Symmetries and solutions to the thin film equations

We study symmetries and solutions of the generalized fourth-order nonlinear partial differential equations which arise from studies of thin liquid films. It is shown that the equations admit extended scaling and rotation symmetries and a class of generalized conditional symmetries for certain coefficient functions. Exact solutions associated to the symmetries are obtained.     

Self-adjoint sub-classes of generalized thin film equations

In this work we consider a class of fourth-order nonlinear partial differential equation containing several un-specified coefficient functions of the dependent variable which encapsulates various mathematical models used, e.g. for describing the dynamics of thin liquid films. We determine the subclasses of these equations which are self-adjoint. By using a general theorem on conservation laws proved by one of the authors (NHI) we find conservation laws for some of these partial differential equations without classical Lagrangians.     

Propagation of perturbations in thin capillary film equations with nonlinear diffusion and convection

We study the evolution of the support of an arbitrary strong generalized solution to the Cauchy problem for the thin film equation with nonlinear diffusion and convection. We find an upper bound exact (in a sense) for the propagation speed of the support of this solution.     

Well-posedness and stability for a mixed order system arising in thin film equations with surfactant

The objective of the present work is to provide a well-posedness result for a capillary driven thin film equation with insoluble surfactant. The resulting parabolic system of evolution equations is not only strongly coupled and degenerated, but also of mixed orders. To the best of our knowledge the only well-posedness result for a capillary driven thin film with surfactant is provided in [4] by the same author, where a severe smallness condition on the surfactant concentration is assumed to prove the result. Thus, in spite of an intensive analytical study of thin film equations with surfactant during the last decade, a proper well-posedness result is still missing in the literature. It is the aim of the present paper to fill this gap. Furthermore, we apply a recently established result on asymptotic stability in interpolation spaces [15] to prove that the flat equilibrium of our system is asymptotically stable.     

On the structure of positive solutions to an elliptic problem arising in thin film equations

We consider the structure of positive radial solutions of

     

Finite Morse index steady states of van der Waals force driven thin film equations

We study positive solutions of the equation

     

A thin film model

Using the theory of fiber bundles we construct a new model of a thin polycrystal film. The quantum equations of motion thereby obtained are used to determine the regularity of the influence of substructural ordering on the physical characteristics of the film.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 36–40.     

Blow-up and critical exponents for parabolic equations with non-divergent operators: dual porous medium and thin film operators

Our first basic model is the fully nonlinear dual porous medium equation with source

Long-wave oscillations and waves in anisotropic beams

The asymptotic integrating method is used to investigate long-wave oscillations and waves in an infinite heterogeneous (with respect to width) anisotropic beam-belt. A dispersion equation of the second-order accuracy with respect to the relative width of the beam-belt is constructed and additional qualitative effects related to the anisotropy are found.     

Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces

We consider the Cauchy problem for the first and the second order differential equations in Banach and Hilbert spaces with an operator coefficient depending on the parameter . We develop discretization methods with high parallelism level and without accuracy saturation; i.e., the accuracy adapts automatically to the smoothness of the solution. For analytical solutions the rate of convergence is exponential. These results can be viewed as a development of parallel approximations of the operator exponential and of the operator cosine family with a constant operator possessing exponential accuracy and based on the Sinc-quadrature approximations of the corresponding Dunford-Cauchy integral representations of solutions or the solution operators.

     

Deformation of thin polycaprolactam film in uniaxial tension

The individual stages of deformation of spherulitic specimens of polycaprolactam have been studied by a microscopic method. An empirical relation between the total deformation of the specimen and the deformation of the individual spherulite is obtained.     

The FENE viscoelastic model and thin film flows

This Note has as objective to determine, in a rigorous way, a simplified expression of the constitutive law for a visco-elastic fluid of FENE type in thin domains. The proof uses the FENE model behavior for long times and the existence of a stationary solution for this behavioral law. Some possible applications of this study are then briefly described in the domains of lubrication, blood flow, microfluidic, boundary layers, …. To cite this article: L. Chupin, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Quasistatic evolution of a brittle thin film

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in L∞. Mathematics Subject Classification (2000) 74K30, 49J45, 74K30, 35R35, 49Q20     

Symmetry analysis for a thin film equation

In this work we give a complete group classification of the thin film equation u_t = –(f (u)_xxx )_x – kg (u)_x . By using Lie classical method the corresponding reductions are performed and some solutions are characterized. Some nonclassical symmetries for an associated potential system are derived. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A dual approach to regularity in thin film micromagnetics

We prove a regularity result in the two-dimensional theory of soft ferromagnetic films. The associated Euler–Lagrange equation is given by a nonlocal degenerate variational inequality involving fractional derivatives. A difference quotient type argument based on a dual formulation in terms of magnetostatic potentials yields a Hölder estimate for the uniquely determined gradient projection of the magnetization field.     

Transient film profile of thin liquid film flow on a stretching surface

In this paper we have studied a non-planar thin liquid film flow on a planar stretching surface. The stretching surface is assumed to stretch impulsively from rest and the effect of inertia of the liquid is considered. Equations describing the laminar flow on the stretching surface are solved analytically. It is observed that faster stretching causes quicker thinning of the film on the stretching surface. Velocity distribution in the liquid film and the transient film profile as functions of time are obtained.