Stress functions for plane problems with couple stresses

Zusammenfassung Die Lösung der zweidimensionalen Gleichgewichtsbedingungen für ein Kontinuum, das Spannungsmomente aufnehmen kann, wird mit Hilfe von willkürlichen Spannungsfunktionen gegeben. Die Spannungsfunktionen werden mit Hilfe der resultierenden Einzelkraft und des resultierenden Momentes gedeutet, welche durch einen Bogen im Körper übertragen werden.     

Analytic solution of boundary-value problems for nonstationary model kinetic equations

A theory for constructing the solutions of boundary-value problems for nonstationary model kinetic equations is constructed. This theory was incorrectly presented in the recent well-known monographs of Cercignani [1,2]. After application of a Laplace transformation to the studied equation, separation of the variables is used, this leading to a characteristic equation. Eigenfunctions are found in the space of generalized functions, and the eigenvalue spectrum is investigated. An existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved. The proof is constructive and gives explicit expressions for the expansion coefficients. An application to the Rayleight problem is obtained, and the corresponding result of Cercignani is corrected.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 127–138, July, 1992.     

KdV equation in domains with moving boundaries

The initial-boundary value problem in a bounded domain with moving boundaries and nonhomogeneous boundary conditions for the Korteweg-de Vries equation is considered. Existence and uniqueness of global strong solutions are proved as well as the exponential decay of small solutions in asymptotically cylindrical domains.     

On the moving plane method for nonlocal problems in bounded domains

We consider a nonlocal problem involving the fractional Laplacian and the Hardy potential in bounded smooth domains. Exploiting the moving plane method and some weak and strong comparison principles, we deduce symmetry and monotonicity properties of positive solutions under zero Dirichlet boundary conditions.     

Symplectic manifolds with contact type boundaries

Summary An example of a 4-dimensional symplectic manifold with disconnected boundary of contact type is constructed. A collection of other results about symplectic manifolds with contact-type boundaries are derived using the theory ofJ-holomorphic spheres. In particular, the following theorem of Eliashberg-Floer-McDuff is proved: if a neighbourhood of the boundary of (V, ) is symplectomorphic to a neighbourhood ofS ^2n–1 in standard Euclidean space, and if vanishes on all 2-spheres inV, thenV is diffeomorphic to the ballB ²ⁿ.Oblatum 19-III-1990Partially supported by NSF grant no: DMS 8803056     

Analytic wavelets in multiply connected domains with circular boundaries

Wavelet bases in Hardy-type spaces of analytic functions in a domain with circular boundaries are constructed from a wavelet basis in the Hardy space of functions analytic in the disk. An estimate of the rate of convergence of the partial sums of the constructed wavelets is also obtained.     

Calculation of moving phase boundaries with latent heat

An integrodifferential equation is derived which describes the motion of a phase transition of nonvanishing latent heat in a homogeneous body from a knowledge of only the Green's function and heat source, if present. As the temperature distribution no longer appears in the final equation for the surface, a reduction of one dimension over and above those afforded by symmetries is achieved. This is practically imperative for computer solutions involving realistic geometries. The method is applicable to problems in such diverse fields as metallurgy, vulcanology, and laser interaction with materials.     

A dynamically consistent model of the contact stresses in the plane motion of a rigid body

The problem of determining dry friction forces in the case of the motion of a rigid body with a plane base over a rough surface is discussed. In view of the dependence of the friction forces on the normal load, the solution of this problem involves constructing a model of the contact stresses. The contact conditions impose three independent constraints on the kinematic characteristics, and the model must therefore include three free parameters, which are determined from these conditions at each instant. When the body is supported at three points, these parameters (for which the normal stresses can be taken) completely determine the model, while indeterminacy arises in the case of a larger number of contact points and, in order to remove this, certain physical hypotheses have to be accepted. It is shown that contact models consistent with the dynamics possess certain new qualitative properties compared with the traditional quasi-static models in which the type of motion of the body is not taken into account. In particular, a dependence of the principal vector and principal moment of the friction forces on the direction of sliding or pivoting of the body, as well as on the magnitude of the angular velocity, is possible.     

On free boundary problems with moving contact points for the stationary two-dimensional Navier-Stokes equations

Solvability of the problem of slow drying of a plane capillary in the classical setting (i. e., with the adherence condition on a rigid wall) is established. The proof is based on a detailed study of the asymptotics of the solution near a point of contact of the free boundary with a moving wall, including estimates of the coefficients in well known asymptotic formulas. It is shown that the only value of the contact angle admitting a solution of the problem with finite energy dissipation equals π. Bibliography: 18 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 213, 1994, pp. 179–205. Translated by E. V. Frolova.     

One-dimensional problems with free boundaries in ecology

Statements of problems with free boundaries are given for nonlinear parabolic equations arising in ecology and medicine. Some constructive methods for their solution are considered. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 7, pp. 993–997, July, 1997.     

Magneto-hydrodynamic stability of plane Poiseuille flow with flexible boundaries

MHD stability of plane Poiseuille flow between parallel flexible walls with coplaner magnetic field is analysed. The study is restricted to sufficiently low values of magnetic Reynolds number. The eigen-value problem so posed is then solved graphically and neutral curves are obtained for various sets of magnetic parameter and flexible wall parameters. The nature of influence on flow stability depends on the values of these parameters.     

Minimizing singularities of generic plane disks with immersed boundaries

A cooriented circle immersion into the plane can be extended to a stable map of the disk which is an immersion in a neighborhood of the boundary and with outward normal vector field along the boundary equal to the given coorienting normal vector field. We express the minimal number of fold components of such a stable map as a function of its number of cusps and of the normal degree of its boundary. We also show that this minimum is attained for any cooriented circle immersion of normal degree not equal to one. The first author is a research fellow of the Royal Swedish Academy of Sciences sponsored by the Knut and Alice Wallenberg foundation.     

Determination of contact stresses in problems on bending of a package of transversely isotropic bars

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 761–778, November–December, 2004.     

Optimal control in nonstationary problems of magnetohydrodynamics

Optimal control problems are under study for the magnetohydrodynamic (MHD) equations of an incompressible viscous fluid. An optimality system is constructed and justified in the 2-dimensional case without restrictions of the smallness or regularity of the optimal flow. Some examples of problems are given where the controls are the external currents and the boundary values of the total pressure.     

The method of non-linear time transformation in boundary-value problems of potential theory with moving boundaries for the non-linear wave equation

A new method for solving boundary-value problems for the wave equation [1–3] with moving boundaries is used to obtain a solution of a boundary-value problem with boundary conditions of three types [4].     

Solvability of some two-dimensional quasistationary free boundary problems for the Navier-Stokes equations with moving contact points

The paper is concerned with some quasistationary two-dimensional free boundary problems of viscous flow with moving contact points and with contact angle equal to π. A typical example of such a flow is filling a capillary tube in the presence of surface tension. The proof of the solvability of these problems is based on the analysis (made by the author and V. V. Pukhnachëv about 10 years ago) of the asymptotic formulas for the solutions of the Navier-Stokes equations in a neigborhood of contact points. Bibliography: 10 titles.