Singular integral equations applied to free surface seepage from non-linear channels

The problem of two-dimensional seepage from a non-linear channel through a homogeneous medium underlain at a finite depth by a drain will be considered. A new approach is given, transforming the seepage problem to a non-linear singular integral equation for which the unique existence of the solution is proved. The free boundaries of the seepage region can be described explicitly.     

Effect of soil capillarity on seepage from a channel with evaporation

The method of Polubarinova-Kochina is applied to solve the seepage problem for a channel with evaporation and capillary soil. The capillary spread of water and the seepage discharge from the channel are investigated as a function of the height of capillary rise and evaporation rate. Results of numerical experiments are presented. The limiting case of flow without evaporation, previously studied by Rizenkampf, is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 40–43, 1986.     

Transformation of complex velocity area in some problems of filtration in presence of evaporation or infiltration to free surface

We study a question about possible transformations of complex velocity area in some problems of filtration theory depending on ranges of change constant of conformal mapping, which is contained in expressions for mapping function. We examine a linear differential equation of the Fuchsian class, which conforms to problem of conformal mapping circular hexagons in polar grid, typical for problems of filtration theory. It has been shown that with fixed parameter, that determines ratio of circles radii, constituting opposite sides of polygons in the complex flow velocity area, configuration and positional relationship of cutset appreciably depend on not only the properties of the functions, according to which we construct partial solution of the equation under consideration, but also depend on ranges of change constants of conformal mapping.     

Investigation of free vibrations of composite beams by using the finite-difference method

The free-vibration behavior of symmetrically laminated fiber-reinforced composite beams with different boundary conditions is examined. The effects of shear deformation and rotary inertia, separately and/or in combination, on the free-vibration properties of the beams are investigated. The finite-difference method is used to solve the partial differential equations describing the free-vibration motion in each case. The effect of shear deformation on the natural frequencies is considerable, especially for higher frequencies, whereas the influence of rotary inertia is less significant. The study includes comparisons with results available in the literature. In addition, the impact of such factors as the span/depth ratio, fiber orientation, stacking sequence, and material type on free vibrations of the composite beams is investigated. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 3, pp. 331–346, May–June, 2006.     

Cavity formation at the inclined separated impact on a circular cylinder under a free surface of a heavy liquid

Under study is a dynamic mixed problem concerning the impact on a circular cylinder and its subsequent movement with a constant speed in an ideal incompressible liquid. The influence is investigated of the physical and geometrical parameters of the problem on the cavity form and the configuration of the external free surface of the liquid for small times. An asymptotic analysis is carried out for the inner free boundary of liquid taking into account the dynamics of separation points. The reaction force of the medium on the cylinder is determined. The necessity is substantiated of introducing additional cavitation zones in the dynamic problem of impact.     

On steady irrotational flow of incompressible ideal fluid in a circular domain with free surface

We study irrotational steady flows of an incompressible ideal fluid in a two-dimensional circular domain, with free surface. Stationary solutions have been studied as bifurcation problems assuming the obstacle exactly circular, and in the presence of surface tension. In this paper, disregarding the surface tension, for non circular obstacles, we analize the set of solutions close to bifurcation points letting the Froud number as driving parameter.

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Sunto Si studiano moti irrotazionali stazionari di un fluido ideale incomprimibile in un dominio bidimensionale circolare con superficie libera. Di recente, le soluzioni stazionarie sono considerate come problemi di biforcazione assumendo che l’ostacolo sia un cerchio, supponendo che vi sia una tensione superficiale sulla superficie libera. In questo lavoro trascurando la tensione superficiale e considerando ostacoli non circolari, analizziamo l’insieme delle soluzioni vicino punti di biforcazione al variare del numero di Fround.

     

Solution of a seepage problem in a nonhomogeneous medium by the method of p-analytical functions

We consider the problem of plane seepage in a nonhomogeneous medium of infinite thickness under reservoirs separated by Zhukovskii cutoff walls. The seepage coefficient of the permeable foundation is assumed to decrease with depth as a power function k(x, y)=(–y) ^–k (k=const > 0). For the case when k is a positive integer, the solution of the problem by the method of p-analytical functions is reduced to the solution of a system of Fredholm integral equations of the second kind. Some particular cases are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 55–66, 1988.     

A singular-degenerate free boundary problem arising from the moisture evaporation in a partially saturated porous medium

Summary We consider a moisture evaporation process in a porous medium which is partially saturated by a fluid. The mathematical model is a singular-degenerate nonlinear parabolic free boundary problem. We first transform the problem into a weak form in a fixed domain and then derive some uniform estimates for the proper approximate solution. The existence of a weak solution is established by a compactness argument. Finally, the regularity of the solution and interfaces are investigated.     

Solution of the bending problem of a circular plate with a free edge by using paired equations

Used in the investigation of the bending of a plate with a free edge is a method including the solution of non-standard paired equations that are trigonometric series, which results in a quasi-completely-regular system of linear algebraic equations. The nature of the plate state of stress and strain is investigated. One of the methods reducing to the solution of paired summation equations is elucidated in /1/ in application to the problem under consideration. A combination of support and clamping is examined in the majority of papers while the case of the free edge is inadequately studied.     

On the kinetics at a free surface

The kinematics required at a fluid interface where tangentialcomponents of surface traction vanish are discussed. This conditionforces one principal axis of strain to remain aligned with thesurface normal. From the rotation rate of the surface normalwe calculate the vorticity required at the interface. Previousresults restricted to special geometries are generalized andthe role of the surface geometry is discussed.     

A note on surface waves created by a rotating circular cylinder

A circular cylinder, placed vertically in a liquid with a free surface, makes oscillatory rotations about its axis. We find that small amplitude, radially progressive waves are created on the surface of the liquid that move a finite distance away from the cylinder.

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Résumé Nous considérons les oscillations axiales d'un cylindre placé vérticalement dans une liquide à surface libre, et démontrons l'existence des ondes radiales à faible amplitude sur la surface s'eloignant à une distance finie du cylindre.

     

Scattering of guided SH-wave by a partly debonded circular cylinder in a traction free plate

A solution of the scattering problem of guided SH-wave by a partly debonded circular cylinder centered in a traction free plate has been set up. The plate is divided up into three regions with two imaginary planes perpendicular to the plate walls. In the central region where the partly debonded cylindrical obstacle is posted, the wave field is expanded into the cylindrical wave modes and Chebyshev polynomials. In the other two exterior regions the fields are expanded into the plate wave modes. A system of fundamental equations to solve the problem is obtained according to the traction free boundary condition on the plate walls and the continuity condition of the traction and the displacement across the imaginary planes. The approximate numerical method termed mode-matching technique is used to construct a matrix equation to obtain curves showing the coefficient of reflection and transmission versus the ratio of the cylinder’s radius to the plate’s half-thickness and the angular width of the debonded region. A comparison of the numerical results between the welded interface condition and the debonded interface condition is made, and the results are discussed.     

Automorphisms of the endomorphism semigroup of a free monoid or a free semigroup

We determine all isomorphisms between the endomorphism semigroups of free monoids or free semigroups and prove that automorphisms of the endomorphism semigroup of a free monoid or a free semigroup are inner or ``mirror inner". In particular, we answer a question of B. I. Plotkin.

     

Purity of the stratification by Newton polygons

Let be a variety in characteristic . Suppose we are given a nondegenerate -crystal over , for example the th relative crystalline cohomology sheaf of a family of smooth projective varieties over . At each point of we have the Newton polygon associated to the action of on the fibre of the crystal at . According to a theorem of Grothendieck the Newton polygon jumps up under specialization. The main theorem of this paper is that the jumps occur in codimension on (the Purity Theorem). As an application we prove some results on deformations of iso-simple -divisible groups.

     

Separating two simple polygons by a sequence of translations

LetP andQ be two disjoint simple polygons havingm andn sides, respectively. We present an algorithm which determines whetherQ can be moved by a sequence of translations to a position sufficiently far fromP without colliding withP, and which produces such a motion if it exists. Our algorithm runs in timeO(mn(mn) logm logn) where (k) is the extremely slowly growing inverse Ackermann's function. Since in the worst case (mn) translations may be necessary to separateQ fromP, our algorithm is close to optimal.Work on this paper by the first author has been supported by National Science Foundation Grant No. DMS-8501947. Work on this paper by the second author has been supported by Office of Naval Research Grant No. N00014-82-K-0381, National Science Foundation Grant No. NSP-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the second and third authors has also been supported by a grant from the joint Ramot-Israeli Ministry of Industry Foundation. Part of the work on this paper has been carried out at the Workshop on Movable Separability of Sets at the Bellairs Research Institute of McGill University, Barbados, February 1986.     

Enumeration of the self-avoiding polygons on a lattice by the Schwinger-Dyson equations

We show how to compute the generating function of the self-avoiding polygons on a lattice by using the statistical mechanics Schwinger-Dyson equations for the correlation functions of theN-vector spin model on that lattice.     

Effect of pressure work on free convection flow along a vertical circular cone with suction and variable surface temperature

Free convection laminar flow from a vertical circular cone maintained a variable surface temperature with suction and pressure work effects has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations are reduced to local non-similarity equations. The governing non-similarity equations are then solved numerically by implicit finite difference method together with Keller box scheme. Numerical results are presented in terms of velocity and temperature profiles of the fluid as well as the local skin-friction coefficients and the local heat transfer rate for different values of Prandtl number Pr, suction parameter ξ, temperature gradient parameter n and the pressure work parameter ∈. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)