Heat conduction in transversally graded laminates with non-uniform microstructure

Two-component laminates made of conductors distributed non-periodically as laminas along one direction are considered in this note. The macroscopic properties of these laminates are described by continuous slowly-varying functions across laminas. In order to analyse heat conduction, the approach, called the tolerance modelling, is used. The aim of this note is to use averaged equations of the tolerance model for transversally graded laminates to analyse stationary heat conduction across laminas. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Viscoelastic Rayleigh waves in a layered structure with a weak lateral inhomogeneity

Rayleigh waves in an almost layered viscoelastic medium are studied by using the “surface” ray method based on real rays. Viscoelasticity is described in terms of the Maxwell-Boltzmann-Volterra model and, for high frequencies, is treated as perturbed perfect elasticity. In addition to the leading term of the ray asymptotics, which corresponds to the balance of energy along rays, the correction term describing anomalous displacements of the Love type is discussed. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 7–13.     

Viscoelastic Love waves in a layered structure with weak lateral inhomogeneity

High-frequency Love surface waves in a linear medium with Maxwell-Boltzmann-Volterra anelasticity are considered. Arbitrary vertical dependences of the material parameters are allowed. The weak lateral inhomqgeneity and anelasticity of the medium, assumed small in the high-frequency range, are treated as perturbations. The leading term of the ray expansion, which corresponds to the balance of energy along real surface rays, is provided. The additional components, i.e., the Rayleigh-type components of the displacement, described by a higher-order correction, are discussed.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 239, 1997, pp. 7–11.This work was supported in part by the Russian State Committee for Higher Education under grant 95-0-13.1-66.     

Heat conduction in nanofluids

We show that macroscale heat conduction in nanofluids is of a dual-phase-lagging type rather than the Fourier type. This leads to models for effective thermal capacity, conductivity and diffusivity of nanofluids and reveals even more anomalous thermal behavior of nanofluids than those reported in the literature. Due to the coupled conduction of the two phases, thermal waves and possibly resonance may appear in nanofluid heat conduction. Such waves and resonance are responsible for the extraordinary conductivity enhancement. The analysis and result are also valid for heat conduction in two-phase systems.     

Heat conduction with radiating boundary conditions

Existence of positive solutions is established for a class of nonlinear boundary value problems that include steady-state heat conduction problems with radiation at the boundary according to the fourth power radiation law. Existence is established using topological methods and a priori bounds.     

Heat conduction with nonlinear boundary condition

The paper is concerned with determination of the solution of the one-dimensional heat equation in a semi-infinite region subject to a nonlinear boundary condition at the surface. The exact solution is obtained and is expressed in infinite series. It is shown that the series is absolutely and uniformly convergent. Two special cases of Newton's cooling and of Stefan-Boltzmann's radiation at the boundary are discussed.

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Zusammenfassung Diese Arbeit befaßt sich mit der Lösung der Gleichung der eindimensionalen Wärmeleitung in einem Halbraum mit nicht-linearen Randbedingungen an der Oberfläche. Es wird eine genaue Lösung gefunden, die die Form einer unendlichen Reihe besitzt. Es wird gezeigt, daß die Reihe absolut und gleichmäßig konvergiert. Als zwei Spezialfälle werden Newton-Kühlung und Stefan-Boltzmann-Strahlung behandelt.

     

Abundant semigroups with a quasi-ideal quasi-adequate transversal

Characterization theorems for abundant sernigroups having a quasi-ideal quasi-adequate transversal are obtained. Our results generalize and amplify the related results of Satio on regular semigroup obtained in 1985 and Kong obtained in 2007 respectively. Some recent results on this topic given by Guo-Shum are strengthened. In particular, the structure of such an abundant semigroup is described.     

Abundant semigroups with a multiplicative quasi-adequate transversal

The concept of quasi-adequate transversals is introduced. Some properties and characterizations for abundant semigroups with a multiplicative quasi-adequate transversal are obtained. In particular, a structure theorem for such abundant semigroups is given. Finally, some special cases are considered. This research was partially supported by the National Natural Science Foundation of China (No. 10571077) and the Natural Science Foundation of Gansu Province (No. 3ZS052-A25-017).     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

Inverse problem of acoustic scattering in a space with a local inhomogeneity

An inclusion of finite size with a variable wave propagation velocity is contained in a homogeneous space. It is exposed to plane waves being propagated in all possible directions. The inverse problem is to restore the velocity by means of the scattering amplitude which is determined in terms of the scattered wave asymptotic for large x. A procedure for restoration is described and a uniqueness theorem is presented in the paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 156, pp. 24–34, 1986.The authors are deeply grateful to the participants of the A. S. Blagoveschenskii seminar for valuable comments and discussion.     

Heat conduction on the ring: Interface problems with periodic boundary conditions

The classical problem of heat conduction in one dimension on a composite ring is examined. The problem is formulated using the heat equation with periodic boundary conditions. We provide an explicit solution of this problem using the Method of Fokas. The location of the interfaces is known, but neither temperature nor heat flux are prescribed there. Instead, the physical assumption of continuity at the interface is imposed.     

Balanced sets and circuits in a transversal space

The study of fully dependent sets (unions of circuits) has played a part in characterizing transversal spaces. In fact, the fully dependent sets satisfy |Δ(F)| = ?(F) in any deltoid representation, and it is with a consideration of this property that we begin the present paper. We study “balanced” sets and from our results draw conclusions about fully dependent sets and circuits in a transversal space. These include upper bounds for the number of circuits, and the result that a non-trivial transversal space can be neither a hereditary circuit space nor the dual of a geometric hereditary circuit space. The paper is reasonably self-contained; all unusual terms are defined as they are encountered.     

A circular inhomogeneity with a mixed-type imperfect interface in anti-plane shear

We present a rigorous study of the problem associated with a circular inhomogeneity embedded in an infinite matrix subjected to anti-plane shear deformations. The inhomogeneity and the matrix are each endowed with separate and distinct surface elasticities and are bonded together through a soft spring-type imperfect interphase layer. This combination is referred to in the literature as a ‘mixed-type imperfect interface’ due to the fact that the soft interphase layer (described by the spring model) is bounded by two stiff interfaces arising from the separate surface elasticities of the inhomogeneity and the matrix. The entire composite is subjected to remote shear stresses and we allow for the presence of a screw dislocation in either the inhomogeneity or the matrix. The corresponding boundary value problem is reduced to two coupled second-order differential equations for the two analytic functions defined in the two phases (as well as their analytical continuations) leading to solutions in either series or closed-form. The analysis indicates that the stress field in the composite and the image force acting on the screw dislocation can be described completely in terms of three size-dependent parameters and a size-independent mismatch parameter. Interestingly, in the absence of the screw dislocation, the size-dependent stress field inside the inhomogeneity is uniform. Several numerical examples are presented to demonstrate the solution for a screw dislocation located inside the matrix. The results show that it is permissible for the dislocation to have multiple equilibrium positions.     

Thwarts in transversal designs

A subset of points in a transversal design is athwart if each block in the design has one of a small number of intersection sizes with the subset. Applications to the construction of mutually orthogonal latin squares are given. One particular case involves inequalities for the minimum number of distinct symbols appearing in an × subarray of an×n latin square. Using thwarts, new transversal designs are determined for orders 408, 560, 600, 792, 856, 1046, 1059, 1368, 2164, 2328, 2424, 3288, 3448, 3960, 3992, 3994, 4025, 4056, 4824, 5496, 6264, 7768, 7800, 8096, and 9336.