Saadet Erbay

LONGITUDINAL WAVE PROPAGATION IN A GENERALIZED THERMOELASTIC CYLINDER

Abstract In this paper the longitudinal wave propagation in a circular infinite cylinder is studied. The infinite circular cylinder is assumed to be made of a generalized thermoelastic material. The dispersion relation is obtained for the case in which the temperature is kept constant on the surface of the cylinder. Because of the complexity of the dispersion relation, the numerical solutions are given. For various values of parameters appearing in the field equations, some dispersion, attenuation, and phase velocity diagrams are presented

Wave propagation in fluid filled nonlinear viscoelastic tubes

SummaryThe present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic tube filled with an incompressible, inviscid fluid. In order to include the geometric dispersion in the analysis, the tube wall inertia effects are added to the pressure-area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is examined. In the long-wave approximation, a general equation is obtained, and it is shown that by a proper scaling this equation reduces to the well-known nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. In the absence of nonlinear viscoelastic effects all the equations reduce to those of the linear viscoelastic tube.

Two-dimensional wave packets in an elastic solid with couple stresses

Abstract The problem of (2+1) (two spatial and one temporal) dimensional wave propagation in a bulk medium composed of an elastic material with couple stresses is considered. The aim is to derive (2+1) non-linear model equations for the description of elastic waves in the far field. Using a multi-scale expansion of quasi-monochromatic wave solutions, it is shown that the modulation of waves is governed by a system of three non-linear evolution equations. These equations involve amplitudes of a short transverse wave, a long transverse wave and a long longitudinal wave, and will be called the “generalized Davey–Stewartson equations”. Under some restrictions on parameter values, the generalized Davey–Stewartson equations reduce to the Davey–Stewartson and to the non-linear Schrodinger equations. Finally, some special solutions involving sech–tanh–tanh and tanh–tanh–tanh type solitary wave solutions are presented.

Global existence and nonexistence results for a generalized Davey-Stewartson system

We consider a system of three equations, which will be called generalized Davey–Stewartson equations, involving three coupled equations, two for the long waves and one for the short wave propagating in an infinite elastic medium. We classify the system according to the signs of the parameters. Conserved quantities related to mass, momentum and energy are derived as well as a specific instance of the so-called virial theorem. Using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic–elliptic–elliptic case.

Thermally induced vibrations in a generalized thermoelastic solid with a cavity

The present work deals with thermally induced vibrations in an infinite solid with a cavity. The medium is assumed to be linear, isotropic, temperature-rate-dependent thermoelastic. The problem is solved for the cases of cylindrical and spherical cavities. The surface of the cavity is assumed to be subjected to a temperature varying harmonically with time, and free of stress. For the cases considered, the coupled field equations admit exact solutions in terms of Hankel and the spherical Hankel functions, respectively. Numerical results are compared with those of classical ther-moelaslicity. The contribution of the second sound parameters in these problems becomes more significant as the frequency of applied termperature increases.

Nonlinear interaction between long and short waves in a generalized elastic solid

Abstract In the present study, nonlinear interaction between long and short waves propagating in a generalized elastic medium is examined. In particular, the case where the phase velocity of the long longitudinal wave is equal to the group velocity of the short transverse waves is studied. By using an asymptotic expansion method, three coupled nonlinear evolution equations are derived for the description of the interaction. In the absence of one of the transverse waves, these equations reduce to the so-called long wave–short wave interaction equations which are also known as Zakharov–Benney (ZB) equations. Furthermore, the nonlinear interaction between a long longitudinal wave and a short longitudinal wave is considered and ZB equations are derived for the description of interaction. Some special solutions to the interaction equations are also presented.

Nonlinear wave propagation in micropolar media. I: The general theory

Abstract In this work, the plane wave propagation in nonlinear micropolar solids is asymptotically investigated. Micropolar theory in the linear approximation predicts a dispersive optical (high-frequency) mode as well as a dispersive acoustical (low-frequency) mode for the harmonic waves in an unbounded medium. The acoustical mode has a weakly dispersive region when the wave number is small. If nonlinearity is also present in this weakly dispersive region and if both effects are small but finite, it may be expected that nonlinearity and dispersive effects can balance each other, and the wave propagation can be asymptotically governed by a nonlinear evolution equation which admits a solitary wave type solution. Using the reductive perturbation method to examine the plane wave propagation in a general nonlinear polar solid, it is found that far-field approximation of wave motion is governed by coupled Modified Korteweg-de Vries equations.

Nonlinear wave modulation in fluid filled distensible tubes

SummaryThe present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic or elastic tube filled with an incompressible, inviscid fluid. Using the reductive perturbation technique, and assuming the weakness of dissipative effects, the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a dissipative nonlinear Schrödinger equation (NLS). In the absence of dissipative effects, this equation reduces to the classical NLS equation. The examination of the coefficients of the dissipative and classical NLS equations reveals the significance of the tube wall inertia to obtain a balance between nonlinearity and dispersion. Some special solutions of the NLS equation are given and the modulational instability of the plane wave solution is discussed for various incompressible hyperelastic materials.

Investigation of solutal convection during the dissolution of silicon in a sandwich system

Abstract This paper considers natural convection due to solutal gradients during the dissolution of silicon in a substrate-solution-substrate ‘sandwich’ system under isothermal conditions. This work is motivated by the need to understand the role of convection in liquid-phase epitaxial growth of semiconductor crystals. Unsteady two-dimensional numerical simulations are presented for two dissolution experiments. Solutal convection is found to be predominant during the initial phase of the process and causes a rapid increase in the dissolution depths of both substrates. However, because convection is mostly confined to the lower half of the sandwich system, lower substrate dissolution depths are about twice as large. This result is in good agreement with available experimental data. Another interesting consequence of convection is the development of a wavy irregular surface along the lower substrate.

Standing waves for a generalized Davey–Stewartson system

In this paper, we establish the existence of non-trivial solutions for a semi-linear elliptic partial differential equation with a non-local term. This result allows us to prove the existence of standing wave (ground state) solutions for a generalized Davey–Stewartson system. A sharp upper bound is also obtained on the size of the initial values for which solutions exist globally.