Coupled System Of Evolution Equations Research Articles

Modified Wavelet Method for Solving Two-dimensional Coupled System of Evolution Equations

Modified Wavelet Method for Solving Two-dimensional Coupled System of Evolution Equations

Plasma-waves evolution and propagation modeled by sixth order Ramani and coupled Ramani equations using symmetry methods

Nonlinear shock waves in plasma was modeled and studied using Ramani equation of sixth order and its coupled form representing the interaction between two waves. A new combined methodology of both Lie infinitesimal transformation and singular manifold methods was exploited to create analytical solutions. The method was extended to investigate a coupled system of evolution equations as same as a single evolution equation. Soliton solutions were created for both models.

Nonlinear dynamics of thin liquid films subjected to mixed-frequency electrical field

The nonlinear dynamics of the interface between both perfect and leaky dielectric liquid films, interposed between two parallel electrodes, are investigated under the effect of mixed-frequency electric fields. A coupled system of evolution equations is derived in dimensionless form, by employing the long-wave approximation. The linear stability analysis is implemented in accordance with the characteristics of each specific case, namely, the constant (DC) and the altering (AC) fields. In particular, the response of the system to the multi-mode AC electrical field is analyzed. Assisted by the conclusions of the theoretical investigation, the initial-boundary-value problem associated with the coupled system of evolution equations is solved numerically for several parameter sets. The system behavior is studied by monitoring the evolution process and by examining the steady/quasi-steady pillar formations in the nonlinear regime. The possibility to generate interface profiles of diverse topological forms, to manipulate their features, and to control the time dependent progress and the film rupture by imposing different combinations of frequencies and/or amplitudes of the corresponding mode is confirmed.

Homogenization of a pore scale model for precipitation and dissolution in porous media

In this article, we employ homogenization techniques to provide a rigorous derivation of the Darcy scale model for precipitation and dissolution in porous media. The starting point is the pore scale model in van Duijn & Pop (2004 ), which is a coupled system of evolution equations, involving a parabolic equation which models ion transport in the fluid phase of a periodic porous medium, coupled to an ordinary differential equations modelling dissolution and precipitation at the grains boundary. The main challenge is in dealing with the dissolution and precipitation rates, which involve a monotone but possibly discontinuous function. In order to pass to the limit in these rate functions at the boundary of the grains, we prove strong two-scale convergence for the concentrations at the microscopic boundary and use refined arguments in order to identify the form of the macroscopic dissolution rate, which is again a discontinuous function. The resulting upscaled model is consistent with the Darcy scale model proposed in Knabner et al . (1995 ).

Electrostatic suppression of the "coffee stain effect".

The dynamics of a slender, evaporating, particle-laden droplet under the effect of electric fields are examined. Lubrication theory is used to reduce the governing equations to a coupled system of evolution equations for the interfacial position and the local, depth-averaged particle concentration. The model incorporates the effects of capillarity, viscous stress, Marangoni stress, elecrostatically induced Maxwell stress, van der Waals forces, concentration-dependent rheology, and evaporation. Via a parametric numerical study, the one-dimensional model is shown to recover the expected inhomogeneous ring-like structures in appropriate parameter ranges due to a combination of enhanced evaporation close to the contact line, and resultant capillarity-induced flow. It is then demonstrated that this effect can be significantly suppressed via the use of carefully chosen electric fields. Finally, the three-dimensional behavior of the film and the particle concentration field is briefly examined.

Electrohydrodynamic interaction of spherical particles under Quincke rotation

Weakly conducting dielectric particles suspended in a dielectric liquid of higher conductivity can undergo a transition to spontaneous sustained rotation when placed in a sufficiently strong dc electric field. This phenomenon of Quincke rotation has interesting implications for the rheology of these suspensions, whose effective viscosity can be controlled and reduced by application of an external field. While previous models based on the rotation of isolated particles have provided accurate estimates for this viscosity reduction in dilute suspensions, discrepancies have been reported in more concentrated systems where particle-particle interactions are likely significant. Motivated by this observation, we extend the classic description of Quincke rotation based on the Taylor-Melcher leaky dielectric model to account for pair electrohydrodynamic interactions between two identical spheres using the method of reflections. A coupled system of evolution equations for the dipole moments and angular velocities of the spheres is derived that accounts for electric dipole-dipole interactions and hydrodynamic rotlet interactions up to order O(R(-5)), where R is the separation distance between the spheres. A linear stability analysis of this system shows that interactions modify the value of the critical electric field for the onset of Quincke rotation: both electric and hydrodynamic interactions can either stabilize or destabilize the system depending on the orientation of the spheres, but the leading effect of interactions on the onset of rotation is hydrodynamic. We also analyze the dynamics in the nonlinear regime by performing numerical simulations of the governing equations. In the case of a pair of spheres that are fixed in space, we find that particle rotations always synchronize in magnitude at long times, though the directions of rotation of the spheres need not be the same. The steady-state angular velocity magnitude depends on the configuration of the spheres and electric field strength and agrees very well with an asymptotic estimate derived for corotating spheres. In the case of freely-suspended spheres, dipolar interactions are observed to lead to a number of distinct behaviors depending on the initial relative configuration of the spheres and on any infinitesimal initial perturbation introduced in the system: in some cases the spheres slowly separate in space while steadily rotating, while in other cases they pair up and either corotate or counterrotate depending on their orientation relative to the field.

Gradient expansion, curvature perturbations, and magnetized plasmas

The properties of magnetized plasmas are always investigated under the hypothesis that the relativistic inhomogeneities stemming from the fluid sources and from the geometry itself are sufficiently small to allow for a perturbative description prior to photon decoupling. The latter assumption is hereby relaxed and predecoupling plasmas are described within a suitable expansion where the inhomogeneities are treated to a given order in the spatial gradients. It is argued that the (general relativistic) gradient expansion shares the same features of the drift approximation, customarily employed in the description of cold plasmas, so that the two schemes are physically complementary in the large-scale limit and for the low-frequency branch of the spectrum of plasma modes. The two-fluid description, as well as the magnetohydrodynamical reduction, is derived and studied in the presence of the spatial gradients of the geometry. Various solutions of the coupled system of evolution equations in the anti-Newtonian regime and in the quasi-isotropic approximation are presented. The relation of this analysis to the so-called separate universe paradigm is outlined. The evolution of the magnetized curvature perturbations in the nonlinear regime is addressed for the magnetized adiabatic mode in the plasma frame.

Multiscale Dynamics of Heterogeneous Media in the Peridynamic Formulation

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation.

Energy decay rates of magnetoelastic waves in a bounded conductive medium

We consider a coupled system of evolution equations modeling the propagation of elastic waves interacting with a magnetic field in a bounded simply connected region of $\mathbb{R}^3$ with boundary of class $C^2$. A nonlinear dissipative mechanism is allowed to be effective in an small subregion of $\Omega$. We prove that the total energy decays as $t \to +\infty$.

Drop manipulation and surgery using electric fields

We study the dynamics of a slender drop sandwiched between two electrodes using lubrication theory. A coupled system of evolution equations for the film thickness and interfacial charge density is derived and simplified for the case of a highly conducting fluid. The contact line singularity is relieved by postulating the existence of a wetting precursor film, which is stabilised by intermolecular forces. We examine the motion of the drop as a function of system parameters: the electrode separation, β, an electric capillary number, C , and a spatio-temporally varying bottom electrode potential. The possibility of drop manipulation and surgery, which include drop spreading, translation, splitting and recombination, is demonstrated using appropriate tuning of the properties of the bottom potential; these results could have potential implications for drop manipulation schemes in various microfluidic applications. For relatively small β and/or large C values, the drop assumes cone-like structures as it approaches the top electrode; the latter stages of this approach are found to be self-similar and a power-law exponent has been extracted for this case.

Droplet spreading, imbibition and solidification on porous media

The spreading, imbibition and solidification of a hot two-dimensional droplet on a cooler porous substrate is studied. Lubrication theory and asymptotic reduction are used to derive a coupled system of evolution equations for the droplet thickness and vertical extents of the solid/melt boundary and the saturation front within the porous medium. This medium is assumed to be a membrane composed of an array of pores having fixed width. A precursor layer model and a disjoining pressure are used to relieve the singularity at the contact line. When the solidification and imbibition time scales are similar to those associated with spreading, the dynamics follow two stages: spreading and imbibition accompanied by solidification within the pores leading to their blockage, followed by contact line arrest and basal solidification of the droplet; the possibility of crust formation at the gas–melt interface is excluded from the present study. The dependence of the dynamics on the relevant system parameters is elucidated. Furthermore, our modelling predictions compare favourably with experimental data also obtained as part of the present work.

Dynamics of thin free films with reaction-driven density and viscosity variations

We study the dynamics of thin free viscous films in the presence of exothermic chemical reactions, which alter the film density and viscosity. We use long-wave theory and the assumption of rapid vertical diffusion to derive a coupled system of evolution equations for the film thickness, axial velocity, temperature, and reactant and product concentrations; appropriate closure relations are chosen for the dependence of the density and viscosity (both dynamic and bulk) on the product concentration and temperature. Analytical and numerical solutions of the evolution equations allow a full parametric study to be conducted. The results indicate that the film dynamics is primarily governed by density variations, which depend on competing factors: the increase in product concentration and temperature (due to the heat of reaction), which act to increase and decrease the density, respectively. Thus the film can collapse and rupture (expand) if the former (latter) effects outweigh the latter (former); here, rupture is driven by van der Waals forces.

Analyse numérique d'un problème de contact viscoélastique sans frottement avec adhérence et endommagement

A model for quasistatic, adhesive, and frictionless contact between two viscoelastic bodies is described. Material damage, which results from tension or compression, is taken into account in the constitutive law. The adhesion process is modelled by introducing the bonding field on the contact surface as a dependent variable. Contact is described with a modified Signorini condition which includes the adhesive normal tensile force. The variational problem is formulated as a coupled system of evolution equations. It is discretized using an explicit scheme for the time derivatives and a nonconforming finite element method based on the mortar projection operator. Error estimates are obtained for the numerical scheme under additional regularity hypotheses. Finally, numerical results for a two-dimensional example are depicted. To cite this article: J.R. Fernández et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Electrically induced pattern formation in thin leaky dielectric films

The stability of the interface between two thin leaky dielectric liquid layers bounded between two flat electrodes is considered. A coupled system of evolution equations is derived for the interfacial location and charge density using lubrication theory. This system is parametrized by the dielectric constants of the two fluids in addition to ratios of their conductivities, viscosities, and thicknesses. A linear stability analysis is conducted and the behavior of the system in the nonlinear regime is also examined. The system is destabilized by electrical stresses that are resisted by capillarity and modified by viscous dissipation. Our results suggest that decreasing the thickness ratio is destabilizing, giving rise to periodic structures of decreasing wavelength. Decreasing the viscosity ratio was also found to lead to the formation of sharp-edged structures whose vertical extent is virtually equal to the gap width between the electrodes. Similar structures were also determined upon increasing the ratio of the dielectric constants and electric conductivities.

Evolution scales for wave regimes in liquid film flow over a spinning disk

We study the spatiotemporal development of a thin viscous film flowing over a spinning disk. A coupled system of evolution equations for the film thickness and volumetric flow rates in the radial and azimuthal directions is derived using the Karman–Polhausen method, assuming a parabolic profile for the film velocity. In the limit of large Eckman number, this system reduces to equations previously used to study the falling film problem. Numerical solutions of the system are obtained starting from initially waveless profiles, which correspond to the Nusselt solution for the case of a spinning disk. Results from these simulations reveal the development of finite-amplitude waves, which, locally, approximate closely to the shape of quasisteady periodic traveling waves. These waves are found to be in good agreement with the predictions of the localized version of the model.

Solitary waves in a two-layer quasigeostrophic model with wind stress forcing

The effects of wind stress forcing on solitary waves in a two-layer quasigeostrophic model are investigated. A coupled system of evolution equations for the wave amplitude and induced mean-flow are derived by a multiple-scale asymptotic method, and solved numerically for an initial value problem in which a solitary wave is moving from a region free of wind stress forcing towards a region subject to localised wind stress forcing. It is found that the response of the wave to the forcing is diverse and complex depending closely on both the strength of the forcing and the initial velocity of the incident wave. For weak forcing the wave can pass through the forcing without any significant change of its properties, while for a moderate forcing the wave is much influenced by the forcing. It may reflect from the forcing with a change of direction, or pass through the forcing with an increase in both its amplitude and velocity, or break down into a very weak wave train. For strong forcing some new waves can be excited in the forcing region, periodically propagating downstream. These numerical results are supported by an asymptotic analysis based on certain conservation laws.

N=4 super KdV hierarchy in N=4 and N=2 superspaces

We present the results of further analysis of the integrability properties of the N=4 supersymmetric Korteweg–de Vries (KdV) equation deduced earlier by two of us [F. Delduc and E. Ivanov, Phys. Lett. B 309, 312 (1993)] as a Hamiltonian flow on N=4 SU(2) superconformal algebra in the harmonic N=4 superspace. To make this equation and the relevant Hamiltonian structures more tractable, we reformulate it in the ordinary N=4 and further in N=2 superspaces. In N=2 superspace it is represented by a coupled system of evolution equations for a general N=2 superfield and two chiral and antichiral superfields, and involves two independent real parameters, a and b. We construct a few first bosonic conserved charges in involution, of dimensions from 1 to 6, and show that they exist only for the following choices of the parameters: (i) a=4, b=0; (ii) a=−2, b=−6; (iii) a=−2, b=6. The same values are needed for the relevant evolution equations, including N=4 KdV itself, to be bi-Hamiltonian. We demonstrate that the above three options are related via SU(2) transformations and actually amount to the SU(2) covariant integrability condition found in the harmonic superspace approach. Our results provide a strong evidence that the unique N=4 SU(2) super KdV hierarchy exists. Upon reduction to N=2 KdV, the above three possibilities cease to be equivalent. They give rise to the a=4 and a=−2 N=2 KdV hierarchies, which thus prove to be different truncations of the single N=4 SU(2) KdV one.

Smoothing Properties, Decay, and Global Existence of Solutions to Nonlinear Coupled Systems of Thermoelastic Type

We consider a nonlinear coupled system of evolution equations, the simplest of which models a thermoelastic plate. Smoothing and decay properties of solutions are investigated as well as the local well-posedness and the global existence of solutions. For the system of standard thermoelasticity it is proved that there is no similar smoothing effect.

Propagative Sine-Gordon solitons in the spatially forced Kelvin-Helmholtz instability

The spatio-temporal evolution of the vortex sheet separating two finite-depth layers of immiscible fluids is examined in the vicinity of threshold when spatially periodic forcing is imposed at the horizontal boundaries. As a result of the Galilean invariance of the problem, the interface deformation is shown to satisfy a coupled system of evolution equations involving not only the usual “short-wave” at the critical wavenumber but also a shallow-water “long-wave” associated with the mean elevation of the interface. The weakly nonlinear model is further studied in the Boussinesq approximation where it reduces to a forced Klein-Gordon equation. Thus, the secondary Benjamin-Feir instability of nonlinear Stokes wavetrains is analysed in the absence of forcing. When spatial forcing is reintroduced, the competition between the imposed external length scale and the natural length scale of the interface is shown analytically to give rise to one-dimensional propagating Sine-Gordon phase solitons. Numerical simulations of the Klein-Gordon evolution model fully confirm this prediction and also lead to the determination of the range of stability of phase solitons.

On a free boundary problem for a nonlinear coupled system of evolution equations of Oceanography

On a free boundary problem for a nonlinear coupled system of evolution equations of Oceanography