Regular, partially invariant solutions of rank 1 and defect 1 of equations of plane motion of a viscous heat-conducting gas

A system of the Navier-Stokes equations of two-dimensional motion of a viscous heat-conducting perfect gas with a polytropic equation of state is considered. Regular, partially invariant solutions of rank 1 and defect 1 are studied. A sufficient condition of their reducibility to invariant solutions of rank 1 is proved. All solutions of this class with a linear dependence of the velocity-vector components on spatial coordinates are examined. New examples of solutions that are not reducible to invariant solutions are obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 23–33, November–December, 2006.     

Invariant and partially invariant solutions of integro-differential equations for linear thermoviscoelastic aging materials with memory

A linear thermoviscoelastic model for homogeneous, aging materials with memory is established. A system of integro-differential equations is obtained by using two motions (a one-dimensional motion and a shearing motion) for this model. Applying the group analysis method to the system of integro-differential equations, the admitted Lie group is determined. Using this admitted Lie group, invariant and partially invariant solutions are found. The present paper gives a first example of application of partially invariant solutions to integro-differential equations.     

Exact solutions of the axisymmetric equations of motion of a viscous heat-conducting perfect gas described by systems of ordinary differential equations

All invariant solutions of rank 1 of the two-dimensional equations of motion of a heat-conducting perfect gas with a polytropic equation of state are described. A sufficient condition for reducibility of regular, partially invariant solutions of rank 1 and defect 1 to invariant solutions is given. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 51–54, September–October, 1999.     

Invariant and Partially Invariant Solutions of the Green-Naghdi Equations

All invariant and partially invariant solutions of the Green-Naghdi equations are obtained that describe the second approximation of shallow water theory. It is proved that all nontrivial invariant solutions belong to one of the following types: Galilean-invariant, stationary, and self-similar solutions. The Galilean-invariant solutions are described by the solutions of the second Painleve equation, the stationary solutions by elliptic functions, and the self-similar solutions by the solutions of the system of ordinary differential equations of the fourth order. It is shown that all partially invariant solutions reduce to invariant solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 26–35, November–December, 2005.     

Nonlinear dynamic response of axially moving,stretched viscoelastic strings

The dynamical response of axially moving, partially supported, stretched viscoelastic belts is investigated analytically in this paper. The Kelvin–Voigt viscoelastic material model is considered and material, not partial, time derivative is employed in the viscoelastic constitutive relation. The string is considered as a three part system: one part resting on a nonlinear foundation and two that are free to vibrate. The tension in the belt span is assumed to vary periodically over a mean value (as it occurs in real mechanisms), and the corresponding equation of motion is derived by applying Newton’s second law of motion for an infinitesimal element of the string. The method of multiple scales is applied to the governing equation of motion, and nonlinear natural frequencies and complex eigenfunctions of the system are obtained analytically. Regarding the resonance case, the limit-cycle of response is formulated analytically. Finally, the effects of system parameters such as axial speed, excitation characteristics, viscousity and foundation modulus on the dynamical response, natural frequencies and bifurcation points of system are presented.     

ON SLOSHING IN A CONTAINER WITH MOVING WALL

The research on the coupled frequencies of a fluid–structure system comprised of a container with a moving wall partially filled with water (Figure 1) was presented in two papers by Lu et al. and Chai et al., but their solutions are different. The aim of this letter is to compare them. The fluid is incompressible and inviscid, and the structure is a mass m[kg m⁻¹] in translation, connected to the Galilean reference by a spring of stiffness k[N m⁻²]; these characteristics are given per unit length in the z direction. The authors linearized the equations and looked for a potential-flow solution for the fluid motion. They obtain the same set of equations.     

Problem of filling of a spherical cavity in a Kelvin-Voigt medium

This paper is concerned with mathematical modeling and solution of the problem of the collapse of a spherical cavity in a viscoelastic medium under the action of constant pressure at infinity. A differential equation of motion for the cavity boundary is constructed and solved numerically. The existence of three modes of motion of the boundary is established, and a map of these modes in the plane of the determining parameters is constructed. Asymptotic forms of the solutions of the problem for all modes are constructed. The problem of cavity collapse with capillary forces taken into account is formulated and solved. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 93–101, September–October, 2008.     

Partially invariant solutions for a submodel of radial motions of a gas

All partially invariant solutions in terms of the group of extensions for a model of radial motions of an ideal gas are found. The solutions are obtained by the method of separation of variables in an equation containing functions of one variable but different functions of different independent variables. The solutions predict different continuous unsteady convergence or expansion of the gas under the action of a piston with a point sink or source. If the sink or source affects all particles simultaneously, a collapse or an explosion occurs. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 5, pp. 26–34, September–October, 2007.     

Regular partially invariant solutions of defect 1 of the equations of ideal magnetohydrodynamics

All irreducible regular partially invariant submodels with one noninvariant function for the equations of ideal magnetohydrodynamics are constructed. The submodels are completed to involution, and partially integrated. The submodels specify Ovsyannikov vortex type motion or motion with homogeneous deformation in some spatial directions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 5–15, March–April, 2009.     

Mathematical model of an incompressible viscoelastic Maxwell medium

Nonstationary motions of incompressible viscoelastic Maxwell continuum with a constant relaxation time are considered. Because in an incompressible continuous medium, pressure is not a thermodynamic variable but coincides with the stress-tensor trace to within a factor, it follows that, separating the spherical part from this tensor, one can assume that the remaining part of the stress tensor has zero trace. In the case of an incompressible medium, the equations for the velocity, pressure, and stress tensor form a closed system of first-order equations which has both real and complex characteristics, which complicates the formulation of the initial-boundary-value problem. Nevertheless, the resolvability of the Cauchy problem can be proved in the class of analytic functions. Unique resolvability of the linearized problem was established in the classes of functions of finite smoothness. The class of effectively one-dimensional motions for which the subsystem of three equations is a hyperbolic one was studied. The results of an asymptotic analysis of the latter imply the possible formation of discontinuities during the evolution of the solution. The general system of equations of motion admits an infinite-dimensional Lie pseudo-group which contains an extended Galilean group. The theorem of the invariance of the conditions on the a priori unknown free boundary was proved to obtain exact solutions of free-boundary problems. The problem of deformation of a viscoelastic strip subjected to tangential stresses applied to the free boundary is considered as an example of application of this theorem. In this problem, a scale effect of short-wave instability caused by the absence of diagonal dominance of the stress tensor deviator was found.     

Exact solutions of the one-dimensional Russo-Smereka kinetic equation

We obtain new classes of invariant solutions of the integrodifferential equations describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid. For all the solutions obtained, trajectories of particle motion in phase space are calculated. The stability of some flows is studied in a linear approximation. In several cases, the construction of solutions reduces to an integrodifferential equation of the second kind, which can be solved by the iteration method. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 4, pp. 21–32, July–August, 2000.     

Galilean-invariant axisymmetric self-similar submodel of gas dynamics without swirling

This paper deals with one insufficiently studied submodel of invariant solutions of rank 1 of the equations of gas dynamics. It is shown that, in cylindrical coordinates, the submodel without swirling reduces to a system of two ordinary differential equations. For the equation of state with additional invariance, a self-similar system is obtained. A pattern of phase trajectories is constructed, and particle motion is studied using asymptotic methods. The obtained solutions describe unsteady flows over axisymmetric bodies with possible strong discontinuities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 46–52, March–April, 2009.     

Group properties of the equations of adiabatic motion of a medium in relativistic hydrodynamics

A group classification is presented and the complete set of invariant solutions is found for the equations of adiabatic motion of a medium in relativistic hydrodynamics.     

Submodel of rotational motions in gas dynamics

An invariant submodel of the equations of gas dynamics constructed on a one-dimensional subalgebra consisting of the sum of operators of rotation and translation in time is studied within the framework of the SUBMODELS program. The system of equations of the submodel is brought to symmetric form. Hyperbolicity conditions for the system are derived. Group analysis is performed and an invariant solution is considered. Isobaric flows are listed. For the simplest of them, characteristics and strong discontinuities are considered. Necessary conditions for existence of solutions without singularities on the axis are derived. Institute of Mechanics, Ural Scientific Center, Russian Academy of Sciences, Ufa 450000. Translated from Prikladnya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 6, pp. 37–45, November–December, 1998.     

Nonlinear vibrations of a viscoelastic plate with concentrated masses

The problem of vibrations of a viscoelastic plate with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate, the action of the concentrated masses is taken into account using Dirac δ-functions. The problem is reduced to solving a system of Volterra type ordinary nonlinear integrodifferential equations using the Bubnov-Galerkin method. The resulting system with a singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The effect of the viscoelastic properties of the plate material and the location and amount of concentrated masses on the vibration amplitude and frequency characteristics is studied. A comparison is made of numerical calculation results obtained using various theories. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 158–169, November–December, 2007.     

One class of partially invariant solutions of the Navier-Stokes equations

A family of partially invariant solutions of the Navier-Stokes equations of rank 2 and defect 2 is considered. These solutions describe the three-dimensional unsteady motions of a viscous incompressible fluid in which the vertical velocity component and the pressure are independent of the horizontal coordinates. In particular, they can be interpreted as flows in a horizontal layer, one boundary of which is the free surface. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 2, pp. 24–33, March–April, 1999.     

Infinite-Dimensional Hyperbolic Sets and Spatio-Temporal Chaos in Reaction Diffusion Systems in $${\mathbb{R}^{n}}$$

The paper is devoted to a rigorous construction of a parabolic system of partial differential equations which displays space–time chaotic behavior in its global attractor. The construction starts from a periodic array of identical copies of a temporally chaotic reaction-diffusion system (RDS) on a bounded domain with Dirichlet boundary conditions. We start with the case without coupling where space–time chaos, defined via embedding of multi- dimensional Bernoulli schemes, is easily obtained. We introduce small coupling by replacing the Dirichlet boundary conditions by strong absorption between the active islands. Using hyperbolicity and delicate PDE estimates we prove persistence of the embedded Bernoulli scheme. Furthermore we smoothen the nonlinearity and obtain a RDS which has polynomial interaction terms with space and time-periodic coefficients and which has a hyperbolic invariant set on which the dynamics displays spatio-temporal chaos. Finally we show that such a system can be embedded in a bigger system which is autonomous and homogeneous and still contains space–time chaos. Obviously, hyperbolicity is lost in this step. Research partially supported by the INTAS project Attractors for Equations of Mathematical Physics, by CRDF and by the Alexander von Humboldt–Stiftung.     

Solving initial–boundary-value creep problems

The Galerkin–Bubnov method with global approximations is used to find approximate solutions to initial–boundary-value creep problems. It is shown that this approach allows obtaining solutions available in the literature. The features of how the solutions of initial–boundary-value problems for oneand three-dimensional models are found are analyzed. The approximate solutions found by the Galerkin–Bubnov method with global approximations is shown to be invariant to the form of the equations of the initial–boundary-value problem. It is established that solutions of initial–boundary-value creep problems can be classified according to the form of operators in the mathematical problem formulation     

Ovsyannikov plane vortex: The equations of the submodel

A submodel of the equations of ideal magnetohydrodynamics is constructed that generalizes the classical motion of an ideal continuous medium with plane waves. It is shown that, in contrast to classical motion, in this submodel the velocity and magnetic-field vectors can change direction in a plane orthogonal to a distinguished spatial direction. The submodel is described by a system of equations with two independent variables and a finite relation specifying the orientation of the vector fields in space. The solutions of the submodel define substantially spatial processes and singularities in the motion of continuous media which cannot be studied in the classical one-dimensional formulation. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 5, pp. 27–40, September–October, 2008.     

Group analysis of integro-differential equations describing stress relaxation behavior of one-dimensional viscoelastic materials

In this paper a recently developed approach of the group analysis method is applied to a system of integro-differential equations describing the stress relaxation behavior of one-dimensional viscoelastic materials. An admitted Lie group is defined by solving determining equations of the system. Using an optimal system of one-dimensional subalgebras, all invariant solutions are obtained.