On one contact problem of plane elasticity for a doubly connected domain: application to a hexagon

The paper addresses a problem of plane elasticity theory for a doubly connected body whose external boundary is a regular hexagon boundary, and the internal boundary is the required full-strength hole including the origin of coordinates. Hexagon’s two vertices are laid at the axis Oy, and the middle points of its two opposite sides are laid at the axis Ox. This full-strength hole is cycle symmetric. It is assumed that to every link of the broken line of the outer boundary of the given body are applied absolutely smooth rigid stamps with rectilinear bases, which are under action of the force P that applies to their middle points. There is no friction between the surface of given elastic body and stamps. The unknown full-strength contour is free from outer actions. Using the methods of complex analysis, the analytical image of Kolosov–Muskhelishvili’s complex potentials (characterizing an elastic equilibrium of the body) and unknown parts of its boundary are determined under the condition that the tangential normal moment arising at it takes a constant value. Such holes are called full-strength holes. Numerical analysis are also performed and the corresponding graphs are constructed.     

Stress concentration control in the problem of plane elasticity theory

In engineering practice, one of the important problems is the problem of finding full-strength contours which permits to control stress concentration at the hole boundary. The article addresses the mixed problem of plane elasticity theory for doubly-connected domain with partially unknown boundary conditions. In the presented work the stress state of the given body and full-strength contours were defined. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Counterexamples to Rational Dilation on Symmetric Multiply Connected Domains

We show that if R is a compact domain in the complex plane with two or more holes and an anticonformal involution onto itself (or equivalently a hyperelliptic Schottky double), then there is an operator T which has R as a spectral set, but does not dilate to a normal operator with spectrum on the boundary of R.     

Homogenization in perforated domains with mixed conditions

In this paper we study the asymptotic behaviour of the Laplace equation in a periodically perforated domain of R ⁿ , where we assume that the period is ε and the size of the holes is of the same order of greatness. An homogeneous Dirichlet condition is given on the whole exterior boundary of the domain and on a flat portion of diameter if (, if n=2) of the boundary of every hole, while we take an homogeneous Neumann condition elsewhere.     

Asymptotics for eigenvalues of the Laplacian in higher dimensional periodically perforated domains

This paper considers the periodic spectral problem associated with the Laplace operator written in \mathbbR^N{\mathbb{R}^N} (N = 3, 4, 5) periodically perforated by balls, and with homogeneous Dirichlet condition on the boundary of holes. We give an asymptotic expansion for all simple eigenvalues as the size of holes goes to zero. As an application of this result, we use Bloch waves to find the classical strange term in homogenization theory, as the size of holes goes to zero faster than the microstructure period.     

An extremal problem for Laplace's equation in a half-space

The use of a minimum condition for a given functional to determine a function harmonic in a half-space whose boundary value is given only on part of the boundary plane is reduced to one or more Dirichlet problems for plane regions. The functional is defined as the L₂-distance between the limiting values of the normal derivative of the harmonic function and a given function.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 101–103.     

Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary

The paper is devoted to the studies of viscous flows caused by a vibrating boundary. The fluid domain is a half‐space, its boundary is a nondeformable plane that exhibits purely tangential vibrations. Such a simple geometrical setting allows us to study general boundary velocity fields and to obtain general results. From a practical viewpoint, such boundary conditions may be seen as the tangential vibrations of the material points of a stretchable plane membrane. In contrast to the classical boundary layer theory, we aim to build a global solution. To achieve this goal we employ the Vishik–Lyusternik approach, combined with two‐timing and averaging methods. Our main result is: we obtain a uniformly valid in the whole fluid domain approximation to the global solutions. This solution corresponds to general boundary conditions and to three different settings of the main small parameter. Our solution always include the inner part and outer part that both contain oscillating and non‐oscillating components. It is shown that the nonoscillating outer part of the solution is governed either by the full Navier–Stokes equations or the Stokes equations (both with the unit viscosity) and can be interpreted as a steady or unsteady streaming. In contrast to the existing theories of a steady streaming, our solutions do not contain any secular (infinitely growing with the inner normal coordinate) terms. The examples of the spatially periodic vibrations of the boundary and the angular torsional vibrations of an infinite rigid disc are considered. These examples are still brief and illustrative, while the core of the paper is devoted to the adaptation of the Vishik–Lyusternik method to the development of the general theory of vibrational boundary layers.     

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

A new method for solution of the evolution of plane curves satisfying the geometric equation v=β(x,k,ν), where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ ? ?² at the point x∈Γ, is proposed. We derive a governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves and prove local in time existence of a classical solution. These equations include a non‐trivial tangential velocity functional governing a uniform redistribution of grid points and thus preventing numerically computed solutions from forming various instabilities. We discretize the governing system of equations in order to find a numerical solution for 2D anisotropic interface motions and image segmentation problems. Copyright © 2004 John Wiley & Sons, Ltd.     

On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications

Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L² tangential fields and then the attention is focused on some particular Sobolev spaces of order $‐{1\over 2}$\nopagenumbers\end . In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Γ is the boundary of a polyhedron Ω, these spaces are important in the analysis of tangential trace mappings for vector fields in H ( curl , Ω) on the whole boundary or on a part of it. By means of these Hodge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright © 2001 John Wiley & Sons, Ltd.     

A spotlike edge state in plane Poiseuille flow

We study the laminar turbulent boundary in plane Poiseuille flow at Re = 1400 and 2180 using the technique of edge tracking. For large computational domains the attracting state in the laminar-turbulent boundary is localized in spanwise and streamwise direction and chaotic. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Homogenization of the Navier-Stokes equations with a slip boundary condition

This paper deals with the homogenization of the Stokes or Navier-Stokes equations in a domain containing periodically distributed obstacles, with a slip boundary condition (i.e., the normal component of the velocity is equal to zero, while the tangential velocity is proportional to the tangential component of the normal stress). We generalize our previous results (see [1]) established in the case of a Dirichlet boundary condition; in particular, for a so-called critical size of the obstacles (equal to ε³ in the three-dimensional case, ε being the inter-hole distance), we prove the convergence of the homogenization process to a Brinkman-type law.     

The Effects of Flexion and Torsion on a Fluid Flow Through a Curved Pipe

We consider an injection of incompressible viscous fluid in a curved pipe with a smooth central curve γ . The one-dimensional model is obtained via singular perturbation of the Navier—Stokes system as ɛ , the ratio between the cross-section area and the length of the pipe, tends to zero. An asymptotic expansion of the flow in powers of ɛ is computed. The first term in the expansion depends only on the tangential injection along the central curve γ of the pipe and the velocity as well as the pressure drop are in the tangential direction. The second term contains the effects of the curvature (flexion) of γ in the direction of the tangent while the effects of torsion appear in the direction of the normal and the binormal to γ . The boundary layers at the ends of the pipe are studied. The error estimate is proved. Accepted 21 March 2001. Online publication 9 August 2001.     

Uniform Regularity in a Wedge and Regularity of Traces of CR Functions

We discuss in Sect. 1 the property of regularity at the boundary of separately holomorphic functions along families of discs and apply, in Sect. 2, to two situations. First, let W\mathcal{W} be a wedge of ℂ ⁿ with C ^ω , generic edge ℰ: a holomorphic function f on W\mathcal{W} has always a generalized (hyperfunction) boundary value bv(f) on ℰ, and this coincides with the collection of the boundary values along the discs which have C ^ω transversal intersection with ℰ. Thus Sect. 1 can be applied and yields the uniform continuity at ℰ of f when bv(f) is (separately) continuous. When W\mathcal{W} is only smooth, an additional property, the temperateness of f at ℰ, characterizes the existence of boundary value bv(f) as a distribution on ℰ. If bv(f) is continuous, this operation is consistent with taking limits along discs (Theorem 2.8). By Sect. 1, this yields again the uniform continuity at ℰ of tempered holomorphic functions with continuous bv. This is the theorem by Rosay (Trans. Am. Math. Soc. 297(1):63–72, 1986), in whose original proof the method of “slicing” by discs is not used.     

Diffraction of a magnetoelastic wave on stress concentrators in conducting bodies (antiplanar deformation)

We consider a two-dimensional boundary-value problem of magnetoelasticity for a half-space weakened by tunnel stress concentrators (cracks, holes) in the presence of a static magnetic field. The mechanical stimulus is taken as a magnetoelastic shear wave incident from infinity or a shear load that varies harmonically in time and is prescribed on the edges of the crack or hole. The problem reduces to a singular integral equation that can be solved numerically by the method of mechanical quadratures. We give the results of computation of the coefficient of stress intensity K₁₁₁ for a slit and the stress concentration on the edge of a hole. We conclude that it is necessary to take account of electromagnetic effects in estimating the strength of diamagnetic or paramagnetic bodies. Four figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 103–110 1991.     

A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach

We consider a periodically perforated domain obtained by making in a periodic set of holes, each of them of size proportional to ε. Then, we introduce a nonlinear boundary value problem for the Lamé equations in such a periodically perforated domain. The unknown of the problem is a vector‐valued function u, which represents the displacement attained in the equilibrium configuration by the points of a periodic linearly elastic matrix with a hole of size ε contained in each periodic cell. We assume that the traction exerted by the matrix on the boundary of each hole depends (nonlinearly) on the displacement attained by the points of the boundary of the hole. Then, our aim is to describe what happens to the displacement vector function u when ε tends to 0. Under suitable assumptions, we prove the existence of a family of solutions {u(ε, ? )}_{ε ∈ ]0,ε ′ [} with a prescribed limiting behavior when ε approaches 0. Moreover, the family {u(ε, ? )}_{ε ∈ ]0,ε ′ [} is in a sense locally unique and can be continued real analytically for negative values of ε. Copyright © 2013 John Wiley & Sons, Ltd.     

On detachment conditions in the problem on the motion of a rigid body on a rough plane

The classical mechanical problem about the motion of a heavy rigid body on a horizontal plane is considered within the framework of theory of systems with unilateral constraints. Under general assumptions about the character of friction, we examine the question on the possibility of detachment of the body from the plane under the action of reaction of the plane and forces of inertia. For systems with rolling, we find new scenarios of the appearing of motions with jumps and impacts. The results obtained are applied to the study of stationary motions of a disk. We have showed the following.

Local existence and uniqueness theorems for a free boundary problem

Free boundary problems are considered, where the tangential and normal components u_t and u_n of an otherwise unknown plane harmonic vector field are prescribed along the unknown boundary curve as a function of the coordinates x, y and the tangent angle θ. The vector field is required to exist either in the interior region G⁺ or in the exterior G^?. In each case the free boundary is characterized by a nonlinear integral equation. A linearised version of this equation is a one-dimensional singular integral equation. Under rather general hypotheses which are easy to check, the properties of the linear equation are described by Noether's theorems. The regularity of the solution is studied and the effect of the nonlinear terms is estimated. A variant of the Nash-Moser implicit-function theorem can be applied. This yields local existence and uniqueness theorems for the free boundary problem in Hölder-classes H^2+μ. The boundary curve depends continuously on the defining data. Finally some examples are given, where the linearised equation can be completely discussed.     

Some remarks on the Dirichlet problem in plane exterior domains

Abstract  In this paper we deal with the Dirichlet problem for the Laplace equation in a plane exterior domain Ω with a Lipschitz boundary. We prove that, if the boundary datum a is square summable, then the problem admits a solution which tends to a in the sense of nontangential convergence, is unique in a suitable function class and vanishes at infinity as r^–k if and only if a satisfies k compatibility conditions, which we are able to explicit when Ω is the exterior of an ellipse. Keywords: Dirichlet problem, Asymptotic behavior, Potential theory Mathematics Subject Classification (2000): 31A05, 31A10     

On the Neumann problem for the Helmholtz equation in a plane angle

We prove that the solution of the Neumann problem for the Helmholtz equation in a plane angle Ω with boundary conditions from the space H_−1/2(Γ), where Γ is the boundary of Ω, which is provided by the well‐known Sommerfeld integral, belongs to the Sobolev space H₁(Ω) and depends continuously on the boundary values. To this end, we use another representation of the solution given by the inverse two‐dimensional Fourier transform of an analytic function depending on the Cauchy data of the solution. Copyright © 2000 John Wiley & Sons, Ltd.     

Boundary Layer Heat Transport in turbulent Rayleigh-Bénard Convection in Air

We study the convective heat transfer in a large-scale Rayleigh-Bénard (RB) experiment which is called the “Barrel of Ilmenau”. We present the results of flow visualization and Particle Image Velocimetry (PIV) measurements of the near wall flow field in a plane perpendicular to the surface of the heated bottom plate. The experiment was run in a smaller rectangular inset that was placed inside the larger barrel. The Rayleigh number amounts to Ra = 1.4 × 10¹⁰. The aspect ratios were Γ_x = 1 in flow direction and Γ_y = 0.26 perpendicular to the vertical flow plane. The measurements have been undertaken using a 2 W continuous wave Laser in combination with a light sheet optics and various cameras. Due to the slender geometry of the cell, the mean wind is confined in one direction where the Laser light sheet is aligned parallel. The flow was seeded with droplets of 1...2 µm size generated using an ordinary fog machine. Flow visualization as well as the PIV data clearly show the intermittent character of the boundary layer flow field that permanently switches between “laminar” and “turbulent” phases. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)