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 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.     

On the Euler flow in ℝ2

We give a global existence and uniqueness theorem for the Euler flow in ² for suitable initial velocity fields, possibly diverging at infinity.     

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},     

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.     

Capturing the baroclinic effect in non-Boussinesq gravity currents

Direct numerical simulations of two-dimensional gravity currents with small and medium density variations are performed using different non-Boussinesq buoyancy approximations. Taking the full low-Machnumber approximation as the reference, the accuracy of several buoyancy terms are examined. It is found that all considered buoyancy terms performed well in the cases with small density variation. In the cases with medium density variation, the classical gravitational Boussinesq’s buoyancy term showed the lack of accuracy, and a simple correction did not make any improvement. In contrast, the recently introduced second-order buoyancy term showed a significantly higher accuracy. The present results and our previous derivations indicate that simple algebraic buoyancy approximations extended from the Boussinesq’s gravitational buoyancy are unlikely to achieve an accuracy beyond first order. Instead, it seems necessary to solve at least one extra Poisson equation for buoyancy terms to capture the higher-order baroclinic effect. An approximate analysis is also provided to show the leading term of the non-Boussinesq effect corresponding to gravity.     

Double-Diffusive Natural Convection Flows with Thermosolutal Symmetry in Porous Media in the Presence of the Soret–Dufour Effects

The double-diffusive natural convection past a vertical plate embedded in a fluid-saturated porous medium is considered in the boundary-layer and Boussinesq approximations. It is assumed that the Soret–Dufour cross-diffusion effects are significant. The heat and mass fluxes on the plate are prescribed as functions of the surface coordinate x. The general similarity reduction of the problem for power-law and exponential variation of the wall fluxes is given. In the case of thermosolutal symmetry, when the similar temperature and concentration fields become coincident, exact analytical as well as numerical solutions are reported and discussed in some detail. For the flows without thermosolutal symmetry, the final similarity equations have been solved numerically, by paying attention to the influence of the Soret and Dufour numbers on the departure from thermosolutal symmetry. The reported results focus on the wall temperatures and concentrations, whose reciprocals are Nusselt and Sherwood numbers, respectively.     

Nonnormal Effects in the Taylor–Couette Problem

This work is devoted to the study of transient growth of perturbations in the Taylor–Couette problem due to linear nonnormal mechanisms. The study is carried out for a particular small gap case and is mostly focused on the linearly stable regime of counter-rotation. The exploration covers a wide range of inner and outer angular speeds as well as axial and azimuthal modes. Significant transient growth is found in the regime of stable counter-rotation. The numerical results are in agreement with former analyses based on energy methods and other independent numerical studies. The optimal energy transient growth factor appears to be consistent with experimental observations. This study might shed some light on the subcritical transition to turbulence which is found experimentally in Taylor–Couette flow when the cylinders rotate in opposite directions. Received 13 February 2001 and accepted 29 March 2002 Published online: 2 October 2002 RID="*" ID="*" This work was supported by the UK EPSRC under Grant GR/M30890. The author thanks Nick Trefethen for fruitful discussions. RID="*" ID="*" Present address: Departament de Fisica Aplicada, Univ. Politecnica de Catalunya, 08034 Barcelona, Spain (alvar@fa.upc.es) Communicated by H.J.S. Fernando     

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 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.     

Impact of the interplanetary magnetic field on thewave flow pattern in the neighborhood of the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The impact of the interplanetary magnetic field on transformation and disintegration of the Earth’s bow shock into a system of magnetohydrodynamic (MHD) shock waves, rotational discontinuities and rarefaction waves under the action of abrupt variations in the solar wind dynamic pressure is simulated in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model using the solution of the MHD Riemann problem of breakdown of an arbitrary discontinuity. This discontinuity arises when a contact discontinuity, on which the solar wind density increases or decreases suddenly and which travels together with the solar wind, impinges on the Earth’s bow shock and propagates along its surface. The interaction pattern is constructed in the quasisteady- state formulation as a mosaic of exact solutions obtained on computer using an original MHD Riemann solver. The wave flow patterns are found for all elements of the surface of the bow shock as functions of their latitude and longitude for various jumps in the density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that when the solar wind dynamic pressure increases, a fast MHD shock wave, that first penetrates into the magnetosheath, is always formed. When the solar wind dynamic pressure decreases, the influence of the interplanetary magnetic field can lead to the development of the leading fast MHD shock wave in certain zones on the surface of the Earth’s bow shock. The solution obtained can be used to interpret measurements on spacecraft in the solar wind at the libration point and in the neighborhood of the Earth’s magnetosphere.     

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.     

Graphical Illustrations for the Nur-Byerlee-Carroll Proof of the Formula for the Biot Effective Stress Coefficient in?Poroelasticity

This work presents a graphically illustrated version of the Nur-Byerlee-Carroll proof of the formula for the Biot effective stress coefficient in poroelasticity. The original elegant proof was provided by Nur and Byerlee (J. Geophys. Res. 76:6414, 1971) for isotropic materials and extended by Carroll (J. Geophys. Res. 84:7510?C7512, 1979) to anisotropic materials. Although the application of this result is in poroelasticity or in the analysis of composite materials, the proof is an analytical thought experiment in linear elasticity, and should be appreciated as such.     

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.     

About the “normal growth” approximation in the dynamical theory of phase transitions

Nonequilibrium phase transitions can often be modeled by a surface of discontinuity propagating into a metastable region. The physical hypothesis of normal growth presumes a linear relation between the velocity of the phase boundary and the degree of metastability. The phenomenological coefficient, which measures the mobility of the phase boundary, can either be taken from experiment or obtained from an appropriate physical model. This linear approximation is equivalent to assuming the surface entropy production (caused by the kinetic dissipation in a transition layer) to be quadratic in a mass flux.In this paper we investigate the possibility of deducing the normal growth approximation from the viscosity-capillarity model which incorporates both strain rates and strain gradients into constitutive functions. Since this model is capable of describing fine structure of a thick advancing phase boundary, one can derive, rather than postulate, a kinetic relation governing the mobility of the phase boundary and check the validity of the normal growth approximation.We show that this approximation is always justified for sufficiently slow phase boundaries and calculate explicitly the mobility coefficient. By using two exact solutions of the structure problem we obtained unrestricted kinetic equations for the cases of piecewise linear and cubic stress-strain relations. As we show, the domain of applicability of the normal growth approximation can be infinitely small when the effective viscosity is close to zero or the internal capillary length scale tends to infinity. This singular behavior is related to the existence of two regimes for the propagation of the phase boundary — dissipation dominated and inertia dominated.     

Boundary Behavior of the Ginzburg–Landau Order Parameter in the Surface Superconductivity Regime

We study the 2D Ginzburg--Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order (in L ² norm) by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address the question of whether this approximation can hold uniformly in the boundary region. We prove that this is indeed the case as a corollary of a refined, second order energy expansion including contributions due to the curvature of the sample. Local variations of the GL order parameter are controlled by the second order term of this energy expansion, which allows us to prove the desired uniformity of the surface superconductivity layer.     

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.     

Solid mechanical problems in the “Risorgimento” generation

We want to give a synthetic idea of the history of some arguments of the solid mechanics drawn up in Italy during the last four decades of the 19th Century (Your virtue is to be corious; you want to penetrate into the causes, to go back to the seeds. … I know how from one of those trifles, that the uninitiated don’t get, you scientists can get out the spark, that gave light to the most remote truths./Voi … avete la virtù di essere curiosi; volete penetrare nelle cause, rimontare al seme. … So come da una di quelle inezie, le quali sfuggono all’attenzione dei profani, voi scienziati potete cavare la scintilla, che rischiara poi le verità più riposte.), Boito (Senso, 1883). The expression Risorgimento Generation is ascribable to Giovan Battista Guccia (1855–1914), with reference to the school of Francesco Brioschi (1824–1897), as inspiration of the foundation (1884) of the Mathematical Society of Palermo.