Relaxation of a Low‐Intensity Atomic Beam in a Quiescent Gas

Relaxation of a lowintensity atomic beam in a gas at rest is examined by means of numerical modeling with the method of test particles. Temperaturefield features in the mixing region are considered. A relation between the relaxation length and the initial velocity and mass of injected particles is obtained. Conditions are found under which the relaxation length is minimal.     

The Motion of Viscous Liquid Column with Finite Length in a Vertical Straight Capillary Tube

This paper deals with the motion of viscous liquid column with finitelength and two free surfaces in a vertical straight capillary tube.It isassumed that fluid is Newtonian.Linearizing the boundary conditions,analytic expressions in the form of infinite series have been obtainedfor velocity,plessure and free surface at low Reynolds number.Thenumerical calculation is carried out for a set of cylinder’s length ofwater and blood.It has been revealed that there are considerable cir-culating currents at the upper and lower meniscuses.Its maximum ve-locity is about57%of the average velocity of the mainstream.Iner-tial effect is also studied in this paper.Using the time-dependent methodin finite difference techniques,numerical solution of the correspondiugnonlinear equation at Re≤24.5 is computed.Comparing it with analyticexact solution at low Reynolds number shows that inertial effect isnegligible provided Re≤24.5.     

Effect of Transitional Nonequilibrium on the Molecular–Dissociation Rate in a Hypersonic Shock Wave

The problem of the influence of a nonequilibrium (non–Maxwellian( distribution of translational energy over the degrees of freedom of molecules on the rate of their dissociation in a hypersonic shock wave is considered. An approximate beam—continuous medium model, which was previously applied to describe a hypersonic flow of a perfect gas, was used to study translational nonequilibrium. The degree of dissociation of diatomic molecules inside the shock–wave front, which is caused by the nonequilibrium distribution over the translational degrees of freedom, is evaluated. It is shown that the efficiency of the first inelastic collisions is determined by the dissociation rate exponentially depending on the difference in the kinetic energy of beam molecules and dissociation barrier.     

Reflection of a Dam–Break Wave at a Vertical Wall. Numerical Modeling and Experiment

Results of the experimental study and numerical modeling of the reflection of a dam–break wave at the vertical end wall of a channel are given. A wave forms with distance from a partition creating the initial level difference of the liquid. It is shown that a numerical calculation based on the Zheleznyak—Pelinovskii nonlinear dispersion model satisfactorily describes the height of the splash–up, the amplitude of reflected waves, and the wave velocity in front of the wall for smooth and dam–break waves. It is also shown that, for smooth and weakly breaking (without significant entrainment of air) incoming waves, the experimental values of the height of the splash–up at the wall agree well with relevant experimental and calculated data for solitary waves.     

Calculation of Linear Stability of a Stratified Gas–Liquid Flow in an Inclined Plane Channel

Linear stability of liquid and gas counterflows in an inclined channel is considered. The full Navier–Stokes equations for both phases are linearized, and the dynamics of periodic disturbances is determined by means of solving a spectral problem in wide ranges of Reynolds numbers for the liquid and vapor velocity. Two unstable modes are found in the examined ranges: surface mode (corresponding to the Kapitsa waves at small velocities of the gas) and shear mode in the gas phase. The wave length and the phase velocity of neutral disturbances of both modes are calculated as functions of the Reynolds number for the liquid. It is shown that these dependences for the surface mode are significantly affected by the gas velocity.     

Wave–wave interactions in a periodic chain with quadratic nonlinearity

Two waves are studied using perturbation analysis for their interactions in an one-dimensional periodic structure with quadratic nonlinearity. A first-order multiple-scales analysis along with numerical simulations on the full chain are used to understand the interaction of two waves when one is the sub- or super-harmonic of the other. The strength of quadratic nonlinearity affects the rate at which the energy is exchanged between the two waves. Depending on parameters and energy states, the interactions between the waves are periodic or whirling and result in quasi-periodic combined propagating waves with either phase drifts or weakly phase-locking properties. The analysis suggests the possibility of the existence of emergent wave harmonics. Due to quadratic nonlinearity, a very small amplitude subharmonic or superharmonic wave mode can drift in its phase, and then burst out with a larger amplitude as it circumnavigates a separatrix. Depending on the parameters and wave numbers, the amplitude of this emergent wave burst can have varying significance.     

Deformation of a Droplet of an Organic Water—Coal Fuel in a Gas Flow

This paper describes an experimental study of deformation of a droplet of an organic water—coal fuel in a gas flow. The experimental data obtained are used to develop a technique that serves as a basis for a numerical study of integral characteristics of deformation of the droplets of organic water—coal fuel 4.5 mm in diameter with a gas flow velocity of 2 m/s and a temperature of 298 K. The characteristic changes of the area of longitudinal and transverse sections of the droplets of the organic water—coal fuel during their gravitational deposition in time are revealed. It is shown that the theoretical and experimental data satisfactorily agree.     

Shock–vortex interactions in a soap film

This work experimentally visualizes the interaction of a quasi-one-dimensional moving shock wave with a two-dimensional vortex in a soap film for the first time. A vertical soap film shock tube was used to generate a quasi-one-dimensional moving shock wave and a NACA-0012 airfoil intruded into the soap film was towed to shed the starting vortex. The interesting interaction phenomena were then visualized using a traditional high-speed flash photography. The concentration of sodium dodecyl sulfate (SDS) used was 0.5 CMC (critical micelle concentration) to keep the surfactant molecules behave as two-dimensional gases. A sequence of pictures shows that the shock is distorted non-symmetrically as it passes through the spiral vortex flow field and the vortex structure is compressed in the direction normal to the shock. These flow features observed in soap films are qualitatively similar to their counterparts in gases. In addition, the visualization of the interactions of a quasi-one-dimensional moving shock wave with a Kármán vortex street are presented.      

One-Dimensional Interaction of a Cylindrical Unloading Wave with a Moving Elastic–Plastic Boundary

The one-dimensional dynamic problem of the theory of large elastic–plastic deformations is considered for the interaction of an unloading wave with an elastic–plastic boundary. It is shown that before the occurrence of the unloading wave, the increasing pressure gradient leads to quasistatic deformation of the elasti©viscoplastic material filling the round tube, which is retained in the tube due to friction on its wall, resulting in the formation of near-wall viscoplastic flow and an elastic core. The unloading wave is initiated at the moment of the onset of slippage of the material along the inner wall of the tube. Calculations were conducted using the ray method of constructing approximate solutions behind strong discontinuity surfaces, and ray expansions of the solutions behind the cylindrical surfaces of discontinuities were obtained.     

Finite displacement static analysis of thin plate with an opening––a variational approach

A generalized work–energy method for the linear and geometrically nonlinear static analysis of thin isotropic plate with a cutout is presented. The plate geometry is divided into few quadrilateral segments. Each segment is defined by four curved edges and the natural coordinates in conjunction with the Cartesian coordinates are used in formulating the stiffness matrix and the load vector. Two different sets of interpolating functions are used for the geometric and displacement representations respectively. The matrix equation of equilibrium is derived from the variational principle. By exploiting the geometric symmetry, numerical results are obtained for the following examples: (a) square plate with circular opening at the centre and (b) circular plate with circular or square inner boundary. The plates are subjected to uniformly distributed load and both the pinned and fixed outside boundary conditions are considered. Very good comparison is observed between the present results and those published in the literature for the fixed square plate without an opening. Effects of the opening size on the displacement are examined in detail.     

Finite differences study of ignition in a methane–air mixture flow

A detailed simulation of the ignition process and premixed flames propagation, taking into account molecular transport, chemical reaction, thermodynamics and convection, is built by making use of the implicit finite difference scheme with the help of the Tridiagonal Matrix Algorithm. The velocity of chemical reaction is expressed by means of Arrhenius form of first order in both fuel and oxygen. The main objective of this work is to define numerically in two cases, u=0.1 m/s and u=0.4 m/s, the ignition temperature of the methane–air mixture with different air excess coefficients in the mixture. In addition, the effect of the thickness of the region ignition and of ignition location on the transient behavior of the flame was studied.     

Gas–Dynamic Structures in Supersonic Combustion of Hydrogen in a System of Plane Jets in a Supersonic Flow

Results of numerical and theoretical studies of supersonic diffusive combustion of a system of plane hydrogen jets in a supersonic air flow are described. It is shown that large–scale vortex structures appear in the mixing zone of the system of hydrogen jets and the cocurrent flow. These vortex structures affect the mechanism of turbulent exchange between the fuel and the oxidizer.     

Asymptotic Modeling of Nonlinear Wave Processes in Shock‐Loaded Elastoplastic Materials

Nonlinear wave processes in shockloaded elastoplastic materials are modeled. A comparison of the results obtained with experimental data and numerical solutions of exact systems of dynamic equations shows that the model equations proposed qualitatively describe the stressdistribution evolution in both the elasticflow and plasticflow regions and can be used to solve one and twodimensional problems of pulsed deformation and fracture of elastoplastic media.     

Rayleigh–Lamb Waves in a Compound Wave Guide. Reflection from and Transmission Through the Vertical Interface between Media with Different Impedances

The method of superposition is used to study the first normal wave reflecting from and transmitting through the interface in a compound waveguide consisting of two rigidly joined elastic half-strips with equal width and different mechanical properties. We study how the impedances of the contacting media influence the transformation of the energy of the incident wave to those of the reflected and transmitted waves. Two cases are considered — propagating waves of higher orders appear in the reflected wavefield earlier than in the transmitted wavefield and propagating waves of higher orders appear in the transmitted wavefield earlier than in the reflected wave field. For both cases, the impedances vary so that the incident wave can propagate in both more rigid and softer media. It is shown that by increasing the impedances of the contacting media, the interface can be made more transparent     

Shock Compression of a Plate on a Wedged–Shaped Target

Problems of compression of a plate on a wedge–shaped target by a strong shock wave and plate acceleration are studied using the equations of dissipationless hydrodynamics of compressible media. The state of an aluminum plate accelerated or compressed by an aluminum impactor with a velocity of 5—15 km/sec is studied numerically. For a compression regime in which a shaped–charge jet forms, critical values of the wedge angle are obtained beginning with which the shaped–charge jet is in the liquid or solid state and does not contain the boiling liquid. For the jetless regime of shock–wave compression, an approximate solution with an attached shock wave is constructed that takes into account the phase composition of the plate material in the rarefaction wave. The constructed solution is compared with the solution of the original problem. The temperature behind the front of the attached shock wave was found to be considerably (severalfold) higher than the temperature behind the front of the compression wave. The fundamental possibility of initiating a thermonuclear reaction is shown for jetless compression of a plate of deuterium ice by a strong shock wave.     

Reflection and Transmission of Plane Waves at a Water–Porous Sediment Interface with a Double-Porosity Substrate

This paper investigates the wave propagation at the interface between the ocean and the ocean floor. The ocean floor is assumed to be composed of covered porous sediment with an underlying double-porosity substrate. For this purpose, plane wave reflection and transmission in the coupled water–porous sediment–double-porosity substrate system are analytically solved in terms of displacement potentials. Using numerical examples, the effects of the material properties of the underlying double-porosity substrate on the reflection coefficients are discussed in detail. Variations in pore and fracture fluid, fracture volume fraction, and permeability coefficients are considered. In addition, two cases of boundary conditions at the porous sediment–double-porosity substrate interface, i.e., sealed-pore boundary and open-pore boundary, are compared in the numerical calculations. Results show that material property variations in the double-porosity substrate may significantly affect the reflected wave in the overlying water if the sandwiched sediment depth is less than the critical value.     

Soliton-Stripe Patterns of a Functional with an Attractive–Repulsive–Attractive Interaction

We study critical points of a Ginzburg–Landau type functional with an attractive–repulsive–attractive nonlocal interaction. Using an appropriate scaling and -convergence method we reduce the problem to a finite dimensional one. In contrast to a similar problem with just an attractive–repulsive interaction, we obtain a richer set of solutions. The soliton-stripe patterns appear as skewed local minimizers of the free energy, and disappear or become symmetric as the number of interfaces reaches a certain threshold. We also show how other critical points can be constructed using results of the diblock copolymer problem.     

Numerical Investigation of Weak Planar Shock—Elliptical Light Gas Bubble Interaction in Shock and Reshock Accelerated Flow

Fluid Dynamics - The computational results of the weak planar shock–elliptical light gas bubble interaction are presented. The influence of different light gases (helium and neon) on the...     

Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork–Hopf bifurcation

Nonlinear Dynamics - The traditional geometric singular perturbation theory and the slow–fast analysis method cannot be directly used to explore the mechanism of the relaxation oscillations...     

Vanishing Viscosity with Short Wave–Long Wave Interactions for Systems of Conservation Laws

Motivated by Benney’s general theory, we propose new models for short wave–long wave interactions when the long waves are described by nonlinear systems of conservation laws. We prove the strong convergence of the solutions of the vanishing viscosity and short wave–long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems. We analyze several of the representative examples, such as scalar conservation laws, general symmetric systems, nonlinear elasticity and nonlinear electromagnetism.