Multiple-scale Expansion Method Research Articles

Influence of impurities on hydrodynamic solitons

The interaction between impurities and solitary waves has been experimentally observed on the surface of a defective water layer subjected to vertical vibration. A slightly rugged surface on one sidewall of the water layer serves as the impurity, making the layer breadth at the defect slightly different elsewhere. A wide-breadth impurity will attract or pin not only the hydrodynamic breather at lower driving frequency, but also the kink at higher driving frequency, while a narrow-breadth one will repel them. Using a multiple scale expansion method, a nonlinear Schro dinger equation with an impurity term (NLSI) was derived from the basic hydrodynamic equation. Furthermore, we present numerical calculations that show good agreement between the NLSI-based theoretical model and the experimental results.

Microstructural effects on the flow law of power-law fluids through fibrous media

In this work, the flow of power-law fluids through anisotropic fibrous media is revisited, upscaling the fluid flow at the pore scale with the homogenization method of multiple scale expansions for periodic structures. This upscaling technique permits a quantitative study of the seepage law by performing numerical simulation with simple two-dimensional periodic arrays of circular solid inclusions. The significant role of the solid fraction, the fluid rheology and the porous media anisotropy on the resulting macroscopic flow law is underlined from the simulation.

Does Klinkenberg’s Law Survive Upscaling?

This work deals with the large-scale mathematical modelling of flow of gas at low pressure in porous media. At the pore scale, this type of flow is characterised by a wall-slip effect, which at the sample scale entails a dependence of permeability upon gas pressure. This latter property is described by Klinkenberg's law. The goal of the present work is to examine the robustness of this law, by determining whether it is still verified on a large-scale: upscaling is thus applied, starting with Klinkenberg's law at the local scale. A Klinkenberg's flow of gas in a two-constituent composite porous medium is considered, and the constituents are firstly assumed to be homogeneous. The cases of low and of high permeability contrast are successively examined. Upscaling is performed using the homogenisation method of multiple scale expansions. In both cases, the large-scale permeability tensor differs from its liquid counterpart. Except in the particular case of equal Klinkenberg factors, Klinkenberg's law is not verified at low permeability contrast. At high permeability contrast, the large-scale gas permeability verifies Klinkenberg's law. The case of heterogeneous constituents is then examined. It is shown that the large-scale permeability differs from its liquid counterpart, but it does not verify Klinkenberg's law.

About Darcy's law in non‐Galilean frame

AbstractThis paper is aimed towards investigating the filtration law of an incompressible viscous Newtonian fluid through a rigid non‐inertial porous medium (e.g. a porous medium placed in a centrifuge basket). The filtration law is obtained by upscaling the flow equations at the pore scale. The upscaling technique is the homogenization method of multiple scale expansions which rigorously gives the macroscopic behaviour and the effective properties without any prerequisite on the form of the macroscopic equations. The derived filtration law is similar to Darcy's law, but the tensor of permeability presents the following remarkable properties: it depends upon the angular velocity of the porous matrix, it verifies Hall–Onsager's relationship and it is a non‐symmetric tensor. We thus deduce that, under rotation, an isotropic porous medium leads to a non‐isotropic effective permeability. In this paper, we present the results of numerical simulations of the flow through rotating porous media. This allows us to highlight the deviations of the flow due to Coriolis effects at both the microscopic scale (i.e. the pore scale), and the macroscopic scale (i.e. the sample scale). The above results confirm that for an isotropic medium, phenomenological laws already proposed in the literature fails at reproducing three‐dimensional Coriolis effects in all types of pores geometry. We show that Coriolis effects may lead to significant variations of the permeability measured during centrifuge tests when the inverse Ekman number Ek−1 is 𝒪(1). These variations are estimated to be less than 5% if Ek−1<0.2, which is the case of classical geotechnical centrifuge tests. We finally conclude by showing that available experimental data from tests carried out in centrifuges are not sufficient to determining the effective tensor of permeability of rotating porous media. Copyright © 2004 John Wiley & Sons, Ltd.

Nonlinear interfacial stability for magnetic fluids in porous media

The weakly nonlinear stability is employed to analyze the interfacial phenomenon of two magnetic fluids in porous media. The effect of an oblique magnetic field to the separation face of two fluids is taken into account. The solutions of equations of motion under nonlinear boundary conditions lead to deriving a nonlinear equation in terms of the interfacial displacement. This equation is accomplished by utilizing the cubic nonlinearity. The method of multiple scale expansion is employed in order to obtain a dispersion relation for the first-order problem and nonlinear Ginzburg–Landau equation, for the higher-order problem, describing the behaviour of the system in a nonlinear approach. Regions of stability and instability are identified for the magnetic field intensity versus the wave number. It is found that the oblique magnetic filed has a stabilizing influence under some certain conditions for the directions of the magnetic fields. The resistance coefficient has a destabilizing influence in the linear description. Further, in the nonlinear scope, the increase of the resistance parameters plays both stabilizing and destabilizing role in the stability criteria.

Solute diffusion in fractured porous media with memory effects due to adsorption

Modelling of solute transport in fractured porous media is a subject of intensive research in many engineering disciplines, such as petroleum engineering, water resources management, civil engineering. Recent field and laboratory experiments show that, in presence of strong adsorption, the behaviour of solute penetrating into the fractured porous medium diverges from classical hypotheses, rendering impossible the adjustment of classical transport models. The aim of this paper is to develop a mathematical continuous model of solute transport, when strong adsorption of solute occurs on the grains of the porous matrix. The macroscopic model is obtained by upscaling the pore and the fracture behaviours, by using the multiple scale expansion method. We obtain a non-standard diffusion behaviour of solute which shows local non-equilibrium between transport in the fractures and in the porous matrix, as well as memory effects. To cite this article: J. Lewandowska et al., C. R. Mecanique 330 (2002) 879–884.

Nonlinear stability in magnetic fluids of cylindrical interface with mass and heat transfer

The nonlinear Rayleigh-Taylor stability of the cylindrical interface between the vapor and liquid phases of a magnetic fluid is studied when the phases are enclosed between two cylindrical surfaces coaxial with the interface, and when there is mass and heat transfer across the interface. The method of multiple scale expansion is used for the investigation. The evolution of amplitude is shown be governed by a nonlinear Ginzburg-Landau equation. The various stability criteria is discussed, and the region of stability is displayed graphically.

Coriolis Effects on Filtration Law in Rotating Porous Media

We investigate the filtration law of incompressible viscous Newtonian fluids in rigid non-inertial porous media, for example, rotating porous media. The filtration law is obtained by upscaling the flow at the pore scale. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour without any prerequisite on the form of the macroscopic equations. For finite Ekman numbers the filtration law is shown to resemble a Darcy's law, but with a non-symmetric permeability tensor which depends on the angular velocity of the porous matrix. We obtain the filtration analog of the Hall effect. For large Ekman numbers the filtration law is a small correction to the classical Darcy's law. The corrector is antisymmetric. In this case we recover a structure of law which is similar to phenomenological laws introduced in the literature, but with a dissimilar effective coefficient.

Localized modes in a one-dimensional sphalerite-structure lattice with anharmonicity

The nonlinear localized vibrational modes of a one-dimensional atomic chain with two periodically alternating masses and force constants are analytically investigated using a discrete multiple-scale expansion method. This model simulates a row of atoms in the -direction of sphalerite, or zinc blende, crystals. Owing to the structural asymmetry; the vibrational amplitude is governed by a perturbed nonlinear Schrodinger equation instead of the standard one found in one-dimensional lattices with two alternating masses but uniform force constant. Although the stationary localized modes with carrier wavevector at the Brillouin-zone boundary are similar to those of ionic lattices, the moving localized modes with wavevectors within the zone are different owing to the perturbation. The calculation shows that the height of the moving localized modes in this lattice dampens with time.

Magnetohydrodynamic flow through porous media

We investigate the filtration law in rigid porous media for steady-state slow flow of an electrically conducting, incompressible and viscous Newtonian fluid in the presence of magnetic field. The seepage law under magnetic field is obtained by upscaling the flow at the pore scale by using the method of multiple scale expansions. The macroscopic magnetic field and electric flux are also obtained. For finite Hartmann number, i.e. ε⪡Ha⪡ε −1 where ε characterizes the separation of scale, the filtration law is shown to resemble a Darcy's law but with an additional term proportional to the electric field. The permeability tensor which strongly depends on the magnetic induction, i.e. Hartmann number, is symmetric, positive and verifies the filtration analog of the Hall effect. Mass and electric fluxes are coupled.

Instability of triple junctions in lamellar eutectic growth

By employing the method of multiple-scale expansions we derive a system of nonlinear ordinary differential equations to describe the dynamics of lamellar eutectic crystals in directional solidification. The equations govern the motions of trijunctions where the liquid and the two solid phases meet, and predict a critical lamellar spacing λ c below which the solidifying fronts are unstable. When the morphological numbers are large this critical spacing agrees with the value predicted by the Jackson and Hunt's theory: Vλ 2 c=constant, where V is the growth speed. As the morphological numbers decrease, the speed-spacing relationship deviates from the power law, with the predicted spacing smaller than that at the minimum undercooling point. The thermal gradient then plays an important role in determining the lamellar structures.

Transport phenomena in saturated porous media undergoing liquid-solid phase change

The macroscopic modelling of transport phenomena occurring during the solidification process in porous media is revisited in this paper. The particular case of binary phase change in metallic alloys is considered. Continuum models for momentum, mass, heat and species transport in metallic saturated porous media undergoing liquid-solid phase change are derived from the description at the pore scale by using an upscaling technique. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour. Different macroscopic descriptions are derived in function of the orders of magnitude of dimensionless numbers that characterize the dominating phenomena and the physical properties of the constituents at the microscopic scale. Among the distinct homogenizable situations, the three richest cases are presented in this paper. The domains of validity of the micro-macrosegregation models, i.e. lever-rule type models and Scheil type models, are shown by means of the order of magnitude of dimensionless numbers. The continuous passage between the different models is investigated.

Interactions of nonlinear acoustic waves in a fluid-filled elastic tube

In the present work, the nonlinear interactions of two acoustical waves propagating in a fluid-filled elastic tube with different wave numbers, frequencies and group velocities are examined. Employing the multiple-scale expansion method, expanding the field quantities into asymptotic series of the smallness parameter and solving the resulting differential equations of various orders of the same parameter, we obtained two coupled nonlinear Schrödinger equations. The nonlinear plane wave solutions to these equations are also given for some special cases.

Filtration law in rotating porous media

We investigate the filtration law in rigid non-galilean porous media of incompressible viscous Newtonian fluids. The filtration law is obtained by upscaling the flow from the pore scale. We use the method of multiple scale expansions which gives rigorously the macroscopic behaviour without any prerequisite at the macroscopic scale. For finite Ekman numbers the filtration law is shown to resemble a Darcy's law, but with a non-symmetric permeability tensor which depends on the angular velocity of the porous matrix. For large Ekman numbers the filtration law is a small correction to the classical Darcy's law. The corrector is antisymmetric. In this case we recover a similar structure of law to phenomenologic laws introduced in the literature, but with a dissimilar effective coefficient.

Amplitude equations for a system with thermohaline convection

The multiple scale expansion method is used to derive amplitude equations for a system with thermohaline convection in the neighborhood of Hopf and Taylor bifurcation points and at the double zero point of the dispersion relation. A complex Ginzburg-Landau equation, a Newell-Whitehead-type equation, and an equation of the $\phi^4$ type, respectively, were obtained. Analytic expressions for the coefficients of these equations and their various asymptotic forms are presented. In the case of Hopf bifurcation for low and high frequencies, the amplitude equation reduces to a perturbed nonlinear Schr\"odinger equation. In the high-frequency limit, structures of the type of "dark" solitons are characteristic of the examined physical system.

On the contribution of higher order terms to solitary waves in fluid filled elastic tubes

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thin tube and the approximate equations of an incompressible inviscid fluid, the propagation of weakly nonlinear waves, in such a medium is studied through the use of the modified multiple scale expansion method. It is shown that the evolution of the lowest order (first order) term in the perturbation expansion may be described by the Korteweg-de Vries equation. The governing equation for the second order terms and the localized travelling wave solution for these equations are also obtained. The applicability of the present model to flow problems in arteries is discussed.

A method for higher-order expansion in non-linear ion-acoustic waves

The basic equations describing the non-linear ion-acoustic waves in a cold collisionless plasma, in the longwave limit, is re-examined through the use of a modified multiple-scale expansion method. Expanding the field quantities into a power series of the smallness parameter ε, a set of evolution equations is obtained for various terms in the perturbation expansion. To illustrate the present derivation, a localized travelling wave solution is studied for the derived field equations and the result is compared with those of Malfliet and Wieers (J. Plasma Phys. 56 (1996) 441–450), who employed the dressed solitary wave approach from the outset of their study.

Note on the Stability of the Nonlinear Mathieu Equation

In order to explain a beating-like phenomenon observed in an experiment by Kono,2) nonlinear effects on a parametrically excitable oscillation is theoretically and numerically investigated in the case of the nonlinear Mathieu equation and the nonlinear Melde's string vibration. It is shown that when the resonance condition is fulfilled or nearly fulfilled, depending on the initial conditions, the solution is in one of two distinctive states, one of which, the pseudo-unstable state, may presumably be ascribed to the observed state. A multiple scales expansion method is utilized to derive a reduced nonlinear equation governing the complex amplitude from fundamental equations. This method has proved to be profitable both in theoretical and numerical investigations.

Nonlinear electrohydrodynamic Rayleigh-Taylor instability with mass and heat transfer subject to a vertical oscillating force and a horizontal electric field

Weakly nonlinear stability of interfacial waves propagating between two electrified inviscid fluids influenced by a vertical periodic forcing and a constant horizontal electric field is studied. Based on the method of multiple-scale expansion for a small-amplitude periodic force, two parametric nonlinear Schrodinger equations with complex coefficients are derived in the resonance cases. A standard nonlinear Schrodinger equation with complex coefficients is derived in the nonresonance case. A temporal solution is carried out for the parametric nonlinear Schrodinger equation. The stability analysis is discussed both analytically and numerically.

The Nonlinear Stability of Mass and Heat Transfer in Magnetic Fluids

AbstractThe nonlinear analysis of the Rayleigh‐Taylor stability of two superposed magnetic fluids with interfacial transfer of mass and heat is presented for two layers each of finite thickness. The system is subjected to a tangential magnetic field. The method of multiple scale expansion is employed for the analysis. It is shown that the evolution of the amplitude is governed by a nonlinear Ginzburg‐Landau equation. There is also obtained a nonlinear diffusion equation describing the evolution of wave packets near the marginal state. Further, the nonlinear Schrödinger equation is obtained when the influence of mass and heat transfer is neglected. The various stability criteria are discussed both analytically and numerically and the stability diagrams are obtained. It is found that, in the linear theory, the stability criterion is independent of mass and heat transfer coefficient. While in the nonlinear theory it is found that, when this coefficient is large enough, the system which would be unstable classically, can be stabilized for finite amplitude disturbances.