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Linear and non-linear stability analysis of the interface between a thin vapor film and a layer of liquid in the presence of a steady heat flux from a metal surface heated to a high temperature, to the vapor film and then from vapor to the subcooled liquid is investigated. The temperature dependence of saturation pressure is taken into account. Boundary conditions on the vapor–liquid interface that generalizes the known correlations on the free surface of liquid in the gravity field are derived. The thermal processes on the phase boundary lead to the generation of weakly decaying periodic surface waves of low amplitude and may cause small length waves (ripple), which are not capillary ones. Thermal processes on the phase interface are capable to provide the stability of a film of lights vapor under a layer of heavy liquid in the gravity field. The explosive instability may arise in the non-linear stage due to a weak variation of the film thickness or superheating of liquids.

Based on compressive tests, static on 810 material test system (MTS) and dynamic on the first compressive loading in split Hopkinson pressure bar (SHPB) tests at different strain rates and temperatures for Lanthanum (La) cylinder specimens, this study determined Johnson-Cook (J-C) type compressive constitutive equation of La metal. The determined compressive constitutive equation of La metal was calibrated in numerically simulating the recorded large deformations of La cylinder specimens generated by the multi-compressive loadings in SHPB tests. Based on tensile tests, static on MTS and dynamic on the first tensile loading in optimized tensile split Hopkinson bar (TSHB) tests at different strain rates for La sheet specimens, this study determined the J-C type tensile constitutive equation of La metal. It was found that the La sheet specimens were fractured during the first tensile loading in TSHB tests. The numerical simulations of transmitted and reflected pulses of TSHB tests for La sheet specimens using strain rate dependent failure strain criterion were consistent with the experimental data. The relation between dynamic failure strength and strain rates was discussed. From scanning electron microscope investigation of fractured La specimens, it was found that damage evolution patterns at high loading rates tend to become regular.

In this work, we apply “the modified reductive perturbation method” to the generalized Boussinesq equation and obtain various form of generalized KdV equations as the evolution equations. Seeking a localized travelling wave solutions for these evolution equations we determine the scale parameters g 1 and g 2 , which corresponds to the correction terms in the wave speed, so as to remove the possible secularities that might occur. Depending on the sign and the values of certain parameters the resulting solutions are shown to be a solitary wave or a periodic solution. The suitability of the method is also shown by comparing the results with the exact travelling wave solution for the generalized Boussinesq equation.

This paper is concerned with asymptotic stability and stabilization of nonlinear uncertain stochastic switched discrete time-delay systems. The system to be considered is subject to interval time-varying delays, which allows the delay to be a fast time-varying function and the lower bound is not restricted to zero. Based on the discrete Lyapunov functional, a switching rule for the asymptotic stability and stabilization for the nonlinear uncertain stochastic discrete time-delay system is designed via linear matrix inequalities. Numerical examples are included to illustrate the effectiveness of the results.

A novel mechanistic two-fluid model (CB model) similar, in spirit, to the Taitel and Dukler [1, 2] (TD model), for the identification of possible multiple solutions of a multi-phase (gas-liquid) stratified flow in slightly inclined pipelines, is proposed. While Blasius-type closures are used in the TD model to represent the wall friction coefficients, the newly-implemented CB model makes use of Colebrook-White-type closures. Moreover, different closures for the interfacial shear are also employed in the CB models. The predictive capabilities of the CB model have been tested by using several experimental data, finding a better agreement between measured and calculated data than that existing when the TD model is used. The region of multiple solutions is influenced by the closures in use, such a dependence is more evident when different interfacial friction factors are used. Moreover, for the CB model also the fluid mixture in use influences the boundaries of the non-uniqueness region, while by using the TD model the multiple-solution region is unchanged. The choice of closures for the interfacial friction strongly influences the holdups, the Andritsos and Hanratty [10] correlation significantly shifting the non-uniqueness region to small values of the inclination parameter. Such a behaviour is more and more significant with the increase of the superficial gas velocity, even if for values of the inclination parameter within the range of inclinations for stratified flows (i.e. less than about 30° from the horizontal [11]), multiple solutions were not found. Finally, for the fluid mixture and flow conditions analyzed, multivalued solutions are obtained only for upward flows. Moreover, the portion of multiple-solution region interested by co-current flow (that occurs for slightly upward and downward pipes) is rather small, so that the operational point unlikely falls within such a region in the case of the studied hydrocarbon gas-liquid mixture.

In this paper, adaptive synchronization of Filippov systems with unknown parameters is investigated via sliding mode control. In order to obtain smooth error systems, Filippov systems are redesigned by the Generalized Hamiltonian system. The sliding mode control is applied to the error systems and stabilizes their zero solutions. Adaptive synchronization of chaotic systems via sliding mode control is investigated with the applications to Chua's system and the Duffing Oscillator with dry friction. Numerical simulations are also given to identify the effectiveness of the theoretical analysis.

This article studies the numerical solution of the two-dimensional sine-Gordon equation (SGE) using a split-step Chebyshev Spectral Method. In our method we split the 2D SGE by considering one dimension at a time, first along x and then along y. In each fractional step we solve a 1D SGE. Time integration is handled by a finite difference scheme. The numerical solution is then compared with many of the known numerical solutions found throughout the literature. Our method is simple to implement and second order accurate in time and has spectral convergence. Our method is both fast and accurate.

The logistic map, in this article, is used to show the propagation of the numerical error for numerically solved logistic ODE's. When this equation is solved numerically (discretization procedure) the solution shows a chaotic behavior by varying the control parameter (integration step-size). However, the logistic ODE dx / dt  = ( k  − 1) x  −  kx 2 also has an analytical solution, i.e., not presenting the signature of chaos. The connection between the two limits, analytical solution and chaos, allows us to establish a region of acceptability for the numerical error. Therefore the limits of the numerical process can be found in between the limits of the analytical and chaotic solutions. This study can be extended to any other ODE or map. For example, there is a similar numerical connection between the chaotic bi-dimensional Cat Map with its respective integrable ODE, which has an analytical solution. Our discussion applied to those particular systems is generic, that is, the solutions obtained from the numerical discretization can be more instable than those obtained from the continuous procedure. The important point is to find a compromise between the step-size and the propagation length in order to obtain the best numerical solution since the instabilities can be unavoidable.

In this paper, by using the factorization technique combined with the method of complete discrimination system for polynomial, some exact travelling wave solutions for the modified Fornberg-Whitham equation are obtained. Moreover, our results extend previously known results in the literature.