Crystallization waves in thin amorphous layers on heat conducting substrates

A crystallization process in thin films is considered, where, driven by the release of the latent heat of fusion, the transformation from an amorphous state to the crystalline state takes place in a progressing wave of invariant shape. The crystallization rate is determined by a rate equation. The influence of the heat loss due to heat conduction into the substrate is taken into account. The resulting system of an ordinary differential equation and an integro-differential equation is solved numerically using a collocation method. The propagation speed of the wave in dependence on a non-dimensional heat loss parameter is determined. It turns out that the existence of a self-sustaining crystallization wave requires the heat loss parameter to be smaller than a certain critical value. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear waves on shallow water under an ice cover. Higher order expansions

Non-linear wave processes on the surface of shallow water under a layer of ice are considered taking bending deformations and tension compression into account. A closed system of equations in the water level perturbations and the velocity potential is derived to describe them. From the consistancy conditions for this system, using the method of multiple scales and perturbation theory, a ninth-order non-linear evolution equation is obtained for describing the perturbations of the water level, taking into account higher order corrections in the small parameters. A periodic solution of the equation obtained is constructed, expressed in terms of Weierstrass elliptic functions. Solutions are obtained in the form of solitary waves, expressed in terms of hyperbolic functions, using a modification of the simplest equations method. It is shown that, for periodic and solitary waves, two forms of wave profiles exist depending on the parameters of the mathematical model.     

The transmission of a plane acoustic wave through a non-uniform thermoelastic layer

The reflection and refraction of a plane acoustic wave by a thermoelastic plane layer, non-uniform in thickness, bounded by non-viscous heat-conducting liquids, generally different, is considered. The system of equations for small perturbations of the thermoelastic medium is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by two methods: the spline-collocation method and the power-series method. Analytic expressions are obtained which describe the wave fields outside the layer. The results of calculations of the intensity transmission coefficient of the acoustic wave are presented.     

Internal Kelvin waves in a two-layer liquid model

The subinertial internal Kelvin wave solutions of a linearized system of the ocean dynamics equations for a semi-infinite two-layer f-plane model basin of constant depth bordering a straight, vertical coast are imposed. A rigid lid surface condition and no-slip wall boundary condition are imposed. Some trapped wave equations are presented and approximate solutions using an asymptotic method are constructed. In the absence of bottom friction, the solution consists of a frictionally modified Kelvin wave and a vertical viscous boundary layer. With a no-slip bottom boundary condition, the solution consists of a modified Kelvin wave, two vertical viscous boundary layers, and a large cross-section scale component. The numerical solutions for Kelvin waves are obtained for model parameters that take account of a joint effect of lateral viscosity, bottom friction, and friction between the layers.     

Direct Sturm–Liouville problem for surface Love waves propagating in layered viscoelastic waveguides

This paper presents theoretical model for shear-horizontal (SH) surface acoustic waves of the Love type propagating in lossy waveguides consisting of a lossy viscoelastic layer deposited on a lossless elastic half-space. To this end, a direct Sturm–Liouville problem that describes Love waves propagation in the considered viscoelastic waveguides was formulated and solved, what constitutes a novel approach to the state-of-the-art. To facilitate the solution of the complex dispersion equation, the Author employed an original approach that relies on the separation of its real and imaginary part. By separating the real and imaginary parts of the resulting complex dispersion equation for a complex wave vector k = k₀ + jα of the Love wave, a system of two real nonlinear transcendental algebraic equations for k₀ and α has been derived. The resulting set of two algebraic transcendental equations was then solved numerically. Phase velocity v_p and coefficient of attenuation α were calculated as a function of the wave frequency f, thickness of the surface layer h and its viscosity η₄₄. Dispersion curves for Love waves propagating in lossy waveguides, with a lossy surface layer deposited on a lossless substrate, were compared to those corresponding to Love surface waves propagating in lossless waveguides, i.e., with a lossless surface layer deposited on a lossless substrate. The results obtained in this paper are original and to some extent unexpected. Namely, it was found that: 1) the phase velocity v_p of Love surface waves increases as a function of viscosity η₄₄ of the lossy surface layer, and 2) the coefficient of attenuation α has a maximum as a function of thickness h of the lossy surface layer. The results obtained in this paper are novel and can be applied in geophysics, seismology and in the optimal design and development of viscosity sensors, bio and chemosensors.     

Continuum models in problems of the hypersonic flow of a rarefied gas around blunt bodiest

All possible continuum (hydrodynamic) models in the case of two-dimensional problems of supersonic and hypersonic flows around blunt bodies in the two-layer model (a viscous shock layer and shock-wave structure) over the whole range of Reynolds numbers, Re, from low values (free molecular and transitional flow conditions) up to high values (flow conditions with a thin leading shock wave, a boundary layer and an external inviscid flow in the shock layer) are obtained from the Navier-Stokes equations using an asymptotic analysis. In the case of low Reynolds numbers, the shock layer is considered but the structure of the shock wave is ignored. Together with the well-known models (a boundary layer, a viscous shock layer, a thin viscous shock layer, parabolized Navier-Stokes equations (the single-layer model) for high, moderate and low Re numbers, respectively), a new hydrodynamic model, which follows from the Navier-Stokes equations and reduces to the solution of the simplified (“local”) Stokes equations in a shock layer with vanishing inertial and pressure forces and boundary conditions on the unspecified free boundary (the shock wave) is found at Reynolds numbers, and a density ratio, k, up to and immediately after the leading shock wave, which tend to zero subject to the condition that (k/Re)^1/2 → 0. Unlike in all the models which have been mentioned above, the solution of the problem of the flow around a body in this model gives the free molecular limit for the coefficients of friction, heat transfer and pressure. In particular, the Newtonian limit for the drag is thereby rigorously obtained from the Navier-Stokes equations. At the same time, the Knudsen number, which is governed by the thickness of the shock layer, which vanishes in this model, tends to zero, that is, the conditions for a continuum treatment are satisfied. The structure of the shock wave can be determined both using continuum as well as kinetic models after obtaining the solution in the viscous shock layer for the weak physicochemical processes in the shock wave structure itself. Otherwise, the problem of the shock wave structure and the equations of the viscous shock layer must be jointly solved. The equations for all the continuum models are written in Dorodnitsyn--Lees boundary layer variables, which enables one, prior to solving the problem, to obtain an approximate estimate of second-order effects in boundary-layer theory as a function of Re and the parameter k and to represent all the aerodynamic and thermal characteristic; in the form of a single dependence on Re over the whole range of its variation from zero to infinity.

An efficient numerical method of global iterations, previously developed for solving viscous shock-layer equations, can be used to solve problems of supersonic and hypersonic flows around the windward side of blunt bodies using a single hydrodynamic model of a viscous shock layer for all Re numbers, subject to the condition that the limit (k/Re)^1/2 → 0 is satisfied in the case of small Re numbers. An aerodynamic and thermal calculation using different hydrodynamic models, corresponding to different ranges of variation Re (different types of flow) can thereby, in fact, be replaced by a single calculation using one model for the whole of the trajectory for the descent (entry) of space vehicles and natural cosmic bodies (meteoroids) into the atmosphere.     

Method of successive approximations for three-dimensional laminar boundary layer problems (locally self-similar case) : PMM vol. 37, n6, 1973, pp. 974–983

We present an analytical method for the computation of problems of incompressible boundary layer theory based on an application of the method of successive approximations. The system of equations is reduced to a form suitable for integration. Parameters characterizing the external flow and the body geometry are contained only in the coefficients of the system and do not enter into the boundary conditions. The transformed momentum equations are integrated across the boundary layer from a current value to infinity with the boundary conditions taken into account. If the integration is made from zero to infinity, then the equations pass over into the Kármán relations. Integrating the system of equations a second time, using the boundary conditions at the wall, we obtain a system of nonlinear integro-differential equations. To solve this system of equations we apply the method of successive approximations. To satisfy the boundary Conditions at infinity we introduce, at each step of the iterations, unknown “governing” functions. From the conditions at the outer side of the boundary layer we obtain additional equations for their determination. With the iterational algorithm formulated in this way, the boundary conditions, both on the body and at the outer side of the boundary layer; are satisfied automatically.We consider a locally self-similar approximation. In this case, relative to the “governing” functions, we obtain an algebraic system of equations. We write out the solution in the first approximation. The results obtained in the first approximation are compared with the results of finite-difference computations for a wide range of problems. The results obtained in this paper are compared with those obtained in [1] for the flow in the neighborhood of a stagnation point. An indication is given of the nonuniqueness of the solutions of the three-dimensional boundary layer equations.     

Near-surface buckling in stability of a system consisting of a moderately rigid substrate, a viscoelastic bond layer, and an elastic covering layer

Within the frame work of the three-dimensional linearized theory of stability of deformable bodies (TLTSDB), the near-surface buckling instability of a system consisting of a half-plane (substrate), a viscoelastic bond layer, and an elastic covering layer is suggested. The equations of the TLTSDB are obtained from the three-dimensional geometrically non linear equations of viscoelasticity theory by using the boundary-form perturbation technique. By employing the Laplace transform, a method for solving the problem is developed. It is supposed that the covering layer has an insignificant initial imperfection. The stability of the system is considered lost if the imperfection starts to increase and grows indefinitely. Numerical results for the critical compressive force and the critical time are presented. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 517–530, July–August, 2006.     

The impedance boundary-value problem of diffraction by a strip

We consider the impedance boundary-value problem for the Helmholtz equation originated by the problem of wave diffraction by an infinite strip with imperfect conductivity. The two possible different situations of real and complex wave numbers are considered. Bessel potential spaces are used to deal with the problem, and the identification of corresponding operators of single and double layer potentials allow a reformulation of the problem into a system of integral equations. The well-posedness of the problem is obtained for a set of impedance parameters (and wave numbers), after the incorporation of some compatibility conditions on the data. At the end, an improvement of the regularity of the solution is derived for the same set of parameters previously considered.     

Green's function approach to study the propagation of SH‐wave in piezoelectric layer influenced by a point source

Green's function technique serves as a powerful tool to find the particle displacements due to SH‐wave propagation in layer of a shape different from the space between two parallel planes. Therefore, the present paper undertook to study the propagation of SH‐wave in a transversely isotropic piezoelectric layer under the influence of a point source and overlying a heterogeneous substrate using Green's function technique. The coupled electromechanical field equations are solved with the aid of Green's function technique. Expression for displacements in both layer and substrate, scalar potential and finally the dispersion relation is obtained analytically for the case when wave propagates along the direction of layering. Numerical computations are carried out and demonstrated with the aid of graphs for six different piezoelectric materials namely PZT‐5H ceramics, Barium titanate (BaTiO₃) ceramics, Silicon dioxide (SiO₂) glass, Borosilicate glass, Cobalt Iron Oxide (CoFe₂O₄), and Aluminum Nitride (AlN). The effects of heterogeneity, piezoelectric and dielectric constants on the dispersion curve are highlighted. Moreover, comparative study is carried out taking the phase velocity for different piezoelectric materials on one hand and isotropic case on the other. Dispersion relation is reduced to well‐known classical Love wave equation with a view to illuminate the authenticity of problem. Copyright © 2017 John Wiley & Sons, Ltd.     

A model for describing near-resonance oscillations in an elastic layer

One-dimensional transverse oscillations in a layer of a non-linear elastic medium are considered, when one of the boundaries is subjected to external actions, causing periodic changes in both tangential components of the velocity. In a mode close to resonance, the non-linear properties of the medium may lead to a slow change in the form of the oscillations as the number of the reflections from the layer boundaries increases. Differential equations describing this process were previously derived. The equations obtained are hyperbolic and the change in the solution may both keep the functions continuous and lead to the formation of jumps. In this paper a model of the evolution of the wave patterns is constructed as integral equations having the form of conservation laws, which determine the change in the functions describing the oscillations of the layer as “slow” time increases. The system of hyperbolic differential equations previously obtained follows from these conservation laws for continuous motions, in which one of the variables is slow time, for which one period of the actual time serves as an infinitesimal quantity, while the second variable is the real time. For the discontinuous solutions of the same integral equations, conditions on the discontinuity are obtained. An analogy is established between the solutions of the equations obtained and non-linear waves propagating in an unbounded uniform elastic medium with a certain chosen elastic potential. This analogy enable discontinuities which may be physically realised to be distinguished. The problem of steady oscillations of an elastic layer is discussed.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

Diffraction of a plane acoustic wave by a non-uniform thermoelastic cylindrical layer bounded by inviscid heat-conducting fluids

The diffraction of sound by a radially laminated isotropic thermoelastic cylindrical shell is considered. The system of equations for small perturbations of a hollow thermoelastic cylinder is reduced to a system of ordinary differential equations, the boundary-value problem for which is solved by the spline-collocation method. Expressions are obtained describing the wave field outside the cylindrical layer. Results of calculations of polar radiation patterns of the amplitude of the scattered acoustic wave in the far zone are presented.     

Resonance excitation and mode selection in smart structures with piezoelectric patch actuators

A system of piezoelectric flexible patch actuators bonded to an elastic layered substrate is considered. An integral equation based model for the smart structure under consideration has been developing. The rigorous solution to the patch–substrate dynamic contact problem extends the range of the model's utility far beyond the bounds of conventional simplified models that rely on plate, beam or shell equations for the waveguide part. The developed approach provides the possibility to reveal the effects of resonance energy radiation associated with higher modes that would be inaccessible using models accounting for the fundamental modes only. Algorithms that correctly account for the mutual wave interaction among the actuators via the host medium, for selective mode excitation in a layer as well as for body waves directed to required zones in a half–space, have also been elaborated and implemented in computer code. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Indentation of an incompressible inhomogeneous layer by a rigid circular indenter

This work deals with an incompressible inhomogeneous layer bondedto a rigid substrate and indented without friction by a rigidcircular indenter. The corresponding mixed boundary-value problemof elasticity is reduced to equivalent dual integral equations.It is shown that the pliability function in these equationsmay be found from a system of nonlinear differential equationsand that its behaviour is peculiar when the elastic medium isincompressible. A novel technique taking into account this peculiarityis developed in order to reduce the dual integral equationsto Fredholm integral equations of the second kind with symmetricstrictly coercive operators. For a homogeneous layer and a flatindenter, the structure of the Fredholm integral equations permitsan approximate analytical solution which is very accurate forany layer thickness. For an indenter of three-dimensional profile,leading asymptotic terms of the solution are derived in thecase of a thin inhomogeneous layer.     

非线性波方程准确孤立波解的符号计算

该文将机械化数学方法应用于偏微分方程领域，建立了构造一类非线性发展方程孤立波解的一种统一算法，并在计算机数学系统上加以实现，推导出了一批非线性发展方程的精确孤立波解．算法的基本原理是利用非线性发展方程孤立波解的局部性特点，将孤立波表示为双曲正切函数的多项式．从而将非线性发展方程（组）的求解问题转化为非线性代数方程组的求解问题．利用吴文俊消元法在计算机代数系统上求解非线性代数方程组，最终获得非线性发展方程（组）的准确孤立波解.     

Contact problem for a layer with two stamps : PMM vol. 42, no. 6, 1978, pp. 1068–1073

A problem of impressing coaxial stamps of circular cross section into the upper and lower surface of a homogeneous elastic layer is studied. The bases of the stamps have axial symmetry. The parts of the layer surfaces lying oustide the contact zone are stress-free, there is no friction or coupling between the layer and the stamps. A system of two integral equations with two unknown functions is obtained, and provides a solution of the problem. The method of separating the singularities provides the way of reducing this system to the Fredholm equations of second kind. An approximate solution of the equations is obtained for the case of flat stamps under the assumptions that the two parameters entering the system are sufficiently small.

Problems of a layer with various boundary conditions were formulated and solved in many papers and books, e.g. [1, 2]. However, to the best of the author's knowledge, in all these problems the conditions at the boundary were assumed different only on one side of the layer; in the present problem the boundary conditions are mixed at both sides of the layer, and this results in a system of two integral equations.     

Wave–Mean-Flow Interactions in a Forced Rossby Wave Packet Critical Layer

This paper describes the nonlinear critical layer evolution of a zonally localized Rossby wave packet forced in mid-latitudes and propagating horizontally on a beta plane in a zonal shear flow. The wave packet has an amplitude that varies slowly in the zonal direction. Numerical solutions of the governing nonlinear equations show that the wave–mean-flow interactions differ from those that would result with a monochromatic forcing. With the localized forcing, the net absorption of the disturbance at the critical layer continues for large time, because there is an outward flux of momentum in the zonal direction. Further insight into the mechanism for this and other aspects of the evolution of the critical layer is obtained through an approximate asymptotic analysis which is valid for large time.     

Traveling Wave Solutions for Combustion in a Porous Medium

Traveling wave solutions are sought for a model of combustion in a porous medium. The problem is formulated as a nonlinear eigenvalue problem for a system of ordinary differential equations of order four, defined over an infinite interval. A shooting method is used to prove existence, and a priori bounds for the solution and parameters are obtained.