction of the front of a reflected shock wave, are uniquely constructed from the distribution of gas-dynamic quantities for t>0. The solution of this problem is a generalization to the case of two independent variable self-similar Sedov's solutions. Ural State Academy of Communications, Ekaterinburg 620034. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 25–38, September–October, 1998.     

Self-similar solutions in a plasma with axial magnetic field (<Emphasis Type="Italic">θ</Emphasis>-pinch)

In this paper, Lie group of transformation method is used to investigate the self-similar solutions for the system of partial differential equations describing a plasma with axial magnetic field (θ–pinch). The arbitrary constants occurring in the expressions for the infinitesimals of the local Lie group of transformations give rise to two different cases of possible solutions i.e. with a power law and exponential shock paths. A particular solution to the problem in one case has been found out.     

Instability of self-similar ideal gas flows with shock waves

The results of the numerical simulation of three problems of ideal gas flow with shock waves, which admit self-similar solutions, are presented. These problems are the double Mach-type reflection of a shock from a wedge, the breakdown of a combined discontinuity on a 90° sharp corner, and the outflow of a supersonic jet from an expanding slot. It is shown that for certain input data the self-similar solution may become unstable and is replaced by a fluctuating flow. The reasons for the generation of these fluctuations and their mechanism are discussed. Volgograd. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 166–175, July–August, 1998.     

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.     

Method of effective length for the boundary layer on a thin filament drawn out of a half-space

Heat transfer from a thin filament pulled from a half-space is considered. On the basis of the self-similar solution, obtained in the study, and using the method of effective length, the friction and heat transfer coefficients on the surface of a thin filament of varying radius and with distributed surface temperature and velocity are determined. The results obtained with this approximate method agree well with the results of a numerical solution of the boundary layer equations. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 23–28, January–February, 1994.     

Thermocapillary motion of a liquid with a free surface with nonlinear dependence of the surface tension on the temperature

The present note considers two problems concerning the thermocapillary motion, due to the existence of a temperature gradient, of a weightless liquid with a parabolic dependence of the surface tension on the temperature. These problems admit self-similar solutions (in the generalized sense) within the framework of the Navier-Stokes equations. It is noted that the solution may not be unique. Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 132–137, September–October, 1988.     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.     

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.     

Self-similar solutions of the problem of displacement of one gas by another in an axisymmetric case with a quadratic drag law

A problem of piston-induced displacement of one gas by another in cracks (porous media) in an axisymmetric case with a quadratic drag law is studied. Self-similar solutions for determining the dynamic characteristics (velocity and pressure) of the displacing and displaced gases are constructed in quadratures. The velocity and pressure are studied as functions of a self-similar variable for several initial conditions and parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 87–92, September–October, 2008.     

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.     

Oil Plume Distribution in an Anisotropic Porous Medium

The saturation distribution—within an anisotropic aquifer—of a pollutant discharging from an underground source is modeled by a two-dimensional, nonlinear diffusion–convection equation. A closed form self-similar solution is obtained for the steady saturation distribution in the immiscible zone. The results may be used to rationalize field data collected for predicting locations of underground leakage sources in aquifers and to understand the influence of the anisotropic permeability’s parameters on the oil distribution in the porous medium. An erratum to this article can be found at     

Mixing dynamics of a vapor-condensate cloud with the ambient air

The mixing of a wet vapor with a gas is studied using analytical and numerical models. The one-dimensional problem of diffusion mixing accompanied by phase transitions is solved in a self-similar formulation. The versions of mixing of the vapor with a cold and warm gas and with a superheated vapor are analyzed. The atmospheric diffusion of immediate emissions containing water vapor and condensate is modeled numerically in a three-dimensional formulation. A study is made of the evolution of hydrodynamic, concentration, and temperature fields as a function of the initial emission parameters (temperature and humidity) and ambient air parameters. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 114–127, May–June, 2008.     

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.     

Stability of a cantilevered skew inhomogeneous plate in supersonic gas flow

This paper considers the vibrations of a skew inhomogeneous plate in gas flow. The plate is clamped in a certain section of one of its sides. Interaction of the flow with the plate is described using piston theory. The problem solution is based on the Hamilton’s variational principle and finite element method. The calculation results are compared with known data of theoretical studies and experiments. For the inhomogeneous plate, similarity parameters were established for the problem, which, in practically important cases, appears to be self-similar for one of the similarity parameters. This allows one to reduce the solution of this problem to the solution of an algebraic eigenvalue problem.