Showing papers in "Zeitschrift für Angewandte Mathematik und Physik in 1973"
TL;DR: In this article, the stability limits for thermally driven acoustic oscillations in a helium-filled tube are given in the form of critical temperature ratios versus a parameter representing the ratio between the tube radius and the extent of the viscous flow region.
Abstract: The stability limits for thermally driven acoustic oscillations in a helium-filled tube are given. Calculations are partly based on a step-by-step integration of the differential equation for the acoustic pressure, but most results refer to a model using a discontinuous temperature distribution; in the latter case a transcendental stability-frequency equation has been numerically solved. Results are presented in the form of critical temperature ratios versus a parameter representing the ratio between the tube radius and the extent of the viscous flow region. Several asymptotic limits of the stability curves are analytically treated. Finally, limitations of the theory are discussed which occur when the solid tube wall specific heats become vanishingly small at the cold end.
175 citations
TL;DR: In this paper, the effect of inclusion length and geometry on the bulk elastic moduli of fiber-or whisker-reinforced composite materials is investigated and expressions for the moduli which are valid for all values of the above parameter are derived from a known general solution.
Abstract: An earlier theoretical, study [3] regarding the effect of inclusion length, and geometry on the bulk elastic moduli of fiber-or whisker-reinforced composite materials is hereby extended. Specifically, with slender-body theory, moduli for materials containing dilute dispersions of aligned inclusions of slender, but otherwise arbitrary, axial geometry are found for the “elastic” range,\(\frac{{E^* }}{E}\kappa ^2 \ln 2/\kappa > 1\), the special case of slender prolate, spheroidal inclusions is considered and expressions for the moduli which are valid for all values of the above parameter are derived from a known general solution obtained by Eshelby. These analytic results are then presented in a compact graphical form which provides a useful means for estimating the moduli when the slender inclusions have non-prolate spheroidal geometries.
91 citations
TL;DR: In this article, the authors give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:
Abstract: In this paper we give conditions sufficient to insure that the following two results hold for eigenfunctions of the fixed membrane problem defined over a bounded region of the plane:
91 citations
TL;DR: In this paper, the first eigenfunction of the fixed membrane problem was extended to the analogous problem in higher dimensions, where the integrals of the L 1 and L 2 integrals were extended to higher dimensions.
Abstract: Previous results of the authors [2] relating theL
1 andL
2 integrals of the first eigenfunction in the two dimensional fixed membrane problem are here extended to the analogous problem in higher dimensions.
38 citations
TL;DR: In this paper, the span-wise aerodynamic loading of a wing with minimum induced drag was derived for prescribed lift and root-bending moment, in the case that the lift and its moment of inertia about the longitudinal axis of the aircraft are given.
Abstract: The spanwise aerodynamic loading of wings having minimum induced drag is derived for prescribed lift and root-bending moment. THis problem is an alternative to Prandtl's solution for the case that the lift and its moment of inertia about the longitudinal axis of the aircraft are given.
32 citations
TL;DR: In this article, it is shown that the problem is mainly conduction dominated provided thatR, the ratio of the radius of the outer cylinder to the inner cylinder, is not too large.
Abstract: Natural convection between infinitely long, horizontal, heated, concentric circular cylinders is considered. It is assumed that the temperature difference between the cylinders is small and also that the radius of the inner cylinder is small. It is shown that the problem is mainly conduction dominated provided thatR, the ratio of the radius of the outer cylinder to the radius of the inner cylinder, is not too large. The criterion derived for a conduction dominated problem is a relation betweenR and the Grashof number.
31 citations
TL;DR: In this article, a method for numerical solution of harmonic mixed boundary value problems in simply connected domains is described, which consists of three successive conformal transformations of which the first is performed numerically.
Abstract: A method is described for the numerical solution of harmonic mixed boundary value problems in simply-connected domains. The method consists of three successive conformal transformations of which the first is performed numerically. In particular the method is applied to problems, defined in polygonal regions, containing re-entrant cornes at which boundary singularities occur.
29 citations
TL;DR: In this article, the pulsatile flow of an incompressible viscous fluid in a cylindrical tube of varying cross section is investigated for small Reynolds numbers and the shear stress distribution on the wall is evaluated and discussed in detail for special geometries like tapered tubes, locally constricted tubes and peristaltic tubes.
Abstract: The pulsatile flow of an incompressible viscous fluid in a cylindrical tube of varying cross section is investigated for small Reynolds numbers. The solutions consist of a stedy and an oscillatory part. The shear stress distribution on the wall is evaluated and discussed in detail for special geometries like tapered tubes, locally constricted tubes and peristaltic tubes. The existence of separation in the flow field is noticed.
29 citations
TL;DR: In this paper, the problem of incompressible viscous laminar flow between two co-axial discs is studied when there is constant injection or suction at the discs and the resultant flow satisfies an ordinary nonlinear differential equation which depends on a suction Reynolds number and this equation is solved by singular perturbation technique.
Abstract: The problem of incompressible viscous laminar flow between two co-axial discs is studied when there is constant injection or suction at the discs. The resultant flow satisfies an ordinary non-linear differential equation which depends on a suction Reynolds number and this equation is solved by singular perturbation technique.
28 citations
TL;DR: In this article, the linear and nonlinear theories of elastic slender curved rods are formulated in a systematic manner and the generalized strains are defined based on the principle of virtual work, and the field equations for finite deformations of curved rods can be simplified in the case of small axial strain and moderately small rotations.
Abstract: The linear and nonlinear theories of elastic slender curved rods are formulated in a systematic manner. Equations of equilibrium for stress resultants and moments are derived. The generalized strains are defined based on the principle of virtual work. Constitutive equations corresponding to small strains are obtained. The field equations for finite deformations of curved rods can be simplified in the case of small axial strain and moderately small rotations. Further simplifications can be made for the case of slightly curved rods. Two examples are presented to illustrate applications of the developed theories.
28 citations
TL;DR: In this paper, the authors investigated the effect of light scattering by small moving particles suspended in a flowing fluid on optical anemometers and derived a dependence of local signal quality and signal strength on the particle size and its optical properties.
Abstract: Theoretical and experimental investigations into the physics of optical anemometry are reported. The theoretical investigations involve considerations of light scattering by small moving particles suspended in a flowing fluid. The analysis yields general expressions for the light intensity at a point on the photodetector if several scattering beams penetrate the measuring control volume in different directions. The derivations also lead to the time dependence of the light intensity caused by the movement of the scattering particles relative to the light source and the photodetector. The relationship between the frequency of the photodetector signal and the instantaneous velocity of the scattering particles is derived on the basis of phase changes of the scattered light waves and is subsequently related to previously published derivations. The azimuthal distributions of signal quality and signal strength are evaluated based on results of computation of scattered light intensity distributions. These yield a dependence of local signal quality and signal strength on the particle size and its optical properties. In addition, a heuristic explanation of this dependence is presented that relates the signal quality influence by the particle size to the apparent fringe spacing in the measuring control volume. The need for finite apertures in optical anemometry is apparent and their influence on optical anemometer signals is discussed. The experimental programme included velocity measurements of rotating discs to demonstrate the correctness of the theoretical derivations. The decay of signal quality with signal mismatch from many particles moving at different velocities is demonstrated and the need for careful control of the measuring region is pointed out.
TL;DR: In this article, a reduced form of the equilibrium equations for the Mooney material was derived from these equations, and the exact solution was derived for a number of mixed boundary value problems of finite elasticity.
Abstract: For plane deformations of isotropic incompressible elastic materials new reduced forms of the equilibrium equations are given. From these equations new exact solutions are deduced for the Mooney material. These solutions are shown to be possible approximate solutions to a number of mixed boundary value problems of finite elasticity.
TL;DR: In this paper, the Lagrangian forms of the conservation equations for total mass and energy and the unsteady version of the Bernoulli equation were used to analyze the mechanics of pulsed flow in a nozzle.
Abstract: The mechanics of pulsed flow in a nozzle is analyzed by using the Lagrangian forms of the conservation equations for total mass and energy and the unsteady version of the Bernoulli equation. The results of the study are presented in simple formulas for predicting pressure profiles within the nozzle during the process as well as the jet velocity as a function of time. Examples are presented illustrating those parameters which are of importance and how they influence the performance of the nozzle.
TL;DR: In this article, the equilibrium equations for the general three dimensional deformation of an isotropic incompressible hyperelastic material are given in a reciprocal form, which can be expressed as approximate solutions of various mixed boundary value problems of finite elasticity.
Abstract: The equilibrium equations for the general three dimensional deformation of an isotropic incompressible hyperelastic material are given in a reciprocal form These reciprocal equations firstly provide an alternative derivation of the inverse deformation results due to Shield [1] and secondly enable new reduced forms of the equilibrium equations to be given for those materials for which the strain-energy function depends upon only one of the principal strain invariants For the neo-Hookean and extreme-Mooney materials these reduced equilibrium equations are shown to admit new exact solutions expressible in terms of Bessel functions These deformation fields can be rendered as approximate solutions of various mixed boundary value problems of finite elasticity Details are given for the problem of the flattening of a portion of a spherical shell which is not necessarily thin
TL;DR: In this article, the problem of combined forced and free convection heat transfer in boundary-layer flow past a heated or cooled, horizontal flat plate is studied in the limits of large and small Prandtl number using the technique of matched asymptotic expansions.
Abstract: The problem of combined forced and free convection heat transfer in boundary-layer flow past a heated or cooled, horizontal flat plate is studied in the limits of large and small Prandtl number using the technique of matched asymptotic expansions It is found that the effects of natural convection are confined to small changes in the basic forced convection velocity and temperature fields providedGr/Re5/2Pr2/3≪1 forPr≫1, orGr/Re5/2Pr1/2≪1 forPr≪1 The governing similarity equations at the first two orders in these small parameters are solved and the results compared with the numerical solutions of Sparrow and Minkowycz (1962)
TL;DR: In this paper, the non-symmetric stokes flow past a spherical cap is solved by the method of complementary integral representations, in particular the drag and couple are evaluated for the hemispherical cup.
Abstract: The non symmetric stokes flow past a spherical cap is solved by the method of complementary integral representations. In particular, the drag and couple are evaluated for the hemispherical cup.
TL;DR: In this paper, it was shown that C∞-smoothness only supports "invariance to all orders" if the frequency changes slowly and analytically smoothly, and the angle variable is split into algebraic and exponential parts.
Abstract: The action of a simple pendulum is shown by an elementary proof to suffer only an exponentially small, total change, if the frequency changes slowly and analytically smoothly.C∞-smoothness only supports ‘invariance to all orders’. The method rests on an asymptotic split of the angle variable into its algebraic and exponential parts.
TL;DR: In this article, the theory of complex variables is used to develop an exact closed-form solution of a transcendental equation basic to the molecular field theory of ferromagnetism.
Abstract: The theory of complex variables is used to develop an exact closed-form solution of a transcendental equation basic to the molecular field theory of ferromagnetism. The analysis yields analytical expressions, in terms of elementary quadratures, for the reduced magnetization ζ as it depends on the temperature and magnetic field.
TL;DR: For axially symmetric deformations of isotropic incompressible hyperelastic materials new reduced forms of the equilibrium equations are given in this article, and these solutions can be rendered as approximate solutions of problems involving the large axially asymmetric deformation of circular discs containing a concentric circular hole.
Abstract: For axially symmetric deformations of isotropic incompressible hyperelastic materials new reduced forms of the equilibrium equations are given. For the neo-Hookean and extreme-Mooney materials these equations are shown to admit exact solutions expressible in terms of Bessel functions of order zero. These solutions can be rendered as approximate solutions of problems involving the large axially symmetric deformations of circular discs containing a concentric circular hole.
TL;DR: In this paper, a linearized theory of the two-and three-dimensional incompressible viscous flows ensuing from locally unstable velocity profiles is presented. But the theory is used to propose a hypothesis for the mechanisms governing formation of the turbulence wedge behind a fixed roughness element.
Abstract: This paper describes a linearized theory of the two- and three-dimensional incompressible viscous flows ensuing from locally unstable velocity profiles. The theory is used to propose a hypothesis for the mechanisms governing formation of the turbulence wedge behind a fixed roughness element.
TL;DR: In this article, the frequency of a simple pendulum varies with time τ slowly and analytically smoothly, and the total change in its action is exponentially small (Part I).
Abstract: If the frequency of a simple pendulum varies with time τ slowly and analytically smoothly, then the total change in its action is exponentially small (Part I). This change is computed here to the first approximation for a class of frequency functions ω (τ) of practical interest.
TL;DR: In this paper, the sinusoidal oscillations of an infinite plate in a mixture of two incompressible Newtonian hemihedral fluids are investigated, and the solution is compared with the previously derived corresponding one for holohedral fluids.
Abstract: The sinusoidal oscillations of an infinite plate in a mixture of two incompressible Newtonian hemihedral fluids are investigated. The solution is compared with the previously derived corresponding one for holohedral fluids.
TL;DR: In this article, a special theory of polar fluids is developed with a view to providing some insight into the questions of the frame-dependence of stress and heat flux, arising out of a recent paper by Muller.
Abstract: A special theory of polar fluids is developed with a view to providing some insight into the questions of the frame-dependence of stress and heat flux, arising out of a recent paper by Muller (1).
TL;DR: In dieser Arbeit wird die Stromung uber eine Grenzflache zwischen zwei anisotropen porosen Korpern betrachtet as mentioned in this paper.
Abstract: In dieser Arbeit wird die Stromung uber eine Grenzflache zwischen zwei anisotropen porosen Korpern betrachtet.
TL;DR: In this paper, the general solution for steady two-dimensional magnetohydrodynamic flows of an incompressible, inviscid, fluid having infinite electrical conductivity when the angle between the magnetic field and the velocity field is every where constant is given.
Abstract: The general solution is given for steady two-dimensional magnetohydrodynamic flows of an incompressible, inviscid, fluid having infinite electrical conductivity when the angle between the magnetic field and the velocity field is every where constant.
TL;DR: In this article, maximum and minimum principles for the steady laminar incompressible flow of an electrically conducting fluid in a rectangular channel under an imposed transverse magnetic field are presented.
Abstract: Maximum and minimum principles are presented for the steady laminar incompressible flow of an electrically conducting fluid in a rectangular channel under an imposed transverse magnetic field. In the case of a nonconducting channel, maximizing and minimizing sequences are produced which converge to the exact solution.
TL;DR: In this paper, the analytische Abhangigkeit der Losungen eines Randwert-and eines Eigenwertproblems gezeigt, sowie des tiefsten EigenWertes λ 1(α) von einem Parameter α, der in den Randbedingungen auftritt.
Abstract: Mit Hilfe der Rellichschen Storungstheorie wird die analytische Abhangigkeit der Losungen eines Randwert- und eines Eigenwertproblems gezeigt, sowie des tiefsten Eigenwertes λ1(α) von einem Parameter α, der in den Randbedingungen auftritt. Als Anwendung ergeben sich Schranken fur das Energieintegral und λ1(α). Ferner wird die Definition der RobinfunktionR(P, Q; α) erweitert auf negative Werte von α, insbesondere singulare Werte −p
j
.
TL;DR: In this paper, it was shown how these ideas can be extended to yield the Onsager-Casimir reciprocal relations, at any rate in certain cases, and it is shown in the present note how these notions can be used to obtain the reciprocal relations for a linear anisotropic viscoelastic material.
Abstract: Day showed that for a linear anisotropic viscoelastic material the kernels in the memory integral are necessarily symmetric, if the dissipation associated with an arbitrary strain history extending from the infinite past to the infinite future is invariant under time-reversal. It is shown in the present note how these ideas can be extended to yield the Onsager-Casimir reciprocal relations, at any rate in certain cases.
TL;DR: Barodiffusion in an isothermal, incompressible, Newtonian, binary mixture set in steady, laminar, axisymmetric motion by an infinite rotating disk is considered in this paper.
Abstract: Barodiffusion (diffusion of matter brought about by pressure gradients) in an isothermal, incompressible, Newtonian, binary mixture set in steady, laminar, axisymmetric motion by an infinite rotating disk is considered. The problem is formulated with isotopic mixtures in view.
TL;DR: In this article, the dependence of the elastic moduli of chopped-fiber reinforced composites on the modulus ratio and the fiber length was studied experimentally at low volume fractions of fibers.
Abstract: The dependence of the elastic moduli of chopped-fiber reinforced composites on the modulus ratio and the fiber length was studied experimentally at low volume fractions of fibers. For this purpose model composites were fabricated from an epoxy casting resin and chopped stainless steel fibers, and the elastic moduli of these samples were measured by an ultrasonic pulse technique. The results confirm in part the analytic expressions for the bulk moduli of such composites previously derived by the authors, but also emphasize the importance of fiber dispersion even at dilute concentrations.