Exact Self-similar Solutions Research Articles

Comparison of difference scheme with customizable dissipative properties and WENO scheme in the case of one-dimensional gas and gas-particle dynamics problems

Subject of Research.The paper presents results of comparative testing of highly stable difference scheme with customizable dissipative properties and WENO5 scheme in the case of one-dimensional gas dynamics and mechanics of gas-suspensions problems. Method. The second order difference scheme is constructed by physical processes splitting into two phases. The first one uses the central differences with adaptive reconstruction of artificial viscosity and semi implicit approximation of the sources. TVD type reconstructions are used in the second phase. Main Results. For the representative collection of test problems, the proposed scheme has confirmed the efficiency and good quality numerical solutions at the level of the WENO5 scheme with substantially lower machine time cost. The wave structure in the decay of initial discontinuity in the non-equilibrium gas suspension and the convergence to exact self-similar solution are considered. Practical Relevance. A detailed resolution of structural peculiarities of gas flows and gas-particle mixtures allows reducing the amount of experimental development of new technologies and technical devices of coating, leakage control when using a trial particulate matter with particles of micro - and nano-sizes and other areas.

Расчетная модель упорного подшипника с пористым покрытием на поверхности направляющей

Introduction. The development of the design model of a thrust sliding bearing with a two-layer porous coating on the way surface running on an electrically conductive lubricant is described. The work objectives are the development and computational justification of the possible increase in oil consumption, growth of the bearing capacity, and reduction of the frictional force, due to their design models refinement. This is based on the formation of the computed hydrodynamic models of thrust bearings taking into account the dependence of the porous layers permeability on the way surface under the steady flow of the electroconductive liquid lubricant. Materials and Methods. New mathematical models that describe a steady flow of the electrically conductive lubricant in the working gap between an inclined slider and a guide with a bilayered porous coating are proposed. The given numerical analysis of the essential performance features has shown that bearings with a two-layer porous coating on the way surface significantly increase the damping characteristics of the supports and bearing capacity of the bearing, and reduce the frictional force. Research Results. A design model of the thrust sliding bearing is formed taking into account the porous coating permeability on the way surface. That is based on the equations of the steady flow of the electrically conductive incompressible liquid lubricant for a “thin layer” in the working gap, continuity and Darcy in the presence of an electromagnetic field. The authors have found the exact self-similar solution to a thrust bearing with a two-layer porous coating on the way surface for the field of velocities and pressures in the lubricating layer and porous coating. Besides, multiparameter expressions for the essential bearing performance with the account for the availability of electromagnetic fields, the permeability of porous layers, and the ratio of the thicknesses of porous layers, are specified. Discussion and Conclusions. The received updated calculation models have made it possible to determine the effect of a number of additional factors, as well as to perform a comparative analysis of the newly obtained results and those already available which confirms the close approximation of the new model to the actual practice. The theoretical models provide the necessary engineering design calculations in a sufficiently wide range of speeds and loads for the application in engineering, aircraft building, instrument making, etc.

Magnetic reconnection: from MHD to QED

The paper examines the prospects of using laser plasmas for studying novel regimes of the magnetic field line reconnection and charged particle acceleration. Basic features of plasma dynamics in the three-dimensional configurations relevant to the formation of current sheets in a plasma are addressed by analyzing exact self-similar solutions of the magneto-hydrodynamics and electron magneto-hydrodynamics equations. Then the magnetic field annihilation in the ultrarelativistic limit is considered, when the opposite polarity magnetic field is generated in collisionless plasma by multiple laser pulses, in the regime with a dominant contribution of the displacement current exciting a strong large-scale electric field. This field leads to the conversion of the magnetic energy into the kinetic energy of accelerated particles inside a thin current sheet. Charged particle acceleration during magnetic field reconnection is discussed when radiation friction and quantum electrodynamics effects become dominant.

Spatiotemporal soliton clusters in strongly nonlocal media with variable potential coefficients

We study analytically and numerically spatiotemporal solitons in three-dimensional strongly nonlocal nonlinear media. A broad class of exact self-similar solutions to the strongly nonlocal Schrodinger equation with variable potential coefficients has been obtained. We find robust soliton cluster solutions of the accessible type, constructed with the help of Kummer and Hermite functions. They are characterized by the set of three quantum numbers. Dynamical features of these spatiotemporal accessible solitons are discussed. The validity of the analytical solutions and their stability is verified by means of direct numerical simulations.

Lagrangian approach to modeling unsteady gas-liquid flow in a well

The purpose of this paper is to develop a numerical method of solving the problem of evolution of the finite gas volume that entered in a liquid flow at a set flow rate. The drift- flux model is used as gas-liquid mixture equations. The velocities of both phases, mixture and gas, are related by the Zuber-Findlay equation which coefficients depend on flow regime and gas void fraction. Lagrangian coordinates are used to simplify the initial equations. The numerical solution scheme is proposed. The numerical solution of the Riemann problem is verified by comparison with the exact self-similar solution. The model and numerical method efficiency is illustrated by examples of gas kick calculations in a vertical well.

Similarity solution of a Stefan drug-release subdiffusion problem

Propagation of a gel/glass transition boundary in a polymer is considered in the context of drug release. Drug molecules are assumed to undergo subdiffusive motion in the gel and be quiescent in the glass region. Exact self-similar solutions for the drug concentration are constructed, and the amount of released drug is determined as a function of time.

Kadomtsev−Petviashvili equation for a flow of highly nonisothermal collisionless plasma

It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev−Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

Nonautonomous vector matter waves in two-component Bose-Einstein condensates with combined time-dependent harmonic-lattice potential

We construct exact self-similar soliton solutions of three-dimensional coupled Gross–Pitaevskii equations for two-species Bose–Einstein condensates (BECs) in a combined time-dependent harmonic-lattice potential. Based on these solutions, we investigate the control and manipulation of solitary waves for three kinds of BECs with changing diffraction and nonlinearity coefficients; the solutions include Ma breathers and Peregrine and Akhmediev soliton solutions. Our results indicate that matter waves readily propagate in this system. It is shown that diffraction and lattice potential factors play important roles in the beam evolution characteristics, such as the peak, the phase offset, the linear phase, and the chirp.

The Riemann problem in the quasi-one-dimensional approximation

The classical one-dimensional Riemann problem is generalized to the quasi-one-dimensional case. A plane slotted channel with a discontinuous cross section is considered. The resulting exact self-similar solution is compared with numerical results obtained for a system of quasi-onedimensional and two-dimensional equations. It is shown that they are in good qualitative agreement and, for certain parameters, also agree well quantitatively.

A THEORETICAL MODEL OF A THINNING CURRENT SHEET IN THE LOW-βPLASMAS

Magnetic reconnection is an important physical process in various explosive phenomena in the universe. In the previous studies, it was found that fast re- connection takes place when the thickness of a current sheet becomes on the order of a microscopic length such as the ion larmor radius or the ion inertial length. In this study, we investigated the pinching process of a current sheet by the Lorentz force in a low-{\beta} plasma using one-dimensional magnetohydrodynam- ics (MHD) simulations. It is known that there is an exact self-similar solution for this problem that neglects gas pressure. We compared the non-linear MHD dynamics with the analytic self-similar solution. From the MHD simulations, we found that with the gas pressure included the implosion process deviates from the analytic self-similar solution as t {\rightarrow} t 0, where t 0 is the explosion time when the thickness of a current sheet of the analytic solution becomes 0. We also found a pair of MHD fast-mode shocks are generated and propagate after the formation of the pinched current sheet as t {\rightarrow} t 0 . On the basis of the Rankine-Hugoniot relations, we derived the scaling law of the physical quantities with respect to the initial plasma beta in the pinched current sheet. Our study could help us to estimate the physical quantities in the pinched current sheet formed in a low-\b{eta} plasma.

Self-similar soliton-like solution for coupled higher-order nonlinear Schrödinger equation with variable coefficients

Self-similar soliton-like solution of a kind of coupled higher-order nonlinear Schrödinger equation with variable coefficients describing the optical soliton pulses propagating in inhomogeneous optical fiber media is investigated in this paper. The balance principle and the F-expansion technique are used to obtain the exact bright and dark self-similar soliton-like solutions analytically. As an example, we present a periodic distributed amplification system to investigate the pulse evolution characteristics by using the split-step Fourier method and the Runge–Kutta method. It is found that the self-similar soliton-like optical pulses can propagate stably in the given coupled system.

Hall MHD and Electron Inertia Effects in Current Sheet Formation at a Magnetic Neutral Line

AbstractAn exact self-similar solution is used to investigate current sheet formation at a magnetic neutral line in incompressible Hall magnetohydrodynamics. The collapse to a current sheet is modelled as a finite-time singularity in the solution for electric current density at the neutral line. We establish that a finite-time collapse to the current sheet can occur in Hall magnetohydrodynamics, and we find a criterion for the finite-time singularity in terms of the initial conditions. We derive an asymptotic solution for the singularity formation and a formula for the singularity formation time. The analytical results are illustrated by numerical solutions, and we also investigate an alternative similarity reduction. Finally, we generalise our solution to incorporate resistive, viscous and electron inertia terms.

Self-similar impulsive capillary waves on a ligament

We study the short-time dynamics of a liquid ligament, held between two solid cylinders, when one is impulsively accelerated along its axis. A set of one-dimensional equations in the slender-slope approximation is used to describe the dynamics, including surface tension and viscous effects. An exact self-similar solution to the linearized equations is successfully compared to experiments made with millimetric ligaments. Another non-linear self-similar solution of the full set of equations is found numerically. Both the linear and non-linear solutions show that the axial depth at which the liquid is affected by the motion of the cylinder scales like t, a consequence of the imposed radial uniformity of the axial velocity at the cylinder surface, and differs from t2/3 known to prevail in surface-tension-driven flows. The non-linear solution presents the peculiar feature that there exists a maximum driving velocity U⋆ above which the solution disappears, a phenomenon probably related to the de-pinning of the contact line observed in experiments for large pulling velocities.

Magnetic flux and heat losses by diffusive, advective, and Nernst effects in magnetized liner inertial fusion-like plasma

The magnetized liner inertial fusion (MagLIF) approach to inertial confinement fusion [Slutz et al., Phys. Plasmas 17, 056303 (2010); Cuneo et al., IEEE Trans. Plasma Sci. 40, 3222 (2012)] involves subsonic/isobaric compression and heating of a deuterium-tritium plasma with frozen-in magnetic flux by a heavy cylindrical liner. The losses of heat and magnetic flux from the plasma to the liner are thereby determined by plasma advection and gradient-driven transport processes, such as thermal conductivity, magnetic field diffusion, and thermomagnetic effects. Theoretical analysis based on obtaining exact self-similar solutions of the classical collisional Braginskii's plasma transport equations in one dimension demonstrates that the heat loss from the hot compressed magnetized plasma to the cold liner is dominated by transverse heat conduction and advection, and the corresponding loss of magnetic flux is dominated by advection and the Nernst effect. For a large electron Hall parameter (ωeτe≫1), the effective diffusion coefficients determining the losses of heat and magnetic flux to the liner wall are both shown to decrease with ωeτe as does the Bohm diffusion coefficient cT/(16eB), which is commonly associated with low collisionality and two-dimensional transport. We demonstrate how this family of exact solutions can be used for verification of codes that model the MagLIF plasma dynamics.

Stratification of lubricating material in radial bearings

The technique of calculating the radial sliding bearings of infinite and finite length running on a double-layer stratified lubricant is presented. The technique is based on the use of the self-simulated variable that allows obtaining the exact self-similar problem solution both in polar and in cylindrical coordinates, as well as the parameter value characterizing the interface of the stratified layers. These tasks are complicated by the simultaneous consideration of the lubricant viscosity-pressure relation, the presence of the friction-adapted supporting profile of the bearing bush, and the effect of the axial lubricant feed in the finite-length bearing. As a result, computational models are obtained for two-layer stratified lubricants the numerical analysis of which allows establishing the impact of variables on the basic bearings performance - components of the bearing capacity vector, friction forces, as well as optimal values of the reference profile parameter, the lubricant supply, and its viscous ratio in the stratified layers.

Hydraulic fracture numerical model free of explicit tip tracking

The paper addresses the problem of modeling a fracture propagation in linear elastic porous media driven by injection of non-Newtonian power-law fluid. The model involves the lubrication theory equation expressing the mass conservation of fluid and a nonlocal singular integral relation for the fracture aperture as a functional of the fluid pressure. We assume only the viscosity-dominated case in which the fracture toughness is neglected. This allows considering the fracture as an opened part of a preexisting closed fracture of larger length. The numerical method consists in solving equations of the model over the whole length of the preexisting fracture without distinguishing the tip region of the opened part. The problem is solved via the finite element method. The weak formulation of the solution allows pressure singularity with the required asymptotics near the fracture tip. Comparison of the numerical and the exact self-similar solutions in the case of a constant flow rate, zero fluid leakoff, and zero fracture toughness reveals the accuracy of the approximate solution to at least O(h1/2), where h is the maximal linear size of the grid cells. As an illustration, we also demonstrate the numerical experiments with periodic fluid injection and variable fluid efficiency.

Self-similar and diffusive expansion of nonextensive plasmas

Exact analytical self-similar solution is presented for free collisionless expansion of a two-component plasma of inertial ions and nonextensive electrons into vacuum, using the generalized nonextensive velocity distribution for electrons. Furthermore, a hydrodynamic model of plasma expansion in the presence of the ambipolar diffusion caused by collisions among the plasma species, such as electrons and ions, is developed and a Fokker-Planck-like generalized diffusion equation for steady-state expansion of a nonextensive electron-ion plasma is derived. For the case of generalized statistics and in the absence of particle diffusion, the density, velocity, electric potential, and field of expansion profiles are exactly obtained and studied in terms of the self-similar parameter. It is found that superthermal electrons lead to an accelerated expansion of plasma compared to that of Maxwellian electrons. It is also revealed that the nonextensivity parameter plays a fundamental role on the density, velocity, electric potential, and field configuration of the expansion. Therefore, one is able to distinguish three different regimes q < 1, q = 1, and q > 1 for expansion corresponding to sub-nonextensive, extensive, and super-nonextensive statistical profiles for electrons, respectively. Current research can provide useful information and suggests techniques for investigation of the involved statistical mechanism on the role of the energetic electron fluid in the expansion of plasma in strong pulsed laser-matter interaction experiments. It is also shown that the particle diffusion expansion mechanism becomes more dominant for relatively large values of the nonextensivity parameter, q.

Numerical studies on the self-similar collapse of the α-patches problem

This paper studies the dynamical evolution of the α-patches problem expressed in self-similar variables. A numerical algorithm is proposed and these equations are numerically explored. Several benchmarks of the code are discussed throughout the paper. Exact self-similar solutions are described and are found to play a role in separating collapsing from non-collapsing initial data: small perturbations around this solution blow up while others do not. Numerical simulations performed near convergent rescaled profiles, such as those described by Córdoba et al. (2005), indicate the absence of a stationary graph in the neighborhood of the rescaled profiles and suggest a more complex scenario for blow up.

Self-Similarity in Two-Phase Curvature Flow

In this work, we present a family of exact self-similar solutions for the distribution of enclosed areas under mean curvature flow on the union of disjoint closed curves in the plane. To our knowledge, this is the first example of closed-form exact self-similar solutions for the mean curvature flow. We perform numerical simulations for large sets of initial data obtained as the zero level set of Gaussian random fields and observe numerical convergence to one particular member of this family.

Groove growth by surface subdiffusion

The investigation of the grain-boundary groove growth by normal surface diffusion was first done by Mullins. However, the diffusion on a solid surface is often anomalous. Recently, the groove growth in the case of surface superdiffusion has been analyzed. In the present paper, the problem of the groove growth is solved in the case of the surface subdiffusion. An exact self-similar solution is obtained and represented in terms of the Fox H-function. Basic properties of the solution are described.