On the Generation of Mean Flows by the Interaction of Görtler Vortices and Tollmien-Schlichting Waves in Curved Channel Flows

There are many fluid flows where the onset of transition can be caused by different instability mechanisms which compete in the nonlinear regime. Here the interaction of a centrifugal instability mechanism with the viscous mechanism which causes Tollmien-Schlichting waves is discussed. The interaction between these modes can be strong enough to drive the mean state; here the interaction is investigated in the context of curved channel flows so as to avoid difficulties associated with boundary layer growth. Essentially it is found that the mean state adjusts itself so that any modes present are neutrally stable even at finite amplitude. In the first instance the mean state driven by a vortex of short wavelength in the absence of a Tollmien-Schlichting wave is considered. It is shown that for a given channel curvature and vortex wavelength there is an upper limit to the mass flow rate which the channel can support as the pressure gradient is increased. When Tollmien-Schlichting waves are present then the nonlinear differential equation to determine the mean state is modified. At sufficiently high Tollmien-Schlichting amplitudes it is found that the vortex flows are destroyed, but there is a range of amplitudes where a fully nonlinear mixed vortex-wave state exists and indeed drives a mean state having little similarity with the flow which occurs without the instability modes. The vortex and Tollmien-Schlichting wave structure in the nonlinear regime has viscous wall layers and internal shear layers; the thickness of the internal layers is found to be a function of the Tollmien-Schlichting wave amplitude.     

Residual stresses in plastic rings reinforced with glass and manufactured with consolidation of the various layers

The residual stresses in thin-wall plastic rings reinforced with glass and prepared by the method of consolidating the layers in combination with various winding regimes have been studied experimentally. It is shown that variation of the winding regime from one layer to another can change the distribution of the stresses through the thickness of the ring.Moscow Energy Institute. Translated from Mekhanika Polimerov, No. 1, pp. 174–176, January–February, 1972.     

Self-similar solutions describing unsteady thermocapillary fluid flows

Unsteady thermocapillary flows in thin layers and layers of infinite thickness with non-uniform heating of the free boundary are investigated at high Marangoni numbers. In the plane and axially symmetric cases, self-similar solutions of the non-linear boundary-layer equations are constructed and asymptotic formulae are presented. It is shown that the self-similar solutions may be non-unique for certain values of the parameters of the problem. The branching points are calculated numerically and the branched solutions are investigated.     

A Hybrid Variational Model for the Analysis of Free‐Corner Effects in Layered Plates

A variational model for the assessment of free‐edge and free‐corner effects [1‐3] in thermally loaded rectangular cross‐ply laminate plates is presented. The physical layers of the plate are discretized by an arbitrary number of mathematical layers through the laminate thickness. A layerwise displacement field with unknown interface functions which depend on the inplane coordinates is formulated wherein an a priori assumed layerwise linear thickness interpolation scheme is employed. The application of the principle of minimum elastic potential yields a set of governing Euler‐Lagrange differential equations for the unknown inplane functions which for the special free‐corner problem in rectangular cross‐ply plates can be solved in a closed‐form analytical manner. Boundary conditions of traction free laminate edges are fulfilled in an average sense. Since the approach utilizes a discretization through the plate thickness yet allows for closed‐form solutions within the layer planes for all state variables it is appropriate to speak of a hybrid analysis approach. The presented method allows easy application, can be run on every standard personal computer and is in favourable agreement with comparative finite element calculations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shear and Flexural Buckling Modes of a Spherical Sandwich Shell in a Centrosymmetric Temperature Field Inhomogeneous across the Thickness

Problems on buckling modes (BMs) are considered for a spherical sandwich shell with thin isotropic external layers and a transversely soft core of arbitrary thickness in a centrosymmetric temperature field inhomogeneous across the shell thickness. For their statement, the two-dimensional equations of the theory of moderate bending of thin Kirchhoff–Love shells are used for the external layers, with regard for their interaction with the core; for the core, maximum simplified geometrically nonlinear equations of thermoelasticity theory, in which a minimum number of nonlinear summands is retained to correctly describe its pure shear BM, are utilized. An exact analytical solution to the problem on initial centrosymmetric deformation of the shell is found, assuming that the temperature increments in the external layers are constant across their thickness. It is shown that the three-dimensional equations for the core, linearized in the neighborhood of the solution, can be integrated along the radial coordinate and reduced to two two-dimensional differential equations, which supplement the six equations that describe the neutral equilibrium of the external layers. It is established that the system of eight differential equations of stability, upon introduction of new unknowns in the form of scalar and vortical potentials, splits into two uncoupled sets of equations. The first of them has two kinds of solutions, by which the pure shear BM is described at an identical value of the parameter of critical temperature. The second system describes a mixed flexural BM, whose realization, at definite combinations of determining parameters of the shell and over wide ranges of their variation, is possible for critical parameters of temperature by orders of magnitude exceeding the similar parameter of shear BM.     

Buckling of laminated plates - A simple finite element based on higher-order theory

A four-noded rectangular element with seven degrees of freedom at each node is developed for buckling analysis of laminated plate structures having any number of layers with a constant thickness of individual layers. The displacement model is so chosen that it can explain adequately the parabolic distribution of transverse shear stresses and the non-linearity of in-plane displacements across the thickness. A geometrical stiffness matrix is developed using in-plane stresses. A wide range of plates from thick to thin are examined under uniaxial loading conditions. The results are compared with the existing analytical and numerical solutions. The present formulations confirm its applicability for buckling analysis of a wide range of plates.     

Nonlinear high-wavenumber Bnard convection

Weakly nonlinear two-dimensional roll cells in Bnard convectionare examined in the limit as the wavenumber a of the roll cellsbecomes large. In this limit the second harmonic contributionsto the solution become negligible, and a flow develops wherethe fundamental vortex terms and the correction to the meanare determined simultaneously, rather than sequentially as inthe weakly nonlinear case. Extension of this structure to Rayleighnumbers O(a³) above the neutral curve is shown to be possible,with the resulting flow field having a form very similar tothat for strongly nonlinear vortices in a centripetally unstableflow. The flow in this strongly nonlinear regime consists ofa core region, and boundary layers of thickness O(a^–1)at the walls. The core region occupies most of the thicknessof the fluid layer and only mean terms and cos az terms playa role in determining the flow; in the boundary layer all harmonicsof the vortex motion are present. Numerical solutions of thewall layer equations are presented and it is also shown thatthe heat transfer across the layer is significantly greaterthan in the conduction state.     

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.     

A posteriori error estimates for subgrid viscosity stabilized approximations of convection–diffusion equations

We derive a posteriori error estimates for subgrid viscosity stabilized finite element approximations of convection–diffusion equations in the high Péclet number regime. Two estimators are analyzed: an asymptotically robust one and a fully robust one with respect to the Péclet number. Numerical results on test cases with boundary layers or internal layers show that the asymptotically robust estimator can be used to construct adaptive meshes.     

A unified formulation of PVD-based finite cylindrical layer methods for functionally graded material sandwich cylinders

A unified formulation of finite cylindrical layer methods (FCLMs) based on the principle of virtual displacements (PVDs) is developed for the quasi-three-dimensional (3D) bending and free vibration analyses of simply-supported, functionally graded material (FGM) sandwich circular hollow cylinders, in which the material properties of the FGM layer are assumed to obey the power-law distributions of the volume fractions of the constituents through the thickness coordinate. In this formulation, the cylinder is divided into a number of cylindrical finite layers, where the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the displacement components of each individual layer, respectively. Because an h-refinement is adopted in this article to yield the convergent solutions, the relative orders used for expansion of the displacement components remain variable, and can be freely chosen as linear, quadratic and cubic ones. The accuracy and convergence rate of a variety of PVD-based FCLMs developed in this article are assessed by comparing their solutions with the available 3D ones.     

The antiplane problem of the motion of a point shear load over a two-layer elastic base at a constant velocity

The problem of modelling the motion of a force disturbance in an elastic medium that is heterogeneous over its depth is investigated. It is in an antiplane formulation in a moving system of coordinates that all possible versions of the ratio of the velocity of motion of the surface point shear load to the velocities of the shear waves in the layers of the two-layer elastic base are examined. Cases of a subsonic regime (SBR) in the upper and lower layers, of a supersonic regime (SPR) in the upper layer and an SBR in the lower layer, and of an SBR in the upper layer and an SPR in the lower layer are studied using the Fourier transform and the theory of residues. The last two cases are extremely interesting from the mathematical point of view, as here, on the boundary between the layers, the solutions of elliptic and hyperbolic equations meet, and previously unknown features arise in the displacements that,it seems, should also occur in the solution of the corresponding plane problem. The case of an SPR in the upper and lower layers is investigated using a special method for successive allowance for the incident, reflected and refracted shock wave fronts. In all cases, expressions are obtained for the displacements in the layers, and their characteristic features are investigated.     

RMVT- and PVD-based finite cylindrical layer methods for the three-dimensional buckling analysis of multilayered FGM cylinders under axial compression

The unified formulations of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) and the principle of virtual displacements (PVD) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under axial compression. The material properties of the FGM layer are assumed to obey the power-law distributions of the volume fraction of the constituents through the thickness coordinate. In these formulations, the cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the primary variables of each individual layer, respectively, as well as the related order of each primary variable can be freely chosen, such as the layerwise linear, quadratic or cubic function distribution through the thickness coordinate. The accuracy and convergence of the RMVT- and PVD-based FCLMs developed in this article are assessed by comparing their solutions with the exact 3D solutions available in the literature.     

Untersuchungen über die partielle Oxydation von pulverförmigem Kupferoxydul zu Kupferoxyd

Summary X-ray and electron-diffraction tests show that, with the aid of a combined oxidizing oven and air sifting apparatus, it is possible to achieve partial oxidation of the Cu₂O particles in a powder in such a way that each Cu₂O particle is encased within a composite CuO layer. The thinnest CuO layer which can still be shown to be composite has a mean thickness of about 120 Å according to an approximate method of calculation. The procedures adopted for this experiment cannot determine whether still thinner layers are also composite. The thickness of the CuO layers increases exponentially with the oxidation temperature. This work has been undertaken in conjunction with research into the ice-forming nucleability of substances.     

Exact solution for Transient bending of a circular plate integrated with piezoelectric layers

This paper presents the exact, explicit solution for the transient motion of a circular plate surface bonded by two piezoelectric layers, based on Kirchhoff plate model. The distribution of eclectic potential along the thickness direction is simulated by a quadratic function so that the Maxwell static electricity equation is satisfied. The piezoelectric layers are electrically grounded over the edge and electrodes at the two surfaces of the piezoelectric layers are shortly connected. The differential equations of motion are solved for simply supported and clamped boundary conditions. The solutions are expressed by elementary Bessel functions and obtained via exact inverse Laplace transform.     

Option price with stochastic volatility for both fast and slow mean-reverting regimes

The Heston model of stochastic volatility has been widely adopted in modern finance, especially in option pricing. Usually, the model can be classified as being in one of two different regimes: the fast mean-reverting regime and the slow mean-reverting regime. Different approximations are needed for each regime. We show a surprising result: the solution in both regimes can be approximated by an identical expression. The predictions of the approximation are in excellent agreement with the numerical solutions of the Heston model in both regimes.     

The fluid dynamic limit of the nonlinear boltzmann equation

Solutions of the nonlinear Boltzmann equation are constructed up to the first appearance of shocks in the corresponding fluid dynamics. This construction assumes the knowledge of solutions of the Euler equations for compressible gas flow. The Boltzmann solution is found as a truncated Hilbert expansion with a remainder, and the remainder term solves a weakly nonlinear equation which is solved by iteration. The solutions found have special initial values. They should serve as “outer expansions” to which initial layers, boundary layers and shock layers can be matched.     

Strength Analyses of Laminated Composite Structures Based on Different Theories

Modern materials, such as composite ones, slowly replace conventional materials in structures of different kind and their growing popularity is caused by their multiple advantages. Through selection of parameters, such as number of layers, thickness of layers, direction of arranging fibers, or material from which internal and outside layers are made, it is possible to control properties of a structure. In the result, structures made of composite materials have high ratio of flexural stiffness to weight. The goal of this paper is to compare three theories of laminated composite plates and shells with the help of the multilayered rectangular plate model subjected to crosswise pressure perpendicular to the surface of a plate. Comparison was made for the classical laminated theory (CLT), first-order laminated theory and third-order laminated theory (TSDT). In all these theories, the number of parameters describing the displacement doesn't depend on the number of layers. For each of these theories strains and displacements were determined. Additionally, the computations time for every method were compared. Obtained results are presented in the form of tables. The analysis of the obtained solutions will be used as the base in choosing the best theory in multicriteria optimization process of composite thin-walled structures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimization of reinforced cylindrical shells with nonuniform thickness

An optimum multilayer shell is designed whose stack of elementary layers has a nonuniform thickness. This optimization problem is solved numerically for the special cases of three-layer cylindrical shells with dynamic and static stability. The optimum variants of layer distribution in this model are compared with the optimum solutions in [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 298–303, March–April, 1976.     

Multidomain spectral method for Schrödinger equations

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied.