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chechan We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrm of a high-contrast random operator. We consider a family of elliptic operators Aε in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator Aε, we show that the point spectrum of A\ε converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator Ahom as. Furthermore, we prove that the eigenfunctions of A\ε decay exponentially at infinity uniformly for sufficiently small \varepsilon. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of hom. [ABSTRACT FROM AUTHOR]

The complex wave structure of annular gas–liquid flow with disturbance waves and liquid entrainment is a result of the evolution of high-frequency initial waves, appearing at the very inlet of the flow, prior to the hydrodynamic stabilization of liquid film. This stage of flow evolution is studied experimentally, using a shadow technique, and theoretically, using a linear stability analysis of the Orr–Sommerfeld equation in both phases. The present work is focused on the comparison of earlier results obtained in air–water downward flow with the new results obtained in upward flow and with more viscous liquids. The flow orientation affects the shape of the liquid film prior to stabilization; the initial film area is thicker but shorter in upward flow. Upward flow orientation also leads to a lower frequency and the increment of growth of initial waves. The viscosity effect is found to be weak if flow rates of both phases are the same. The model is mostly able to reproduce the qualitative trends, but the quantitative agreement is not reached. Experimental observations indicate that the liquid flow within the initial area is significantly different from the stabilized flow of gas-sheared liquid film, which is used in the model. This difference could explain the discrepancy; further development of the model should be aimed at taking into account the evolution of the velocity profile inside the liquid film during the stage of hydrodynamic stabilization. [ABSTRACT FROM AUTHOR]

Using a suitable stochastic version of the compactness argument of [Zhikov VV. On an extension of the method of two-scale convergence and its applications. Sb Math. 2000;191(7-8):973-1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled 'macroscopic' and 'microscopic' components of the field, giving rise to an analogue of the 'Zhikov function', which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit. [ABSTRACT FROM AUTHOR]

Following a number of recent studies of resolvent and spectral convergence of nonuniformly elliptic families of differential operators describing the behavior of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect"": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period ε goes to zero, has a band-gap structure. We consider a sequence of eigenvalues λε that are induced by the defect and converge to a point λ0 located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are "extremely"" localized to the defect, in the sense that the localization exponent behaves as exp(-v/ε), v > 0, which has not been observed in the existing literature. In two- and three-dimensional configurations, whose one-dimensional cross sections are described by the setting considered, this implies the existence of propagating waves that are localized to a vicinity of the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly. [ABSTRACT FROM AUTHOR]

We study the homogenisation of geometrically nonlinear elastic composites with high contrast. The composites we analyse consist of a perforated matrix material, which we call the "stiff" material, and a "soft" material that fills the remaining pores. We assume that the pores are of size 0 < ε << 1 and are periodically distributed with period ε. We also assume that the stiffness of the soft material degenerates with rate ε2γ,γ > 0, so that the contrast between the two materials becomes infinite as ε ↓ 0. We study the homogenisation limit ε ↓ 0 in a low energy regime, where the displacement of the stiff component is infinitesimally small. We derive an effective two-scale model, which, depending on the scaling of the energy, is either a quadratic functional or a partially quadratic functional that still allows for large strains in the soft inclusions. In the latter case, averaging out the small scale-term justifies a single-scale model for high-contrast materials, which features a non-linear and non-monotone effect describing a coupling between microscopic and the effective macroscopic displacements. [ABSTRACT FROM AUTHOR]

We show that nonlinearly elastic plates of thickness $$h\rightarrow 0$$ with an $$\varepsilon $$ -periodic structure such that $$\varepsilon ^{-2}h\rightarrow 0$$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional elasticity: in general, their effective stored-energy density is 'discontinuously anisotropic' in all directions. The proof relies on a new result concerning an additional isometric constraint that deformation fields must satisfy on the microscale. [ABSTRACT FROM AUTHOR]

The wavy structure of liquid film in annular gas-liquid flow was studied using a high-speed modification of the laser-induced fluorescence (LIF) technique, which was adapted for three-dimensional measurements. The three-dimensional structure of different types of waves in regimes with and without liquid entrainment was investigated. A comparison of the circumferential size of different types of waves was performed. Disturbance waves at high liquid Reynolds numbers were shown to be circumferentially non-uniform, and it was shown that this non-uniformity affects the generation of ripples. [ABSTRACT FROM AUTHOR]

We consider an eigenvalue problem for a divergence-form elliptic operator Aε that has high-contrast periodic coefficients with period ε in each coordinate, where ε is a small parameter. The coefficients are perturbed on a bounded domain of “order one” size. The local perturbation of coefficients for such an operator could result in the emergence of localized waves—eigenfunctions whose corresponding eigenvalues lie in the gaps of the Floquet–Bloch spectrum. For the so-called double porosity-type scaling, we prove that the eigenfunctions decay exponentially at infinity, uniformly in ε Then, using the tools of twoscale convergence for high-contrast homogenization, we prove the strong two-scale compactness of the eigenfunctions of Aε. This implies that the eigenfunctions converge in the sense of strong two-scale convergence to the eigenfunctions of a two-scale limit homogenized operator A0, consequently establishing “asymptotic one-to-one correspondence” between the eigenvalues and the eigenfunctions of the operators Aε and A0. We also prove, by direct means, the stability of the essential spectrum of the homogenized operator with respect to local perturbation of its coefficients. This allows us to establish not only the strong two-scale resolvent convergence of Aε to A0 but also the Hausdorff convergence of the spectra of Aε to the spectrum of A0, preserving the multiplicity of the isolated eigenvalues. [ABSTRACT FROM PUBLISHER]

Annular gas–liquid flow was investigated using high-speed modification of LIF technique. The evolution of liquid film surface was studied with high spatial and temporal resolution. It was found that all waves may be classified into primary and secondary waves; all the secondary waves arise due to instability of the back fronts of primary waves. The areas of inception and velocity of secondary waves were compared for annular flow with and without entrainment. The similarities and differences of wavy structure in both cases were demonstrated. [ABSTRACT FROM AUTHOR]