On the Theory of the α–β Phase Transition in Quartz

The – phase transition in quartz has features which workers have long found puzzling. My purpose is to explore using a theory for this that is rather different conceptually from those used by other workers. It does suggest different kinds of experiments, likely to shed light on aspects of the transition, and a possible modification of theory which might be helpful in understanding some subtleties.     

On the Asymptotic Behaviour of the Solutions of the Equations of Linear Elastodynamics in Unbounded Domains

We prove, among other things, that if the acoustic tensor satisfies a suitable growth condition at infinity (the hyperbolicity condition) and the total initial energy is summable with a suitable weight, then the solution to the initial boundary value problem of linear elastodynamics in unbounded domains decays at infinity, at every instant, with a rate depending on the weight. Moreover, we show that the hyperbolicity condition is necessary and sufficient for the equipartition in mean of the total energy.     

On a Theory of the Infusion Method of Measuring the “Opening Pressure” in the Urethra

At rest the muscles which control the urethra (urine duct) are contracted and its lumen is practically equal to zero over its entire length. To open the urethra, a mechanical effort (due, for example, to a pressure rise in the bladder) must be applied. Reduced contractile activity of the muscles may be one of the reasons for incontinence (enuresis). A widespread method of estimating the blocking capability of the urethra consists in inserting a catheter with lateral perforations near the end. The catheter enters the bladder and is then removed at a constant velocity while a fluid is constantly pumped (infused) into it by a syringe pump at a steady rate and then flows out through the gap between the catheter and the urethral wall. The pumping pressure is considered to be a local measure of the blocking capability, and its dependence on the location of the catheter is regarded as an important diagnostic characteristic.Below, we will consider the simple, longitudinally homogeneous model system formed by an elastic tube pulled over a catheter segment when the initial stresses in the tube are constant over its length. An incompressible viscous fluid flows out of the perforations and percolates in a thin layer along the catheter. In solving the model problem, we will use the lubricating layer approximation under the assumption of small layer curvature. On the basis of an analysis of the results and a comparison of the model with a practical intraurethral measurement procedure, we discuss, firstly, the relationship between the measured quantities and the real characteristics of the urethra and, secondly, the possible formulation of a more realistic model problem.     

On the bifurcations of the Lamé solutions in plane-strain elasticity

We consider the in-plane bifurcations experienced by the Lamé solutions corresponding to an elastic annulus subjected to radial tension on the curved boundaries. Numerical investigations of the relevant incremental problem reveal two main bifurcation modes: a long-wave local deformation around the central hole of the domain, or a material wrinkling-type instability along the same boundary. Strictly speaking, the latter scenario is related to the violation of the Shapiro–Lopatinskij condition in an appropriate traction boundary-value problem. It is further shown that the main features of this material instability mode can be found by using a singular-perturbation strategy.     

Stability of the Camassa–Holm Multi-peakons in the Dynamics of a Shallow-Water-Type System

Consideration herein is the stability issue of a variety of superpositions of the Camassa–Holm peakons and antipeakons in the dynamics of the two-component Camassa–Holm system, which is derived in the shallow water theory. These wave configurations accommodate the ordered trains of the Camassa–Holm peakons, the ordered trains of Camassa–Holm antipeakons and peakons as well as the Camassa–Holm multi-peakons. Using the features of conservation laws and the monotonicity properties of the local energy, we prove the orbital stability of these wave profiles in the energy space by the modulation argument.     

Regularity Criteria of Weak Solutions in terms of the Pressure in Lorentz Spaces to the Navier–Stokes Equations

We consider the incompressible Navier–Stokes equations in Ω ×?(0, T), where Ω is a domain in ${\mathbb{R}^3}$ . We give regularity criteria in terms of the pressure in Lorentz spaces with the corresponding small norm. In particular, our results extend previous ones to the Lorentz space with respect to temporal variable.     

On the law of energy conservation in the classical electrodynamics of deformable media—A generalization of the Poynting theorem

This paper establishes the generalized Poynting theorem for the electrodynamics of deformable media with a view to shedding some light on the detailed mechanism of energy transfer between the electromagnetic field and the deformable media. Global field equations are chosen as the starting point and specialized forms of the theorem are derived based on the special postulates for the electromagnetic body force.     

Opening of a system of cracks—on the mechanism of the cyclic lateral eruption of the St. Helens volcano in 1980

The dynamic behavior of a magma melt filling a slot channel (crack) in a closed explosive hydrodynamic structure is considered. The explosive hydrodynamic structure includes the volcano focal point with a connected vertical channel (conduit) closed by a slug and a system of internal cracks (dikes) near the dome, as well as a crater open into the atmosphere. A two-dimensional model of a slot eruption is constructed with the use of the Iordanskii–Kogarko–van Wijngaarden mathematical model of two-phase media and the kinetics that describes the basic physical processes in a heavy magma saturated by the gas behind the decompression wave front. A numerical scheme is developed for analyzing the influence of the boundary conditions on the conduit walls and scale factors on the melt flow structure, the role of viscosity in static modes, and dynamic formulations with allowance for diffusion processes and increasing (by several orders of magnitude) viscosity. Results of the numerical analysis of the initial stage of cavitation process evolution are discussed.     

On the Stability in Weak Topology of the Set of Global Solutions to the Navier–Stokes Equations

Let X be a suitable function space and let ${\mathcal{G} \subset X}$ be the set of divergence free vector fields generating a global, smooth solution to the incompressible, homogeneous three-dimensional Navier–Stokes equations. We prove that a sequence of divergence free vector fields converging in the sense of distributions to an element of ${\mathcal{G}}$ belongs to ${\mathcal{G}}$ if n is large enough, provided the convergence holds “anisotropically” in frequency space. Typically, this excludes self-similar type convergence. Anisotropy appears as an important qualitative feature in the analysis of the Navier–Stokes equations; it is also shown that initial data which do not belong to ${\mathcal{G}}$ (hence which produce a solution blowing up in finite time) cannot have a strong anisotropy in their frequency support.     

The use of Volterra series in the analysis of the nonlinear Schrödinger equation

A Volterra series analysis is used to analyse the dispersive behaviour in the frequency domain for the non-linear Schrödinger equation (NLS). It is shown that the solution of the initial value problem for the nonlinear Schrödinger equation admits a local multi-input Volterra series representation. Higher order spatial frequency responses of the nonlinear Schrödinger equation can therefore be defined in a similar manner as for lumped parameter non-linear systems. A systematic procedure is presented to calculate these higher order spatial frequency response functions analytically. The frequency domain behaviour of the equation, subject to Gaussian initial waves, is then investigated to reveal a variety of non-linear phenomena such as self-phase modulation (SPM), cross-phase modulation (CPM), and Raman effects modelled using the NLS.     

Some Aspects of the Asymptotic Dynamics of Solutions of the Homogeneous Navier–Stokes Equations in General Domains

In the first part of the paper we study decays of solutions of the Navier–Stokes equations on short time intervals. We show, for example, that if w is a global strong nonzero solution of homogeneous Navier–Stokes equations in a sufficiently smooth (unbounded) domain Ω ⊆ R³ and β ∈[1/2, 1) , then there exist C₀ > 1 and δ₀ ∈ (0, 1) such that

On the Dynamics of Solitons in the Nonlinear Schr?dinger Equation

We study the behavior of the soliton solutions of the equation i\frac?y?t = - \frac12m Dy+ \frac12W_e^￠(y) + V(x)y,i\frac{\partial\psi}{{\partial}t} = - \frac{1}{2m} \Delta\psi + \frac{1}{2}W_{\varepsilon}^{\prime}(\psi) + V(x){\psi},     

Air–water flow in a vertical pipe: experimental study of air bubbles in the vicinity of the wall

This study deals with the influence of bubbles on a vertical air–water pipe flow, for gas-lift applications. The effect of changing the bubble size is of particular interest as it has been shown to affect the pressure drop over the pipe. Local measurements on the bubbles characteristics in the wall region were performed, using standard techniques, such as high-speed video recording and optical fibre probe, and more specific techniques, such as two-phase hot film anemometry for the wall shear stress and conductivity measurement for the thickness of the liquid film at the wall. The injection of macroscopic air bubbles in a pipe flow was shown to increase the wall shear stress. Bubbles travelling close to the wall create a periodic perturbation. The injection of small bubbles amplifies this effect, because they tend to move in the wall region; hence, more bubbles are travelling close to the wall. A simple analysis based on a two-fluid set of equations emphasised the importance of the local gas fraction fluctuations on the wall shear stress.     

Solution of the Incompressible Unsteady Navier–Stokes Equations Only in Terms of the Velocity Components

A method which uses only the velocity components as primitive variables is described for solution of the incompressible unsteady Navier–Stokes equations. The method involves the multiplication of the primitive variable-based Navier–Stokes equations with the unit normal vector of finite volume elements and the integration of the resulting equations along the boundaries of four-node quadrilateral finite volume elements. Therefore, the pressure term is eliminated from the governing equations and any difficulty associated with pressure or vorticity boundary conditions is avoided. The equations are discretized on four-node quadrilateral finite volume elements by using the second-order-accurate central finite differences with the mid-point integral rule in space and the first-order-accurate backward finite differences in time. The resulting system of algebraic equations is solved in coupled form using a direct solver. As a test case, an impulsively accelerated lid-driven cavity flow in a square enclosure is solved in order to verify the accuracy of the present method.     

Application of Moiré Interferometry to the Characterization of Orthotropic Materials in the Antisymmetric Configuration using the Virtual Fields Method

Moiré interferometry is an effective full-field deformation measurement technique and has been utilized for mechanical behavior analysis of materials and structures. For isotropic materials, Moiré patterns can be obtained by performing standard tests, such as, tensile and bending tests, to calculate the displacement and strain. Then, the mechanical properties can be characterized. However, standard tests are not sufficient to characterize the mechanical parameters of anisotropic materials due to the complexity of their material properties. Thus, in this work, Moiré interferometry was combined with the Virtual Fields Method to obtain the four in-plane elastic constants (Q₁₁, Q₂₂, Q₁₂, and Q₆₆) of orthotropic materials in the form of a diametrically compressed disk. Firstly, according to finite element method simulation results, optimized parameters can be achieved when the principal direction of the material does not coincide with the loading direction, making the loading configuration antisymmetric. Therefore, Moiré interferometry experiment was simulated to demonstrate the feasibility of measurement in the antisymmetric configuration. Finally, the Q₁₁, Q₂₂, Q₁₂ and Q₆₆ values of a unidirectional carbon fiber composite were measured in a real Moiré interferometry experiment using the proposed method, yielding results that agreed closely with those obtained using the strain gauges.     

The Mechanical Properties of the Slurry and the Resistance Force of a Sphere Moving With Uniform Velocity in the Slurry

It is commonly considered that the mechanical properties of the slurryare different from that of ordinary Newtonian fluid,and can be describedby that of Bingham fluid.Hence its shearing stress should be described bythe formula of the shearing stress of Bingham fluid.But the author holdsthe contrary opinion and firmly believes that the slurry is a highly viscousfluid with very long relaxation time such as those of asphalt,glass,etc.In this article,we have discussed the mechanical properties of the slurryand the reslstance of a sphere moving with uniform veloclty in the slurry.In the process of discussion.we use Stokes solution of the vlscous fluidpassing around a sphere.When the sphere is in equilibrium under theaction of gravitational force,the force of buoyancy and the resistance,we get the velocity of sedimentation.When the velocity of sedimentationis equal to zero,we get the relation between the yield stress of Binghamfluid and the diameter of the particles which will not sink.The theoreticalresults     

On the Existence of Periodic Solutions of the Navier–Stokes Equations in a Thin Domain Using the Topological Degree

We use the method of the topological degree, the theory of fractional powers of positive operators, and the Grisvard formula together with results proved by G. Raugel and G. R. Sell to study the periodic solutions of the incompressible Navier–Stokes equations in a thin three-dimensional domain.     

On the determination of the force distribution in overconstrained cable-driven parallel mechanisms

This paper addresses the determination of the force distribution in the cables of a redundantly actuated cable-driven parallel mechanism. First, the static model of cable-driven parallel mechanisms is derived based on the wrench matrix. Then, four performance indices are considered in order to solve the underdetermined problem associated with the distribution of the forces. A simple numerical example is then developed in order to provide insight into the problem, which leads to a geometric interpretation of the results. Based on the presented results, it is proposed to use a p-norm (e.g. a 4-norm) to optimize the distribution of the forces in a cable-driven parallel mechanism in order to minimize the largest deviations from the median forces (or other target values) while maintaining continuity in the solution. A non-iterative polynomial formulation is then proposed for the 4-norm. It is also pointed out that this formulation leads to a unique real solution.