Adiabatic invariants,diffusion and acceleration in rigid body dynamics

The onset of adiabatic chaos in rigid body dynamics is considered. A comparison of the analytically calculated diffusion coefficient describing probabilistic effects in the zone of chaos with a numerical experiment is made. An analysis of the splitting of asymptotic surfaces is performed and uncertainty curves are constructed in the Poincaré–Zhukovsky problem. The application of Hamiltonian methods to nonholonomic systems is discussed. New problem statements are given which are related to the destruction of an adiabatic invariant and to the acceleration of the system (Fermi’s acceleration).     

A method of asymptotic expansions of the solutions of the steady heat conduction problem for laminated non-uniform anisotropic plates

Outer asymptotic expansions of the solutions of the steady heat conduction problem for laminated anisotropic non-uniform plates for different boundary conditions on the faces are constructed. The two-dimensional resolvents obtained are analysed and the asymptotic properties of the solutions of the heat-conduction problem are investigated. Estimates are obtained of the accuracy with which the temperature in the plate outside the limits of the boundary layer can be assumed to be piecewise-linearly or piecewise-quadratically distributed over the thickness of the laminated structure. A physical justification for certain features of the asymptotic expansions of the temperature is given.     

On the formulation and investigation of a spatial contact problem for elastic bodies under mixed friction conditions

A spatial contact problem is formulated and investgated for rough elastic bodies which touch each other under mixed friction conditions: the elastic bodies are separated in one part of the contact domain by a layer of viscous incompressible liquid (lubricant), while in the other they are in direct contact (such conditions are characteristic for roller bearings, gear transmissions, etc.). The problem is reduced to a system of nonlinear integro-differential and integral equations and inequalities in the contact domain, part of the external boundary, and a number of inner boundaries that are unknown in advance, but separate the lubricated and unlubricated zones. Special cases are problems of dry and completely lubricated contact. A formulation is given for the problem for the case when the materials of the bodies are identical. The problem of mixed friction is considered in strongly drawn out contact. Sections of the contact domain in which the interaction between the bodies is direct or by means of the lubrication layer are investigated using asymptotic methods.     

The solution of the Hertz axisymmetric contact problem

The main terms of the asymptotic form of the solution of the contact problem of the compression without friction of an elastic body and a punch initially in point contact are constructed by the method of matched asymptotic expansions using an improved matching procedure. The condition of unilateral contact is formulated taking account of tangential displacements on the contact surface. An asymptotic solution of the problem for the boundary layer is constructed by the complex potential method. An asymptotic model is constructed, extending the Hertz theory to the case where the surfaces of the punch and elastic body in the vicinity of the contact area are approximated by paraboloids of revolution. The problem of determining the convergence of the contacting bodies from the magnitude of the compressive force is reduced to the problem of calculating the so-called coefficient of local compliance, which is an integral characteristic of the geometry of the elastic body and its fixing conditions.     

Asymptotic analysis for a dynamic piezoelectric shallow shell

The paper deals with the asymptotic formulation and justification of a mechanical model for a dynamic piezoelastic shallow shell in Cartesian coordinates. Starting from the three‐dimensional dynamic piezoelastic problem and by an asymptotic approach, the authors study the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. In order to obtain a nontrivial limit problem by asymptotic analysis, we need different scalings on the mass density. The authors show that the transverse mechanical displacement field coupled with the in‐plane components solves an problem with new piezoelectric characteristics and also investigate the very popular case of cubic crystals and show that, for two‐dimensional shallow shells, the coupling piezoelectric effect disappears. Copyright © 2017 John Wiley & Sons, Ltd.     

A confinement problem for a linearly elastic Koiter's shell

In this Note, we propose a natural two-dimensional model of “Koiter's type” for a general linearly elastic shell confined in a half space. This model is governed by a set of variational inequalities posed over a non-empty closed and convex subset of the function space used for modeling the corresponding “unconstrained” Koiter's model. To study the limit behavior of the proposed model as the thickness of the shell, regarded as a small parameter, approaches zero, we perform a rigorous asymptotic analysis, distinguishing the cases where the shell is either an elliptic membrane shell, a generalized membrane shell of the first kind, or a flexural shell. Moreover, in the case where the shell is an elliptic membrane shell, we show that the limit model obtained via the asymptotic analysis of our proposed two-dimensional Koiter's model coincides with the limit model obtained via a rigorous asymptotic analysis of the corresponding three-dimensional “constrained” model.     

Steplike contrast structure in a singularly perturbed system of equations with different powers of small parameter

A singularly perturbed boundary value problem for a system of equations with different powers of a small parameter is considered in the one-dimensional case. The asymptotic behavior and existence of a solution with an internal transition layer are analyzed. The asymptotics are substantiated using the asymptotic method of differential inequalities.     

Self-similar solutions of the magnetohydrodynamic boundary layer system for a non-dilatable fluid

A rigorous mathematical analysis is given for a magnetohydrodynamic boundary layer problem, which arises in the study of self-similar solutions of the two-dimensional steady laminar boundary layer flow for an incompressible electrically conducting non-dilatable fluid (i.e., a Newtonian fluid or a pseudo-plastic one) along an isolated surface in the presence of an exterior magnetic field orthogonal to the flow. For this problem, only a normal solution has the physical meaning. The uniqueness, existence, and nonexistence results for normal solutions are established. Also the asymptotic behavior of the normal solution at the infinity is displayed. Received: January 10, 2007; revised: September 6, 2007, April 21, 2008     

Perturbation analysis of a modified second grade fluid over a porous plate

A modified second grade non-Newtonian fluid model is considered. The model is a combination of power-law and second grade fluids in which the fluid may exhibit normal stresses, shear thinning or shear thickening behaviors. The flow of this fluid is considered over a porous plate. Equations of motion in dimensionless form are derived. When the power-law effects are small compared to second grade effects, a regular perturbation problem arises which is solved. The validity criterion for the solution is derived. When second grade effects are small compared to power-law effects, or when both effects are small, the problem becomes a boundary layer problem for which the solutions are also obtained. Perturbation solutions are contrasted with the numerical solutions. For the regular perturbation problem of small power-law effects, an excellent match is observed between the solutions if the validity criterion is met. For the boundary layer solution of vanishing second grade effects however, the agreement with the numerical data is not good. When both effects are considered small, the boundary layer solution leads to the same solution given in the case of a regular perturbation problem.     

A convergence analysis of the iterative aggregation method with one parameter

A method for accelerating linear iterations in a Banach space is studied as a linear iterative method in an augmented space, and sufficient conditions for convergence are derived in the general case and in ordered Banach spaces. An acceleration of convergence takes place if an auxiliary functional is chosen sufficiently close to a dual eigenvector associated with a dominant simple eigenvalue of the iteration operator; in this case, the influence of this eigenvalue on the asymptotic rate of convergence is eliminated. Quantitative estimates and bounds on convergence are given.     

Interaction of a Solitary Wave with an External Force Moving with Variable Speed

The interaction of a solitary wave with an external force moving with constant acceleration is studied within the forced Korteweg-de Vries equation. For the case of a weak isolated force an asymptotic model based on equations for the amplitude and position of the solitary wave is developed. Phase portraits for this asymptotic system are obtained analytically and numerically. Analysis has shown that an accelerated force of either sign can capture a solitary wave if the acceleration is less than a certain critical value, depending on the forcing amplitude (for the case of a constant force speed only a positive force can capture a solitary wave). Direct numerical simulation of the forced Korteweg-de Vries equation has confirmed the predictions of the asymptotic model. Also, it is shown numerically that the accelerated force can capture more than one solitary wave.     

Mathematical model of a kinetic boundary layer

The two-dimensional (plane) problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a homogeneous polyatomic gas flow with no dissociation or electron excitation is considered assuming that energy exchange between translational and internal molecular degrees of freedom is easy. (The approximation of a hypersonic kinetic boundary layer arises from the kinetic theory of gases and, within the thin-layer model, takes into account the strong nonequilibrium of the hypersonic flow with respect to translational and internal degrees of freedom of the gas particles.) A method is proposed for constructing the solution of the given kinetic problem in terms of a given solution of an equivalent well-studied classical Navier-Stokes hypersonic boundary layer problem (which is traditionally formulated on the basis of the Navier-Stokes equations).     

One singularly perturbed Stefan problem describing destruction of the fuel layer in a laser target

A mathematical model of degradation of a deuterium-tritium (D-T) fuel layer located on the interior wall of a spherical shell is proposed. Such a shell with a solid layer frozen on it is a laser target that is employed in controlled thermonuclear fusion. As the laser target is delivered from the cryogenic chamber to the focus of the laser beam, the chamber stays, during a certain time interval, in a warm-gas cloud. During this time interval, the D-T layer degrades, and, in particular, its surface becomes nonideal. The mathematical model is formulated as a Stefan problem for a system of parabolic equations with nonlinear initial-boundary conditions. Small-parameter methods are applied to obtain an analytic solution to this problem, and the time during which the changes in the geometric parameters of the target’s fuel layer do not exceed technologically admissible values is estimated.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

Asymptotics of the Solution to a Stationary Piecewise-Smooth Reaction-Diffusion-Advection Equation*

A singularly perturbed boundary value problem for a piecewise-smooth nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems in the case when the discontinuous curve which separates the domain is monotone with respect to the time variable is considered. The existence of a smooth solution with an internal layer appearing in the neighborhood of some point on the discontinuous curve is studied. An efficient algorithm for constructing the point itself a...     

On the contact interaction between a strip-shaped punch and a linearly-deformable base through a covering of variable thickness

The contact problem of the frictionless penetration of a punch with strip-shaped section into the surface of a linearly-deformable base protected by a thin elastic layer (covering) of variable thickness, the stiffness of which is comparable to or smaller than that of the supporting elastic body, is investigated. A Fredholm integral equation of the second kind is obtained for the unknown contact pressure with a coefficient in front of the leading term that is a fairly arbitrary function of the longitudinal coordinate. To solve it the Bubnov-Galerkin projection method is used in which the coordinate elements are chosen to be a system of orthogonal polynomials and delta-shaped functions [1, 2] (variational-difference method), together with an algorithm for the required asymptotic expansions [3] when the above-mentioned coefficient is small. In the special case of an elastic half-space protected by a covering of constant thickness, the results obtained are compared with the corresponding characteristics given in [4].     

Analysis of the structure of the boundary layer in the problem of the torsion of a laminated spherical shell

The structures of the boundary layer in the problem of the torsion of a radially stratified spherical segment (shell) with an arbitrary number of alternating hard and soft layers are investigated. It is shown that weakly attenuating boundary-layer solutions exist. Despite the fact that a stress state, self-balanced in the section, corresponds to these elementary solutions, they may penetrate fairly deeply and considerably change the stress–strain state pattern far from the ends. Using an asymptotic analysis of the problem, an applied theory of torsion is proposed which takes into account weakly attenuating boundary-layer solutions.     

A boundary layer problem arising in gravity-driven laminar film flow of power-law fluids along vertical walls

A rigorous mathematical analysis is given for a boundary layer problem for a third-order nonlinear ordinary differential equation, which arises in gravity-driven laminar film flow of power-law fluids along vertical walls. Firstly, the problem is transformed into a singular nonlinear two-point boundary value problem of second order. Next, the latter is proved to have a unique positive solution, for which some estimates are established. Finally, the result above-mentioned is turned over to the original problem. The conclusion of this paper is that the boundary layer problem has a unique normal solution if the power-law index is less than or equal to one and a generalized normal solution if the power-law index is greater than one. Also the asymptotic behavior of the normal solution at the infinity is displayed.The work was supported by NNSF of China.     

On the stability and the attraction domain of the stationary solution of a singularly perturbed parabolic equation with a multiple root of the degenerate equation

We construct and justify the asymptotics of a boundary layer solution of a boundary value problem for a singularly perturbed second-order ordinary differential equation for the case in which the degenerate (finite) equation has an identically double root. A specific feature of the asymptotics is the presence of a three-zone boundary layer. The solution of the boundary value problem is a stationary solution of the corresponding parabolic equation. We prove the asymptotic stability of this solution and find its attraction domain.