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The aim of this paper is to study the dimension reduction analysis of an elastic plate with small thickness reinforced with increasing number of thin ribbons developing fractal geometry. We prove the$\Gamma $-convergence of the energy functionals to a two-dimensional effective energy including singular terms supported within the Sierpinski carpet.

A computational approach to the investigation of bifurcations, based on the use of a special type of Hermite–Padé approximant, is presented. The first part of this study is a review of a singularity extraction technique based on the assumption that the given series is the local representation of an algebraic function in the independent variable. The principal merit of the procedure is its ability to reveal the underlying problem of the branches solution which are represented by the original series. In the final section, numerical results are presented for Dean flow and two problems coming from heat transfer modelling and whose solutions are obtained by means of a regular perturbation method.

We theoretically investigate the deformation of a perfect dielectric drop suspended in a second dielectric liquid subject to a uniform electric field. Axisymmetric equilibrium shapes are found by solving simultaneously the Young–Laplace equation at the interface and Laplace equation for the electric field. Analytical solutions are constructed for the governing nonlinear boundary-value problem using domain perturbation method together with a special type of Hermite–Pade approximation. The results show the existence of a critical electric capillary number beyond which no axisymmetric figure is possible.

In this work, we perform preliminary group classification of equations governing the flow of a thermo-dependent fluid in a porous medium. The detailed model is a coupled system of nonlinear partial differential equations containing two arbitrary functions. The principal Lie algebra admitted by the coupled system is given. The study also demonstrates that the forms of arbitrary functions provide an extension of this algebra via an optimal system of one-dimensional Lie subalgebras.

We consider elastic materials containing rigid micro-inclusions located along self-similar fractals. We use $$\Gamma $$ -convergence methods in order to study the asymptotic behaviour of the structure with respect to vanishing parameters describing the thickness of the inclusions. We derive effective elastic behaviours of the elastic materials with boundary conditions describing discontinuities in the tractions across the self-similar fractal interfaces.

This paper investigates axisymmetric shapes of a ferrofluid drop immersed in a uniform magnetic field. Approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method. Bifurcation and turning points are located by applying a special type of Hermite–Pade approximation.

We study the homogenization of a slow viscous two-phase incompressible flow in a domain consisting of a free fluid domain, a porous medium, and the interface between them. We take into account the capillary forces on the fluid-fluid interfaces. We construct boundary layers describing the flow at the interface between the free fluid and the porous medium. We derive a macroscopic model with a viscous two-phase fluid in the free domain, a coupled Darcy law connecting two-phase velocities in the porous medium, and boundary conditions at the permeable interface between the free fluid domain and the porous medium.

This paper deals with the homogenization of a composite material built with an elastic matrix periodically reinforced by elastic notched fibres. We assume that the Lame constants in the fibres are highly contrasted with respect to those of the matrix. We derive a class of energy functionals involving additional degrees of freedom implying nonlocal description of localized geometrical discontinuities due to the fibre-notches, nonlocal effects associated with thin boundary layer phenomena, and nonlocal terms including macroscopic fields of extension or flexion.

In this paper, the symmetry group is performed for the heat transfer flow in a porous medium, this flow is described by coupled partial differential equations. Thanks to the method of the symmetry group, the symmetries of the coupled equations are given. The similarity variables and reduction equations generated from the symmetry transformations are provided. Such similarity reductions are computed and exact solutions are given such solutions are important in engineering applications and on the theory of nonlinear science.

In this paper we study the stability of the magnetostatic equilibrium through a relaxation of a magnetic field B in perfectly conducting compressible and viscous fluid. We establish stability criterion of a large class of Beltrami flows to any admissible displacement about the equilibrium configuration. We show that the field is stable to any displacement with the same 2π-periodicity as the basic flow, except the case where perturbations with wavelength much greater than the scale of the basic flow are included.

International audience; This paper is devoted to the equilibrium shapes of rotating liquid drops. Families of axisymmetric shapes are found using a perturbation technique for the governing nonlinear equation. Bifurcation and turning points are located by applying a special type of Hermite–Padé approximation.

In this paper, the Lie symmetry analysis is performed for the heat transfer flow of a second grade fluid between two parallel heated plates. The symmetries for the coupled equations are given. The exact solutions and similarity reductions generated from the symmetry transformations are provided. Furthermore, translational symmetries were utilised to find a family of travelling wave solutions of the governing nonlinear problem.

A model of forest fire propagation, including drying, pyrolysis, heat convection, heat conduction, and radiative transfer, is presented. This model has been obtained by asymptotic expansion and has the generic form of a system of reaction-diffusion equations. One of the aims of the present work is to link this type of reaction diffusion models to models that rely on the concept of rate of spread. The rate of spread is defined as the solution to a nonlinear eigenvalue problem. The solution to this eigenvalue problem is obtained as a perturbation expansion in the thermal conductivity of the solid phase. The calculated rates of spread for two flame models compare favorably to experimental values.

International audience; The aim of this work is to apply the homotopy perturbation method for solving the steady state equations of the exothermic decomposition of a combustible material obeying Arrhenius, Bimolecular, and Sensitised laws of reaction rates. These equations are formulated on some Class A geometries (an infinite cylinder, an infinite slab, and a sphere). We also investigate the effect of Frank-Kamenetskii parameter on bifurcation and thermal criticality by means of the Domb-Sykes graphical method.