Volume 14, Issue 7-8

This paper presents a research for plastic-viscoelastic magnetohydrodynamic (MHD) flow with radiation thermal transfer. A new model is firstly proposed by modifying the classical Maxwell viscoelastic fluid model with the power-law fluid as the damping fluid to substitute the Newtonian fluid as a dashpot. The serial connection gives the nonlinear partial differential equation models. Taylor expansion is used to reduce the nonlinear equation models into linear ones and the analytical solutions are obtained for velocity and temperature fields in terms of G function and Pochhammer polynomial. Moreover, some graphs are presented for associated parameters and the corresponding flow and heat transfer characteristics are analyzed. The ideas and methods of this paper may be used to study other fluid model, such as the generalized Oldroyd-B Fluid or the generalized Burgers' fluid.

The first problem of Stokes for an incompressible Oldroyd-B fluid with fractional derivatives is studied using Fourier sine and Laplace transforms. The solutions that have been obtained, unlike the known solutions from the literature, are presented as a sum between the Newtonian solutions and the non-Newtonian contributions. The non-Newtonian contributions, as expected, tend to zero for α = β and λ → λ r . Furthermore, the solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell, fractional and ordinary second grade fluid, performing the same motion, are also obtained as limiting cases of general solutions. The present solution for ordinary Oldroyd-B and second grade fluids are verified by comparison with the previously well known results. Finally, the influence of material and fractional parameters on the fluid motion, as well as a comparison among fractional Oldroyd-B, fractional Maxwell, fractional second grade and Newtonian fluids, is also analyzed by graphical illustrations.

A modified variational iteration method (MVIM) is proposed for nonlinear oscillators. Different from the existing VIM, the algebraic equation deduced at each iteration stage is always linear when the proposed approach is applied to conservative oscillators. Therefore, highly accurate approximations can be easily obtained. The Duffing and the van der Pol equations are presented to validate the MVIM. Numerical examples show that the presented method is valid for both weakly and strongly nonlinear oscillators. In addition, it reveals that the MVIM is equivalent to a modified Mickens iteration scheme. Furthermore, the MVIM is extended by means of an auxiliary parameter. This parameter provides us with a convenient way to control and adjust the convergence of VIM approximations when necessary.

Existing models of continuous nonlinear functions used for generating multi-scroll chaotic attractors are based on a piece-wise linear (PWL) approach. These models, although relatively easy to build, do not include any information related to the performance parameters of active devices, in the context of a possible physical implementation. This is a serious drawback, since the use of a PWL model introduces a level of inaccuracy into a numerical analysis which is more evident when numerical and experimental results are compared. This paper proposes a methodology to generate the behavioral model of continuous nonlinear functions, but unlike PWL approaches, real physical active device parameters are herein taken into account. In particular, we generate the behavioral model of a nonlinear function called saturated nonlinear function series (SNFS), but in general, the proposed approach can be used to generate the behavioral model of other continuous nonlinear functions. Our results indicate that the proposed approach yields a more realistic and accurate behavioral model than PWL models. As a consequence, not only the generation of chaotic attractors is more precise, but the metrics used to measure the complexity of a chaotic system can also be better predicted. Numerical and electrical simulation results at both domains, phase and time, illustrate the benefits of the new proposed model.

This paper presents approximate analytical solutions for systems of fractional differential equations using the homotopy perturbation method. The fractional derivatives are described in the Caputo sense. The application of homotopy perturbation method, developed for differential equations of integer order, is extended to derive approximate analytical solutions of systems of fractional differential equations. The solutions of our model equations are calculated in the form of convergent series through easily computable components. Using symbolic computation, some examples are solved as illustrations. The numerical results show that the approach is accurate and easy to implement, when applied to systems of fractional differential equations. The method introduces a promising tool for solving many linear and nonlinear fractional differential equations.

In this paper, an attempt is made to obtain the effect of a uniform magnetic field on the solution of non-stationary motion of micropolar fluid in the half-plane when the normal stresses and tangential velocities are given on the boundary. The Laplace-Fourier transform technique is used to point out the solution by quadratures. Numerical results of the physical quantities such as tangential and normal velocities, pressure, microrotation, stresses and momentums included the influence of a uniform magnetic field are obtained and displayed graphically. The problem could be met in the study of the vibrations of a memberance or a plane contacting with the micropolar fluid.

High-order modified Navier-Stokes (NS) equations are introduced to describe the transitional flow derived from Burnett equations considering the inaccurate simulation of shock structure using traditional NS continuum methods, particularly at high Mach numbers. This process is performed by defining the effective transportation coefficient, which is associated with Mach number, molecule propriety, and continuum failure criterion. Within this method, the linear Stokes and Fourier relationships are replaced by nonlinear constitutive relationships to describe transitional flow. The traditional NS equation formulation is preserved. The shock structures of both monatomic and diatomic gases are concentrated with different Mach numbers to validate the accuracy and effectiveness of the derived method. The comparison of shock structure with DSMC and Burnett equations shows that the high-order modified NS equations obtain better results than traditional NS equations in terms of shock wave, particularly at high Mach numbers. The modified NS equations also capture the significant features of Burnett equations in shock structure simulation.

In the present paper, a very new technique, Coupled Fractional Reduced Differential Transform, has been implemented to obtain the numerical approximate solution of coupled time fractional kdV equations. The fractional derivatives are described in the Caputo sense. By using the present method we can solve many linear and nonlinear coupled fractional differential equations. The obtained results are compared with the exact solutions. Numerical solutions are presented graphically to show the reliability and efficiency of the method.

It is shown that the Kronecker product can be applied to construct new nonlinear integrable coupling system of soliton equation hierarchy in this paper. A direct application to the KN spectral problem leads to a novel KN soliton equation hierarchy of nonlinear integrable coupling system, which is different from any one obtained before. Furthermore, we present the Hamiltonian structures of nonlinear integrable couplings of KN hierarchy by using a pairs from special non-semisimple matrix Lie algebras and variational identities over the associated loop algebras. It is also indicated that the study of nonlinear integrable couplings using the Kronecker product is an efficient and straightforward method.

In this study natural convection heat transfer in a cold outer circular enclosure containing a hot inner sinusoidal circular cylinder is investigated numerically using the Control Volume based Finite Element Method (CVFEM). The fluid in the enclosure is Cu-water nanofluid. Both circular enclosure and inner cylinder are maintained at constant temperature. The governing equations of fluid motion and heat transfer in their vorticity stream function form are used to simulate the fluid flow and heat transfer. The effective thermal conductivity and viscosity of nanofluid are calculated using the Maxwell–Garnetts (MG) and Brinkman models, respectively. The calculations were performed for different governing parameters such as the Rayleigh number, values of the number of undulations of the inner cylinder and nanoparticle volume fraction. The results show that for all Rayleigh number, increasing number of undulations causes the average Nusselt number to augment except that the average Nusselt number for N  = 4 is greater than N  = 5.

In this article, a novel synchronization scheme is proposed to achieve hybrid modulus-phase synchronization in hyperchaotic complex systems, which means different synchronization schemes can be achieved simultaneously in modulus and phase spaces of complex systems. A general controller is designed to implement modified projective synchronization in modulus space and anti-synchronization in phase space. And the theoretical proof is given based on the Lyapunov stability theory. By utilizing the proposed synchronization scheme, an improved digital secure communication scheme is presented, the main advantage of which is to develop the transmission efficiency and reduce the design complexity. Numerical simulations are carried out to verify the effectiveness and feasibility of the proposed schemes.