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We investigate numerically the collision dynamics of elliptically polarized solitons of the System of Coupled Nonlinear Schrodinger Equations (SCNLSE) for various different initial polarizations and phases. General initial elliptic polarizations (not sech-shape) include as particular cases the circular and linear polarizations. The elliptically polarized solitons are computed by a separate numerical algorithm. We find that, depending on the initial phases of the solitons, the polarizations of the system of solitons after the collision change, even for trivial cross-modulation. This sets the limits of practical validity of the celebrated Manakov solution. For general nontrivial cross-modulation, a jump in the polarization angles of the solitons takes place after the collision ('polarization shock'). We study in detail the effect of the initial phases of the solitons and uncover different scenarios of the quasi-particle behavior of the solution. In majority of cases the solitons survive the interaction preserving approximately their phase speeds and the main effect is the change of polarization. However, in some intervals for the initial phase difference, the interaction is ostensibly inelastic: either one of the solitons virtually disappears, or additional solitons are born after the interaction. This outlines the role of the phase, which has not been extensively investigated in the literature until now.

We show that the linearized equations of the incompressible elastic medium admit a `Maxwell form' in which the shear component of the stress vector plays the role of the electric field, and the vorticity plays the role of the magnetic field. Conversely, the set of dynamic Maxwell equations are strict mathematical corollaries from the governing equations of the incompressible elastic medium. This suggests that the nature of `electromagnetic field' may actually be related to an elastic continuous medium. The analogy is complete if the medium is assumed to behave as fluid in shear motions, while it may still behave as elastic solid under compressional motions. Then the governing equations of the elastic fluid are re-derived in the Eulerian frame by replacing the partial time derivatives by the properly invariant (frame indifferent) time rates. The `Maxwell from' of the frame indifferent formulation gives the frame indifferent system that is to replace the Maxwell system. This new system comprises terms already present in the classical Maxwell equations, alongside terms that are the progenitors of the Biot--Savart, Oersted--Ampere's, and Lorentz--force laws. Thus a frame indifferent (truly covariant) formulation of electromagnetism is achieved from a single postulate that the electromagnetic field is a kind of elastic (partly liquid partly solid) continuum., accepted

In recent works of the author [Found. Phys. 36 (2006) 1701-1717; Math. Comput. Simul. 74 (2007) 93-103], the argument has been made that Hertz's equations of electrodynamics reflect the material invariance (indifference) of the latter. Then the principle of material invariance was postulated in lieu of Lorentz covariance, and the respective absolute medium was named the metacontinuum. Here, we go further to assume that the metacontinuum is a very thin but very stiff 3D hypershell in the 4D space. The equation for the deflection of the shell along the fourth dimension is the ''master'' nonlinear dispersive equation of wave mechanics whose linear part (Euler-Bernoulli equation) is nothing else but the Schrodinger wave equation written for the real or the imaginary part of the wave function. The wave function has a clear non-probabilistic interpretation as the actual amplitude of the flexural deformation. The ''master'' equation admits solitary-wave solutions/solitons that behave as quasi-particles (QPs). We stipulate that particles are our perception of the QPs (schaumkommen in Schrodinger's own words). We show the passage from the continuous Lagrangian of the field to the discrete Lagrangian of the centers of QPs and introduce the concept of (pseudo)mass. We interpret the membrane tension as an attractive (gravitational?) force acting between the QPs. Thus, a self-consistent unification of electrodynamics, wave mechanics, gravitation, and the wave-particle duality is achieved.

For the Coupled Nonlinear Schrodinger Equations (CNLSE) we construct a conservative fully implicit scheme (in the vein of the scheme with internal iterations proposed in [C.I. Christov, S. Dost, G.A. Maugin, Inelasticity of soliton collisions in system of coupled nls equations, Physica Scripta 50 (1994) 449-454.]). Our scheme makes use of complex arithmetic which allows us to reduce the computational time fourfold. The scheme conserves the ''mass'', momentum, and energy. We investigate collisions of solitary waves (quasi-particles or QPs) with linear polarization in the initial configuration. We elucidate numerically the role of nonlinear coupling on the quasi-particle dynamics. We find that the initially linear polarizations of the QPs change after the collision to elliptic polarizations. For large values of cross-modulation parameter, an additional QP is created during the collision. We find that although the total energy is positive and conserved, the energy only of the system of identifiable after the collision QPs is negative, i.e., the different smaller excitations and radiation carry away part of the energy. The effects found in the present work shed light on the intimate mechanisms of interaction of QPs.

A material-invariant (frame indifferent) version of the Maxwell–Cattaneo law is proposed in which the relaxation rate of the heat flux is given by Oldroyd’s upper-convected derivative. It is shown that the new formulation allows for the elimination of the heat flux, thus yielding a single equation for the temperature field. This feature is to be expected from a truly frame indifferent description.

The present paper deals with the effect of dissipation on the propagation of wave packets governed by a wave equation of Jeffrey type. We show that all packets undergo a shift of the central frequency (the mode with maximal amplitude) towards the lower frequencies (“redshift” in theory of light or “baseshift” in acoustics). Packets with Gaussian apodization function do not change their shape and remain Gaussian but undergo redshift and spread. The possible applications of the results are discussed.

Boussinesq's original derivation of his celebrated equation for surface waves on a fluid layer opened up new horizons that were to yield the concept of the soliton. The present contribution concerns the set of Boussinesq-like equations under the general title of ‘Boussinesq's paradigm’. These are true bi-directional wave equations occurring in many physical instances and sharing analogous properties. The emphasis is placed: (i) on generalized Boussinesq systems that involve higher-order linear dispersion through either additional space derivatives or additional wave operators (so-called double-dispersion equations); and (ii) on the ‘mechanics’ of the most representative localized nonlinear wave solutions. Dissipative cases and two-dimensional generalizations are also considered. To cite this article: C.I. Christov et al., C. R. Mecanique 335 (2007).

We prove that, when linearized, the governing equations of an incompressible viscoelastic continuum can be rendered into a form identical to that of Maxwell's equations of electrodynamics. The divergence of deviator stress tensor is analogous to the electric field, while the vorticity (the curl of velocity field) is interpreted as the magnetic field. The elastic part of constitutive relation explains Maxwell's displacement current, and is responsible for the propagation of gradient (shear) waves. In turn, the viscous part is associated with the Ampere's and Ohm's laws for the current. This analogy is extended further and the nonlinearity of the material time derivative (the advective part of acceleration) is interpreted as the Lorentz force. The classical wave equations of electrodynamics are also derived as corollaries. Thus an interesting and far reaching analogy between the viscoelastic continuum and the electrodynamics is established.

We present a preliminary examination of a new approach to a long-standing problem in non-Newtonian fluid mechanics. First, we summarize how a general implicit functional relation between stress and rate of strain of a continuum with memory is reduced to the well-known linear differential constitutive relations that account for "relaxation" and "retardation." Then, we show that relaxation and retardation are asymptotically equivalent for small Deborah numbers, whence causal pure relaxation models necessarily correspond to ill-posed pure retardation models. We suggest that this dichotomy could be a possible way to reconcile the discrepancy between the theory of and certain experiments on viscoelastic liquids that are conjectured to exhibit only stress retardation., Comment: 5 pages, 1 figure, elsarticle format, v2 corrects some typograpgical errors, published in Mechanics Research Communications

Maxwell accounted for the apparent elastic behavior of the electromagnetic field by augmenting Ampere’s law with the so-called displacement current, in much the same way that he treated the viscoelasticity of gases. Maxwell’s original constitutive relations for both electrodynamics and fluid dynamics were not material invariant. In the theory of viscoelastic fluids, the situation was later corrected by Oldroyd, who introduced the upper-convective derivative. Assuming that the electromagnetic field should follow the general requirements for a material field, we show that if the upper convected derivative is used in place of the partial time derivative in the displacement current term, Maxwell’s electrodynamics becomes material invariant. Note, that the material invariance of Faraday’s law is automatically established if the Lorentz force is admitted as an integral part of the model. The new formulation ensures that the equation for conservation of charge is also material invariant in vacuo. The viscoelastic medium whose apparent manifestation are the known phenomena of electrodynamics is called here the metacontinuum.

A study of nonlinear acoustic waves in a homentropic perfect gas is presented. Conservation laws for the Euler and Lighthill–Westervelt equations are constructed and solved numerically using a Godunov-type finite-difference scheme. Simulations are carried out in the context of two initial-boundary-value problems (IBVP)s—one resulting in finite-time, and the other in infinite-time, blow-up at the wavefront. Additionally, analytical results are presented to support the numerical findings.

Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

The flow of inviscid liquid in a shallow layer with free surface is revisited in the framework of the Boussinesq approximation. The unnecessary approximations connected with the moving frame are removed and a Boussinesq model is derived which is Galilean invariant to the leading asymptotic order. The Hamiltonian structure of the new model is demonstrated. The conservation and/or balance laws for wave mass, energy and wave momentum (pseudo-momentum) are derived. A new localized solution is obtained analytically and compared to the classical Boussinesq sech. Numerical simulation of the collision of two solitary waves is conducted and the impact of Galilean invariance on phase shift is discussed.

Interaction between a base thermocapillary flow and a time-dependent buoyant force is studied for a slot geometry. A temperature gradient applied along a fluid-filled slot with thermocapillarity at a free surface produces a base parallel flow. The system is subjected to streamwise gravitational acceleration that varies harmonically in time. Grassia and Homsy [Phys. Fluids. 10, 1273 (1998)] have shown that in the limit of zero frequency modulation, coupling of the thermocapillary flow with long wave convective modes leads to singularities at critical points corresponding to the Rayleigh–Benard eigenvalues. In the case of small but finite frequency modulation studied here, inertial effects moderate the singularities which are replaced by a response that scales exponentially with the inverse of the dimensionless modulation frequency. An O(1) delay is observed in the onset of the resonant response even for small modulation frequencies. The response is also found to scale exponentially with the inverse Prandtl number for large Prandtl numbers and to be independent of Prandtl number for small Prandtl numbers. Relaxation oscillations are observed in certain parameter ranges as a result of the coupling between the fluid and thermal fields. A Galerkin approximation is used to reduce the problem to an equivalent dynamical system, the analysis of which gives analytical support to and insight into the numerical results.

We consider the long-time evolution of the solution of an energy-consistent system with dispersion and nonlinearity, which is the progenitor of the different Boussinesq equations. Unlike the classical Boussinesq models, the energy-consistent one possesses Galilean invariance. As initial condition we use the superposition of two analytical one-soliton solutions. We use a strongly dynamical implicit difference scheme with internal iterations, which allows us to follow the evolution of the solution at very long times. We focus on the behavior of traveling localized solutions developing from critical initial data. The main solitary waves appear virtually undeformed from the interaction, but additional oscillations are excited at the trailing edge of each one of them. We track their evolution for very long times when they tend to adopt a self-similar shape. We test a hypothesis about the dependence on time of the amplitude and the support of Airy-function shaped coherent structures. The investigation elucidates the mechanism of evolution of interacting solitary waves in the energy-consistent Boussinesq equation.

A higher-order dispersive equation is introduced as a candidate for the governing equation of a field theory. A new class of solutions of the three-dimensional field equation are considered, which are not localized functions in the sense of the integrability of the square of the profile over an infinite domain. For this new class of solutions, the gradient and/or the Hessian/Laplacian are square integrable. In the linear limiting case, an analytical expression for the pseudolocalized solution is found and the method of variational approximation is applied to find the dynamics of the centers of the quasi-particles (QPs) corresponding to these solutions. A discrete Lagrangian can be derived due to the localization of the gradient and the Laplacian of the profile. The equations of motion of the QPs are derived from the discrete Lagrangian. The pseudomass ("wave mass") of a QP is defined as well as the potential of interaction. The most important trait of the new QPs is that at large distances, the force of attraction is proportional to the inverse square of the distance between the QPs. This can be considered analogous to the gravitational force in classical mechanics., Comment: 19 pages, 10 figures, elsarticle format; v2 includes revision in response to referees; accepted for publication in Wave Motion for the special issue on Mathematical Modeling and Physical Dynamics of Solitary Waves: From Continuum Mechanics to Field Theory

A generalized wave equation containing production and dissipation of energy is derived in a heuristic fashion so as to have in the dissipationless limit the (two-way wave) Boussinesq equation, while for the slowly evolving in a moving frame (one-way) wave system, it reduces to the dissipation modified KDV equation (KDV-KSV) with the same energy-balance law. The new equation allows investigating the head-on collision of dissipative localized structures. A special difference scheme is devised which faithfully represents the balance law for energy. The numerical simulations show that if the production-dissipation rate is of order of a small parameter, the coherent structures upon collisions preserve their localized character and within a time interval proportional to the inverse of the small parameter they behave like (imperfect) solitons. The collisions are almost ideal without phase shift. The only difference from the strictly soliton collision is that during the time of interaction the dissipative structures are "aging" and changing their shapes.

The present work presents our research activity concerning the problems of modeling and control of a rigid‐body motion and its applications in multi‐body mechanical systems. Since the treatment of the above topic is quite sensitive with respect to the different parameterizations of the rotation group in three dimensional space SO(3) and since the properties of the parameterization more or less influence on the efficiency of the dynamic model, a full analysis of the different ways of SO(3) group representation is suggested.

The stationary Couette gas flow between rotating inner cylinder and stationary outer one is studied using DSMC method and numerical solution of a continual model. Different cases were studied by varying the temperature of the stationary cylinder and the Knudsen number. The continuum model results are obtained by setting a local value of Knudsen number in the corresponding first order slip boundary conditions. The comparison results showed that the slip boundary conditions with local Knudsen number included improves the accuracy of the continual model.

Basic equations of continuum mechanics reflect deep symmetry properties of Euclidean space and medium, which moves in it. As a corollary, these equations admit a wide Lie group that allows us to apply group‐theoretical methods for continuum motion study. In this paper, both gravitational heat convection and thermocapillary convection equations are considered from the group‐theoretical point of view. It turned out that classical Ostroumov and Birikh's solutions have a group nature. We present a number of new solutions, which describe a non‐isothermal viscous liquid motion in channels, tubes and layers under action of gravity, centrifugal and thermocapillary forces.

3D numerical method is presented for simulation of foam formation and dynamics in viscous flows in the presence of insoluble surfactant and under the influence of van der Waals forces. The mathematical model is based on the Stokes equations in the fluid phases, coupled with velocity and stress boundary conditions at the interfaces. A nonuniform surfactant concentration on the interfaces, governed by a convection‐diffusion equation, leads to a gradient of the interfacial tension, which in turns leads to an additional tangential stress on the interfaces. The presented numerical method is a semi‐implicit coupling of a boundary‐integral method for the velocity in the fluid phases with a finite‐volume method for the surfactant concentration on the interfaces. Additional elements of the method are: Nonsingular contour integration of the singular layer potential; Higher order approximation of the interface position; Dynamic mesh regularization.

Consider a suspension of dilute spherical particles transported in the flow of a viscous fluid along a wall. A Navier slip condition applies on the wall, in view of applications to hydrophobic walls in microchannels. The Stokes flow problem for a sphere in a quadratic shear flow is solved here by the method of bispherical coordinates. It is shown how the infinite linear system for four coupled series of coefficients may be solved with a cost that is simply proportional to the number of coefficients. Results are the force and torque on a fixed sphere and the velocities of translation and rotation of a freely moving sphere in a quadratic flow near a slip wall. The stresslet, that is the symmetric moment of surface stresses on the sphere, is also derived in view of applications to suspension rheology. Finally, expansions of the various quantities for a large distance from the wall and for a low slip length are performed on the basis of the analytical solutions, using computer algebra. Pade approximants calculated therefrom provide a good approximation.

We study the propagation and interaction of solitons modeled by the equation governing the flexural deformation of an elastic rod on elastic foundation (EREFE). The model equation is only of fourth order, but because of the presence of the linear restoring force, it exhibits some of the phenomenology of the 6GBE, especially when the nonmonotonic shapes are concerned. To do that, we apply the Christov spectral method in L2(−∞,∞). Initially, we solve the problem in the moving frame and consider the obtained solutions as initial condition for the time dependant problem and study the propagation and interaction of solitons.

The N‐soliton interactions of two classes of vector NLS (VNLS) equations are analyzed. It is shown that the N‐soliton interactions of the VNLS related to symmetric spaces of BD.I‐type, just like in the scalar case, reduce to shifts of the center of masses and of the phases, while the polarization vectors remain unchanged. The same approach applied to the standard VNLS (or the Manakov model) shows nontrivial effects of the interaction on the polarization vectors.

This is a further note on the (Guass‐Maxwell) force‐flux construct proposed previously (Goddard, J. D., A note on Eringen's moment balances, Int. J. Eng. Sci., in the press, 2011). Motivated in part by its promise as a homogenization technique for constructing micromorphic continua, the present work is focused rather on some additional representations and on novel applications, such as the derivation of dissipative thermodynamic potentials from force‐flux relations.

A fractional oscillator with a power‐law memory kernel subjected to an external periodic force is considered. The influence of the fluctuating viscoelastic environment is modeled by a multiplicative dichotomous noise (fluctuating damping) and an additive internal noise. The main purpose of this work is to provide exact formulas for the analytic treatment of the dependence of spectral amplification on system parameters: viz. the noise correlation time, noise amplitude, memory exponent, and driving frequency. Based on those exact expressions we demonstrate that stochastic resonance is manifested in the dependence of the spectral amplification upon the noise parameters. Moreover, a critical memory exponent is found which marks the transitions between different dynamical regimes of the oscillator.

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/ annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro-structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef-Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction-diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model, the first one comprising linear diffusion of the density of mobile dislocations and the second one, with nonlinear diffusion of said variable. Subsequently, the paper focus on a finite difference, second order in time Cranck-Nicholson semi-implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer 0, 35 , 87, 2002) for solving the model evolution equations in two dimensions. The discussion on the WA model and on the numerical scheme was already presented on a conference paper by the authors (Pontes et al., AIP Conference Proceedings, Vol. 1301 pp. 511-519, 2010). The first results of four simulations, one with linear diffusion of the mobile dislocations and three with nonlinear diffusion are presented. Several phenomena were observed in the numerical simulations, like the increase of the fundamental wavelength of the structure, the increase of the walls height and the decrease of its thickness.

Provided here is numerical evidence of localized solutions, solitary waves, in a model equation for Benard convection driven by interfacial stresses (Marangoni effect).

The internal phase dynamics of a quantum system interacting with an electromagnetic field is revealed in details. Theoretical and experimental evidences of a causal relation of the phase of the wave function to the dynamics of the quantum system are presented sistematically for the first time. A dynamics‐statistical interpretation of the quantum mechanics is introduced.

This paper presents a well‐balanced numerical scheme for solving free surface flows involving wetting and drying. The proposed algorithm combines a component‐wise approach with hydrostatic reconstruction strategy to compute flows over wet or dry surfaces and to satisfy the steady state condition of still water. The robustness of the proposed scheme is verified under several benchmark hydraulic tests.

In this work, we investigate the 2D flow in a rectangular cavity subject to both vertical and horizontal temperature gradients. The linearized model is studied and the effect of thermal relaxation, as described by the Maxwell‐Cattaneo law of heat conduction is examined. To this end, a spectral numerical model is created based on a Galerkin expansion. The basis is the Cartesian product of systems of beam functions and trigonometric functions.The natural modes of the system are derived for both the Fourier and non‐Fourier models. The results are compared to earlier works for the plain Fourier law. Our computations show that for the same set of parameters, the Maxwell‐Cattaneo law yields modes which are quantitatively different from the Fourier. It is found that the real parts of the eigenvalues increase with the Straughan number Sg, which quantifies the non‐Fourier effects. This confirms the destabilizing effect of the MC‐law on the convective flow.

Physical mesomechanics is a relatively recent theory of deformation and fracture. Based on the physical principle known as gauge invariance, it has derived field equations that describe the displacement behavior in plastically deforming materials. By the nature of the way the gauge invariance is applied, the theory is capable of describing all stages of deformation on the same theoretical basis. Previously, we analyzed this formalism from the mathematical point of view, and discussed the dynamics resulting from the field equations. In this paper, we focus on the physical meaning of the field equations and related dynamics. It is shown that plastic deformation is characterized as the dynamics where materials’ longitudinal resistive force is proportional to the local velocity as opposed to the elastic regime where the longitudinal resistive force is proportional to the displacement. Consequently, in the plastic regime, part of the work done by external force is not stored in the material as potential energy...

Two different forms of relativistic dynamics, the instant and the light‐cone form, for the pure SU(2) Yang‐Mills field theory in 4‐dimensional Minkowski space are examined under the supposition that the gauge fields depend on the time evolution parameter only. The obtained under that restriction of gauge potential space homogeneity mechanical matrix model, sometimes called Yang‐Mills classical mechanics, is systematically studied in its instant and light‐cone form of dynamics using the Dirac’s generalized Hamiltonian approach. In the both cases the constraint content of the obtained mechanical systems is found. In contrast to its well‐known instant‐time counterpart the light‐cone version of SU(2) Yang‐Mills classical mechanics has in addition to the constraints generating the SU(2) gauge transformations the new first and second class constraints also. On account of all of these constraints a complete reduction in number of the degrees of freedom is performed. In the instant form of dynamics it is shown th...

The aim of the present work is to investigate numerically phases of soliton‐like solutions coupled to non‐linear electrodynamics in the scalar‐tensor theories of gravity. As a result of the numerical experiments new non‐unique solutions of the corresponding boundary value problem and their general properties were found.

Using solitonic techniques we construct a new exact stationary axisymmetric solution to the 5D Einstein equations in vacuum in asymptotically Kaluza‐Klein spacetime. The solution describes rotating black ring with a single angular momentum surrounded by two Kaluza‐Klein bubbles. It is generated by applying a 2‐soliton Backlund transformation on a static seed. The solution properties are discussed, and its physical characteristics, such as mass, tension, angular velocity and angular momentum, are derived.

Solving electromagnetic scattering problems involving non‐uniformly moving objects or media requires an approximate but consistent extension of Einstein’s Special Relativity theory, originally valid for constant velocities only. For moderately varying velocities a quasi Lorentz transformation is presented. The conditions for form‐invariance of the Maxwell equations, the so‐called “principle of relativity”, are shown to hold for a broad class of motional modes and time scales. A simple example of scattering by a harmonically oscillating mirror is analyzed in detail. Application to generally orbiting objects is mentioned.

The modeling of the free thin film dynamics and stability is usually performed in the framework of the extended lubrication approach, including inertial, viscous, capillary and intermolecular van‐der‐Waals forces. For films with mobile surfaces, this approach leads to a system of nonlinear PDE for the film thickness and lateral velocities. In the present work the 1D variant of this system is studied numerically in the case of a laterally bounded free film by a frame, with a prescribed wetting angle. A linear and non‐linear stability analysis of this problem is performed and their results generalize the stability results obtained for periodic free films by other authors. It is shown that the film rupture can be caused either by the non‐existence of a static film shape or by the instability of an existing static shape. It is found that the static film shapes are always stable when subjected to symmetrical disturbances, but can be unstable if asymmetrical disturbances are imposed on them. The latter fact depends on the wetting angle in combination with the action of the van‐der‐Waals attraction and of the surface tension.

Modeling of collision processes of colliding bodies with different properties is of particular theoretical and practical interest e.g. for modeling of plastic deformation processes. Best results regarding the parameters of the collision processes (forces, stress, strain, duration of contact, etc.) are obtained using the finite elements method (FEM). This paper presents the results of FEM modeling of the collision process after a free fall of a sphere on an elastic‐plastic plate of specific dimensions under the additional application of an external force. Two cases are described: the application of an external force during the time of contact of the colliding bodies, and without application of an external force. Correlations were obtained for the change of the strength of the impact, the stress and strain as function of time for the sphere and the plate. The strain of the plate were defined for various values of the external force. These results allow us to define the conditions, necessary to perform the s...

Micro mechanical devices are rapidly emerging technology, where new potential applications are continuously being found. A simulation of internal and external gas flows in or around these devices is important for their design. The gas flows are characterized with areas of low speed flows (low Reynolds numbers). The flows can go from high speed supersonic to very low speed regimes down to the incompressible limit. This made the pressure based numerical methods very suitable to be used for calculation of this kind of gas flows. In this paper we present a short review of a finite volume method, developed by the authors, and demonstrate its application for calculation of some unsteady state gaseous flows. We study a flow past square‐shaped particle confined in a micro‐channel filled with a hard‐sphere monatomic gas. The problem is solved using the finite volume method SIMPLE‐TS (a modification of SIMPLE created by the authors). The flow motion is described on the basis of the Navier‐Stokes‐Fourier compressible equations with diffusion coefficients determined by the first approximation of the Chapman‐Enskog theory for the low Knudsen numbers. First order velocity slip and temperature jump boundary conditions are used. The target of the research is to find the Knudsen number (corresponding Reynolds number) for which the microflow becomes unstable and some effects of flow separation and Karman vortices appear. The investigation is accomplished for subsonic (Mach number M = 0.1) and supersonic ( M = 2.43) speeds, leading to different regimes of the gas microflow.

The cylindrical Couette flow of a rarefied gas is studied in the case of two cylinders rotating with different velocities. Velocity, density and temperature profiles are investigated by the Direct Monte Carlo Simulation method and a numerical solution of the Navier‐Stokes equations for compressible flow is found. We have obtained numerical results for gas velocity and temperature at the walls for different values of rotation velocities of the cylinders. The gas lags or outstrips in comparison with the walls or it has the elastic rigid body behavior. These results are important for applications in non‐planar microfluidic problems. The results obtained by both methods are in a good agreement at the smaller Knudsen number Kn = 0.02 and differs notably at the larger Kn = 0.1.

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion along with accompanying production/annihilation processes of dislocations lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result into the development of various types of dislocation micro‐structures, such as dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc., depending on the externally applied loading and the intrinsic lattice constraints. The Walgraef‐Aifantis (WA) (Walgraef and Aifanits, J. Appl. Phys., 58, 668, 1985) model is an example of a reaction‐diffusion model of coupled nonlinear equations which describe 0 formation of forest (immobile) and gliding (mobile) dislocation densities in the presence of cyclic loading. This paper discuss two versions of the WA model and focus on a finite difference, second order in time 1‐Nicolson semi‐implicit scheme, with internal iterations at each time step and a spatial splitting using the Stabilizing, Correction (Christov and Pontes, Mathematical and Computer Modelling, 35, 87, 2002) for solving the model evolution equations in two dimensions. The results of two simulations are presented. More complete results will appear in a forthcoming paper.

In this paper the mathematical model of of a body movement in a viscous medium is investigated. The problem of control for a solid body moving from initial position to the given one is considered. Movement occurs at Reynolds great numbers that generates effects of failure of the laminar boundary layer, caused by return difference of a gradient of pressure. Thus behind a body the vortex path is formed. Frequency of failure of whirlwinds is expressed in the form of the dimensionless parameter. Asymmetrical formation of whirlwinds leads to occurrence periodic cross‐section to speed of power influences on a body. Oscillatory movements, especially as a result develop if frequency of formation of whirlwinds comes nearer to own frequency of fluctuations of a body.

Based on the field theoretical approach developed by physical mesomechanics and recent analysis on the associated gauge field, deformation of solid‐state materials is formulated as a linear transformation of infinitesimal line elements in a deforming object. This formalism is capable of describing all stages of deformation including fracture on the same theoretical basis. The elastic and plastic stages are differentiated by whether the transformation is global or local. From the gauge invariance of the transformation and the least action principle applied to the gauge field, dynamics of plastic deformation is obtained being characterized as decaying, transverse wave behavior of the displacement field. Fracture is characterized as the final stage of deformation where the material becomes totally dissipative and the displacement wave decays completely. Optical interferometric experiment has been carried out to analyze in‐plane displacement of a metal specimen under tensile loading. The result of analysis indicates that the plastic deformation and fracture criteria derived from the present formalism are able to reveal the transition from the elastic to plastic stage and predict the initiation of crack growth.

Phases of scalar‐tensor black holes coupled to non‐linear electrodynamics where recently obtained and thermodynamical analysis of the obtained solutions was performed. Our task is to study numerically the linear stability of the solutions which has trivial scalar field. The obtained singular Sturm‐Liouville problem is solved with a modifications of the shooting method. Results are obtained and they are in a good agreement with the predictions of the thermodynamical analysis.

We study the effect of the Maxwell‐Cattaneo law of heat conduction (MCHC) on the 1D flow in a vertical slot subject to both vertical and horizontal temperature gradients. The gravitational acceleration is allowed to oscillate, which provides an opportunity to investigate the quantitative contribution of thermal inertia as epitomized by MCHC. The addition of the time derivative in MCHC increases the order of the system. We use a spectral expansion with Rayleigh’s beam functions as the basis set, which is especially suited to fourth order boundary value problems (BVP). We show that the time derivative (relaxation of the thermal flux) has a dissipative nature and leads to the appearance of purely real negative eigenvalues. Yet it also increases the absolute value of the imaginary part and decreases the absolute value of the real part of the complex eigenvalues. Thus, the system has a somewhat more oscillatory behavior than the one based on Fourier’s heat conduction law (FHC).

In the present work static and spherically symmetric magnetically charged black holes coupled to Euler‐Heisenberg type non‐linear electrodynamics in the scalar‐tensor theories of gravity with massive scalar field are investigated numerically. For the numerical solving of the corresponding boundary‐value problem with free boundary the Continuous Analogue of Newton Method is used. As a result of the numerical experiments some general properties of the solutions were found.

In the present, work we find new numerical solutions describing soliton‐like objects coupled to non‐linear electrodynamics in the scalar‐tensor theories of gravity. Approximate solutions have been also obtained and a comparison to the numerical results has been made. The non‐existence of black holes for the specific choice of lagrangian of the nonlinear electrodynamics has been also proved.

We study the dynamics of the sine-Gordon equation's kink soliton solutions under the coarse-grain description via two "collective variables": the position of the "center" of a soliton and its characteristic width ("size"). Integral expressions for the interaction potential and the quasi-particles' cross-masses are derived. However, these cannot be evaluated in closed form when the solitons have varying widths, so we develop a perturbation approach with the velocity of the faster soliton as the small parameter. This enables us to derive a system of four coupled second-order ODEs, one for each collective variable. The resulting initial-value problem is very stiff and numerical instabilities make it difficult to solve accurately, so a semi-empirical iterative approach to its solution is proposed. Then, we demonstrate that, even though it appears the solitons pass through each other, the quasi-particles actually "exchange" their pseudomasses during a collision.

By treating the centers of solitons as point particles and studying their discrete dynamics, we demonstrate a new approach to the quantization of the soliton solutions of the sine-Gordon equation, one of the first model nonlinear field equations. In particular, we show that a linear superposition of the non-interacting shapes of two solitons offers a qualitative (and to a good approximation quantitative) description of the true two-soliton solution, provided that the trajectories of the centers of the superimposed solitons are considered unknown. Via variational calculus, we establish that the dynamics of the quasi-particles obey a pseudo-Newtonian law, which includes cross-mass terms. The successful identification of the governing equations of the (discrete) quasi-particles from the (continuous) field equation shows that the proposed approach provides a basis for the passage from the continuous to a discrete description of the field., 10 pages, 3 figures (6 images); v2: revised and improved the presentation, updated the references, fixed typos; v3: corrected a few minor mistakes and typos, version accepted for publication in Phys. Lett. A