A cochlear model with transversally inhomogeneous cross-section

The transversally inhomogeneous cross-section of the model is constructed from the data obtained by Retzius, Hardesty, Wever, and Miller. On the basis of a three-dimensional linear mechanical WKB model of a cochlea without negative friction we obtain high frequency selectivity and a phase shift in the domain of the peak of the amplitude function comparable with the measurements taken by Sellick and Robles.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 128–138, 1989.     

Thermoelastic damping in flexural vibration of bilayered microbeams with circular cross-section

Predicting of thermoelastic damping (TED) is crucial in the design of micro-resonators with composite structures. Several analytical models were developed to evaluate TED in bilayered and three-layered microbeams in the past. However, the previous models focus on the microbeams with rectangular cross-section. This paper aims to study the TED in a bilayered microbeam with circular cross-section. The temperature field is approximated by using sine series and Bessel series in the circular cross-section. An analytical full model for TED in flexural vibration of bilayered microbeam is presented in the form of an infinite series. A simple model is also developed by retaining only the first term. The simulation results of the present model have a good agreement with those of the finite element method (FEM). The results indicate that well-resolved two peaks of TED are typically observed in the bilayered microbeams in which the value of k₂/C₂ is much less or higher than that of k₁/C₁. The present model is not a rapidly converging infinite series for most of the bilayered microbeams. The present model is more suitable for the long slender beams.     

Bending-torsion vibration of a partially submerged cylinder with an arbitrary cross-section

An analytical method is presented to investigate the bending-torsion vibration characteristics of a cylinder with an arbitrary cross-section and partially submerged in water. The compressibility and the free surface waves of the water are considered simultaneously in the analysis. The exact solution of structure–water interaction is obtained mathematically. Firstly, the analytical expression of the velocity potential of the water is derived by using the method of separation of variables. The unknown coefficients in the velocity potential are determined by the longitudinal and circumferential Fourier expansions along the outer surface of the cylinder and are expressed in the form of integral equations including the unknown dynamic bending deflection and torsional angle of the cylinder. Secondly, the force and torque acting on the cylinder per unit length, provided by the water, are obtained by integrating the water dynamic pressure along the circumference of the cylinder. The general solution of bending-torsion vibration of the cylinder under the water dynamic pressure is derived analytically. The integral equations included in the velocity potential of the water can be solved exactly. Finally, the eigenfrequency equation of cylinder–water interaction is obtained by means of the boundary conditions of the cylinder. Some numerical examples for elliptical columns partially submerged in water are provided to show the application of the present method.     

Solitary waves in a fluid-filled thin elastic tube with variable cross-section

The present work treats the arteries as a thin walled prestressed elastic tube with variable cross-section and uses the longwave approximation to study the propagation of weakly nonlinear waves in such a fluid-filled elastic tube by employing the reductive perturbation method. By considering the blood as an incompressible inviscid fluid, the evolution equation is obtained as the Korteweg–de Vries equation with a variable coefficient. It is shown that this type of equations admits a solitary wave type of solution with variable wave speed. It is observed that, for soft biological tissues with an exponential strain energy function the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section

For the purpose of the design and optimization of piezoelectric transducers, the modified double orthogonal polynomial series method is proposed to investigate guided waves in functionally graded piezoelectric(FGP) cylindrical structures with sectorial cross-section. The real, imaginary and complex solutions are obtained simultaneously without iterative process. The real solutions represent propagative waves; the imaginary and complex solutions are evanescent waves. The boundary conditions are incorporated into the constitutive equations by virtue of the Heaviside function. Subsequently, the amplitudes are expanded into the double orthogonal polynomial series, and the motion equations are converted into a matrix eigenvalue problem about complex wavenumber. Numerical comparison with available reference result confirms the validity of the present method. Dispersion curves and the Poynting vector distributions are illustrated. The influences of angular measure, radius-thickness ratio and graded index on dispersion curves are analyzed. Results show that there exist some evanescent guided wave modes that have higher velocities than that of the propagative wave modes and simultaneously have low attenuation at high frequencies. These results can be utilized to improve the performance of transducers.     

The critical cross-section of a vortex

Summary This paper discusses the problem of critical-flow cross-sections in vortex flows. It is shown that there are two different types of vortex flows, A-type and B-type vortices (say). An A-type vortex approaches its critical flow state as its cross-sectional area increases and departs from the critical state as the cross-sectional area is decreased. This property is associated with the particular dependence of total pressure and circulation on the stream function, and it holds for both subcritical and supercritical A-type vortices. On the other hand, both subcritical and supercritical B-type vortices approach their critical flow states as their cross-sectional areas are decreased and depart from their critical states for increasing cross-sectional area. As was shown by Benjamin, setting the first variation of the flow force with respect to the stream function equal to zero leads to Euler's equation of motion. The second variation also vanishes if the corresponding flow state is critical. In this case the sign of the third variation decides whether the flow is an A-type or a B-type vortex. Within the framework of inviscid-fluid flow theory the type of a vortex is preserved unless vortex breakdown occurs. Making use of the knowledge that vortex flows are controlled by two different types of critical-flow cross-sections a variety of vortex flow phenomena are investigated, including the two types of inlet vortices that are observed upstream of jet engines, the behavior of vortex valves, the flow characteristics of liquid-fuel atomizers and the bath tub vortex.     

Thermally driven acoustic oscillations, part IV: Tubes with variable cross-section

The problem of thermally driven acoustic oscillations is treated for tubes with variable cross-section, with particular emphasis on the possible reduction of the necessary temperature ratio for excitation. Tubes with optimal conditions in the vicinity of the temperature jump, and with big cross-sections in parts with constant temperature are found to give the best performance in this respect. Included in the family of devices which were treated is the classical Sondhauss-tube. Experiments which give a striking confirmation of the theory are reported.     

Thermally driven acoustic oscillations,part IV: Tubes with variable cross-section

Summary The problem of thermally driven acoustic oscillations is treated for tubes with variable cross-section, with particular emphasis on the possible reduction of the necessary temperature ratio for excitation. Tubes with optimal conditions in the vicinity of the temperature jump, and with big cross-sections in parts with constant temperature are found to give the best performance in this respect. Included in the family of devices which were treated is the classical Sondhauss-tube. Experiments which give a striking confirmation of the theory are reported.

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Zusammenfassung Das Problem der thermisch getriebenen akustischen Schwingungen wird für Rohre mit veränderlichem Querschnitt behandelt, mit besonderer Berücksichtigung der Frage der Reduktion des für die Anfachung nötigen Temperaturverhältnisses. Rohre, die optimale Radien in der Umgebung des Temperatursprunges haben, und möglichst grosse Querschnitte in Teilen mit konstanter Temperatur aufweisen, erweisen sich am besten in dieser Hinsicht. In der Familier der untersuchten Anordnungen its das klassische Sondhauss-Rohr miteinbezogen. Experimente gaben überzeugende Bestätigung für die verwendeten Theorien.

     

The complex three-dimensional elastoplastic bending of a rod with a square cross-section

The complex three-dimensional bending of a long rod (with square cross-section) made of an isotropic ideally plastic and ideally cyclical material is investigated. The bending of the rod occurs due to the action of two moments, applied to its ends in such a way that the longitudinal deformation of the middle fibres of two neighbouring sides is described by a dashed line.     

A randomized maximum cross-section method to simulate random structure systems with distributed transitions

In this paper, random structure systems with distributed transitions are considered. A theorem on the form of conditional structure distributions is proved. To simulate these distributions, a statistical algorithm using a randomized method of maximum cross-section is constructed. Also, a modified version of this algorithm using simulation with a single random number is constructed. The algorithms are used to simulate the numerical solution of random structure systems with distributed transitions. A theorem on weak convergence of a numerical solution obtained by the algorithms is proved.     

Numerical analysis of dissipative instabilities in toroidal plasma configurations with a noncircular cross-section

Translated from Pryamye i Obratnye Zadachi Matematicheskoi Fiziki, pp. 198–203, 1991.     

Non-quasielliptic boundary-value problems in a cylinder with regularly degenerate model problem on the cross-section

The author introduces a class of boundary-value problems in cylindrical domains. These problems are neither elliptic, nor parabolic, nor quasielliptic, but their main properties are inherited from problems of the said three types, namely, in a proper scale of weighted functional spaces, the operators of these problems are of Fredholm type and their solutions admit asymptotic expansions at infinity with power-exponential terms. The proof is based on precise estimates of solutions of some regularly degenerate model problems on the cross-section of the cylinder in stepwise norms depending on a small parameter.     

e-e Scattering cross-section measurements using page type geometry

Experiments to measure differential scattering cross-sections in electron-electron and positron-electron scattering have been performed. Page type geometry has been employed because of its line focussing pro-perty. Moderately intense radioactive sources could, therefore, be used. Measurements of cross-sections have been carried out over a continuous range of values ofq, the kinetic energy transfer. This paper reports on the measurements of electron-electron scattering cross-sections for energies in the range (0·2–0·65) Mev using thin mylar and aluminium foils of different thickness. M?ller formula has been verified within + 4%.     

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section is explored. A constitutive equation superposing elasticity on the viscoplastic behavior through a linear combination of the non-linear simplified Phan–Thien–Tanner constitutive model of viscoelasticity and the non-linear Bingham constitutive equation of viscoplasticity is developed. The equation of motion is solved analytically for the longitudinal field for steady flow in tubes of non-circular cross-section. The evolution of the plug and stagnant zones, hallmarks of viscoplastic behavior, is studied when elasticity is present, and the rate of flow is determined in terms of the Weissenberg and Bingham numbers. Elasticity tends to enhance the rate of flow for given viscoplastic conditions, and stagnant and plug zone configurations are substantially altered.     

Some applications of cross-section measures in Minkowski spaces

Summary The purpose of this paper is to establish extremal values for inner and outer radii of the unit ball of a Minkowski space for the Holmes--Thompson and Busemann measures. Furthermore, we confirm a conjecture of C. A. Rogers and G. C. Shephard on ellipsoids.     

Normal stresses in flow through tubes of non-circular cross-section

Zusammenfassung Eine visko-elastische Flüssigkeit fliesst durch ein zylindrisches Rohr von nichtkreisförmigem Querschnitt. Die Theorie zweiter Ordnung der visko-elastischen Flüssigkeiten wird auf die Bestimmung der Variation der auf die Rohrwandung ausgeübten Normalspannungen angewandt. Explizite Lösungen für elliptische, dreieckige und rechteckige Querschnitte werden angegeben.     

Dusty viscous flow through a cylinder of triangular cross-section

Unsteady laminar flow of a dusty viscous, incompressible fluid through a cylindrical tube of triangular cross-section is considered in two cases: (i) when the pressure gradient varies harmonically with time and (ii) when it varies exponentially. The velocity fields for the fluid and dust particles have been determined. Flux and skin-friction drag on the walls of the cylinder have been calculated and particular cases discussed.     

Equilibrium and bifurcation of varying cross-section microcantilevers subject to the atomic force

This paper treats a microcantilever with varying cross-section subject to interatomic force derived from the Lennard-Jones potential. Such a microcantilever may be used in the atomic force microscope. The equilibrium configurations are calculated via the Bernoulli–Euler beam theory. The effect of the microcantilever length and its working distance from the sample on the number and the stability of the equilibrium configurations are analyzed via the Lagrange–Dirichlet theorem.     

Numerical investigation of a trapezoidal cross-section ridge optical waveguide

The mode dispersion of trapezoidal cross-section narrow ridge dielectric optical waveguides is calculated by the Weighted Residual Method (WRM). In the sloping region, the field is expressed in terms of Airy functions, thus having no arbitrariness at the boundaries. The result is compared with previously reported experimental and theoretical results. Agreement between the experimental and the predicted results is good. © 1996 John Wiley & Sons, Inc.