Experimental classification of nonlinear dynamics in microfluidic bubbles’ flow

A nonlinear analysis method based on the evaluation of d-infinite and largest Lyapunov exponent was used to study the complex dynamics of air bubbles carried by water and flowing in a microfluidic snake channel. A rich variety of nonlinear dynamics and flow patterns was found through the experimental observation of bubbles’ motion. The results and their graphical representation show the capability of the proposed set of dimensionless parameters to classify the nonlinearity of the process showing also its sensitivity to input flow variations.     

Wave equation in fractional Zener-type viscoelastic media involving Caputo–Fabrizio fractional derivatives

Meccanica - We investigate propagation of waves in the Zener-type viscoelastic media through a model which involves fractional derivatives with a regular kernel. The restrictions on the...     

Pattern formation in a nonequilibrium phase transition for a generalized Burgers–Fisher equation

A nonequilibrium phase transition of a generalized Burgers–Fisher equation describing biological pattern formation with a periodic boundary condition is examined. In the presence of a weak external force, some approximate bifurcation solutions near a critical point and new spatially periodic patterns are obtained by using the perturbation method in an infinite-dimensional space. The result shows that the external force delays the bifurcation.     

Exact solutions for generalized Burgers’ fluid in an annular pipe

This paper deals with some unsteady unidirectional transient flows of generalized Burgers’ fluid in an annular pipe. Exact solutions of some unsteady flows of generalized Burgers’ fluid in an annular pipe are obtained by using Hankel transform and Laplace transform. The following two problems have been studied: (1) Poiseuille flow due to a constant pressure gradient; (2) axial Couette flow in a annulus. The well known solutions for Navier-Stokes fluid, as well as those corresponding to a Maxwell fluid, a second grade fluid and an Oldroyd-B fluid appear as limiting cases of our solutions.     

Nonlinear boundary control of the unforced generalized Korteweg–de Vries–Burgers equation

In this paper, we consider the boundary control problem of the unforced generalized Korteweg–de Vries–Burgers (GKdVB) equation when the spatial domain is [0,1]. Three control laws are derived for this equation and the L ²-global exponential stability of the solution is proved analytically. Numerical results using the finite element method (FEM) are presented to illustrate the developed control schemes.     

An efficient differential quadrature method for fractional advection–diffusion equation

Nonlinear Dynamics - This article studies a direct numerical approach for fractional advection–diffusion equations (ADEs). Using a set of cubic trigonometric B-splines as test functions, a...     

Enriched reproducing kernel particle method for fractional advection–diffusion equation

The reproducing kernel particle method(RKPM) has been efficiently applied to problems with large deformations, high gradients and high modal density. In this paper, it is extended to solve a nonlocal problem modeled by a fractional advection–diffusion equation(FADE), which exhibits a boundary layer with low regularity. We formulate this method on a moving least-square approach. Via the enrichment of fractional-order power functions to the traditional integer-order basis for RKPM,leading terms of the solution to the FADE can be exactly reproduced, which guarantees a good approximation to the boundary layer. Numerical tests are performed to verify the proposed approach.     

On the MHD flow of fractional generalized Burgers＇ fluid with modified Darcy＇s law

This work is concerned with applying the fractional calculus approach to the magnetohydrodynamic (MHD) pipe flow of a fractional generalized Burgers’ fluid in a porous space by using modified Darcy’s relationship. The fluid is electrically conducting in the presence of a constant applied magnetic field in the transverse direction. Exact solution for the velocity distribution is developed with the help of Fourier transform for fractional calculus. The solutions for a Navier–Stokes, second grade, Maxwell, Oldroyd-B and Burgers’ fluids appear as the limiting cases of the present analysis.     

An improved theoretical model for the dynamics of a free–clamped cylinder in axial flow

A new, improved linear analytical model is presented in this paper for the dynamics of a slender cantilevered cylinder with an ogival free end, subjected to axial flow directed from its free end towards the clamped one. In the present model the fluid-dynamic forces at the free end of the cylinder are analysed in a meticulous manner. The model predicts that the cylinder loses stability at relatively low flow velocity by flutter, and then at higher flow velocity by static divergence. This agrees with the dynamical behaviour observed in experiments. Moreover, quantitative agreement in the critical flow velocities for flutter and divergence between this improved theory and experiment is fairly good.     

Differential transform method for solving Richards’ equation

An approximate solution to Richards’ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method (DTM) with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.     

Bilinear form and soliton interactions for the modified Kadomtsev–Petviashvili equation in fluid dynamics and plasma physics

In this paper, we investigate the modified Kadomtsev–Petviashvili (mKP) equation for the nonlinear waves in fluid dynamics and plasma physics. By virtue of the rational transformation and auxiliary function, new bilinear form for the mKP equation is constructed, which is different from those in previous literatures. Based on the bilinear form, one- and two-soliton solutions are obtained with the Hirota method and symbolic computation. Propagation and interactions of shock and solitary waves are investigated analytically and graphically. Parametric conditions for the existence of the shock, elevation solitary, and depression solitary waves are given. From the two-soliton solutions, we find that the (i) parallel elastic interactions can exist between the (a) shock and solitary waves, and (b) two elevation/depression solitary waves; (ii) oblique elastic interactions can exist between the (a) shock and solitary waves, and (b) two solitary waves; (iii) oblique inelastic interactions can exist between the (a) two shock waves, (b) two elevation/depression solitary waves, and (c) shock and solitary waves.     

Nonlinear dynamics in mechanics and engineering: 40 years of developments and Ali H. Nayfeh’s legacy

Nonlinear Dynamics - Nonlinear dynamics of engineering systems has reached the stage of full maturity in which it makes sense to critically revisit its past and present in order to establish an...     

A Volterra series approach to the frequency domain analysis of non-linear viscous Burgers’ equation

In this paper, higher order frequency response functions, based on the Volterra series, are employed to characterise the input-output behaviour of the non-linear viscous Burgers?? equation subject to sinusoidal excitation. First, a formal Volterra series representation for each spatial location is derived for the solution of Burgers?? equation with a boundary condition as the input to the system. Then a systematic method is presented to obtain the higher order frequency kernels of the Volterra series at each spatial location by solving a series of ordinary differential equations. It is shown that the convergence region of the individual harmonics with respect to the magnitude of the input excitation can be estimated by using these higher order kernels. The frequency characteristics of Burgers?? equation is investigated and compared with numerical simulation.     

Nonlinear analysis of a simple amplitude–phase motion equation for power-electronics-based power system

Nonlinear Dynamics - With large-scale application of a large number of renewable energy sources, such as wind turbines, photovoltaics, and various power electronic equipment, the power electric...