Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

Measure-theoretic foundations of statistical mechanics

The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles” and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions appear as special cases of the general theory.     

A numerical study based on a weakly compressible formulation for thermosolutal convection in vertical cavities

Laminar thermosolutal convection in cavities with uniform, constant temperature and mass fraction profiles at the vertical side is studied numerically. The study is conducted in the case where an inert carrier gas (species “1”) present in the cavity is not soluble in species “2”, and do not diffuse into the walls. A mass flux of species “2” into the cavity occurs at the hot vertical wall and a mass flux out of the cavity occurs at the opposite cold wall. The weakly compressible model proposed in this work was used to investigate the flow fields, and heat and mass transfer in cavities filled with binary mixtures of ideal gases. The dimensionless form of the seven governing equations for constant thermophysical properties, except density, show that the problem formulation involves ten dimensionless parameters. The results were validated against numerical results published in the literature for purely thermal convection, and thermodynamic predictions for transient thermosolutal flows. A parametric study has been performed to investigate the effects of the initial conditions, molecular weight ratio, Lewis number, and aspect ratio of the cavity for aiding or opposing buoyancy forces. For the range of parameters considered, the results show that variations in the density field have larger effects on mass transfer than on heat transfer. For opposing buoyancy forces, the numerical simulations predict complex flow structures and possible chaotic behavior for rectangular vertical cavities according to the value of the molecular weight ratio.     

Level set method for numerical simulation of a cavitation bubble,its growth,collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating. The project supported by the National Natural Science Foundation of China (10272032 and 10672043). The English text was polished by Keren Wang.     

A rheological approach to drill-in fluids optimization

The aim of this work is to propose design criteria, based on rheological characterisation for improving drill-in fluids performance. In particular, it reports an example in which rheological approaches helped improve drill-in fluids resistance to temperature. As a starting system a commercial drill-in fluid containing xanthan gum and calcium carbonate was chosen and evaluated. Different samples were then prepared by changing the initial formulation in order to increase the system's stability to temperature. Drill-in fluids' performance have been compared by considering their “damaging potential”, filtration properties and, “cakes”. All drill-in fluids have been tested before and after aging at a given temperature with “hot rolling tests”. The systems' gel structure was characterized by measuring dynamic moduli (G′ and G′′) in the linear viscoelastic range and all samples were compared by evaluating their “melting” temperature and gel network strength during time cure tests. The results obtained from this work suggest that the rheological tests carried out on the whole drill-in fluid can provide insights into fluids' damaging potential and “cake” structure. In particular, rheology proved to be able to provide quantitative information about gel strength and temperature stability that permitted one to improve drill-in fluids' formulation in order to preclude formation damage and to meet industrial requirements. Received: 6 February 2000 Accepted: 15 November 2000     

Effect of branching in cross-slot flow: the formation of “W cusps”

The sensitivity of the principal stress difference (PSD) profile to molecular architecture is demonstrated for flow in a cross-slot geometry. For materials with low levels of branching, the pattern along the outlet centre line exhibited “single cusps”, while an increase in molecular branching was found to lead to “W cusps”. The formation of these W cusps was found to be independent of extensional rate for the conditions probed, and they were formed initially at the stagnation point and travelled along the outlet centre line with time. Comparison with simulations performed using a multi-mode “pom-pom” model failed to predict W cusps, although the general level of PSD was accurately captured.     

Numerical modeling of the dynamics of a viscous-fluid droplet

A numerical model developed on the basis of the level set method is proposed. This makes it possible to describe both the nonlinear oscillations of a single viscous-fluid droplet and the fragmentation and coalescence processes. The Navier-Stokes equations written in “velocity-pressure” variables on a rectangular uniform grid in cylindrical coordinates are solved using the method of splitting into physical processes. Non-oscillating solutions for two-phase media with a characteristic density ratio of less than 10⁻³ and Re > 1000 are obtained. The possibilities of the approach proposed are demonstrated with the reference to the problem of a droplet falling from a capillary (detachment from the capillary, formation of a “Plato ball”, droplet motion, collision with a plane wall, droplet oscillations on the wall, and droplet spreading). A comparison of the numerical results with the known calculation models and experimental data shows satisfactory agreement with respect to both the phases and the shape of the droplet.     

Some properties of three-dimensional Biltrami flows

The investigation of Beltrami flows is important for the research on the mechanism of turbulent structure. In this paper the general solutions of the Beltrami flows are given, which depend explicitly on the solutions of three independent Helmholtz equations with scalar unknowns. Velocity fields of Beltrami flows can then be obtained explicitly after the application of some curl operations on the solutions of Helmholtz equations. On the basis of the exact solutions of Euler equations given above, we obtain one kind of exact solutions of non-steady Navier-Stokes equations which are also the Beltrami flows. Some interesting examples of Beltrami flows other than “ABC flows”, “Kolmogolov flows”, “Rayleigh-Bernard flows”, “Q-flows” are given. The detailed analytic results of these examples will be published in the near future.     

Evolution of perturbations of a charged interface between immiscible inviscid fluids in the interelectrode gap

Perturbations of the interface between two immiscible ideal fluids of finite thickness (the lower and upper fluids are the conductor and the dielectric, respectively) located in the gap between two electrodes are considered. In the cases of the “shallow” and “deep” upper fluid the dispersion relations of linear waves and their longwave expansions are found. The methods of determining the space-time evolution of an initial surface perturbation are developed on the basis of the linear approximation. In the cases of the “shallow” and “deep” upper fluid examples of the development of an initial perturbation of the “step” type are given. The development of an initial perturbation of the “step” type are also considered in the near-critical electric fields and in the case of degeneration of cubic dispersion.     

Investigation of high-speed combustible gas ignited by a hot gas jetproduced in the shock tube

The high-speed combustible gas ignited by a hot gas jet, which is induced by shock focusing, was experimentally investigated. By use of the separation mode of shock tube, the test section of a single shock tube is split into two parts, which provide the high-speed flow of combustible gas and pilot flame of hot gas jet, respectively. In the interface of two parts of test sections the flame of jet was formed and spread to the high-speed combustible gas. Two kinds of the ignitions, 3-D “line-flame ignition” and 2-D “plane-flame ignition”, were investigated. In the condition of 3-D “line-flame ignition” of combustion, thicker hot gas jet than pure air jet, was observed in schlieren photos. In the condition of 2-D “plane-flame ignition” of combustion, the delay time of ignition and the angle of flame front in schlieren photos were measured, from which the velocity of flame propagation in the high-speed combustible gas is estimated in the range of 30–90m/s and the delay time of ignition is estimated in the range of 0.12–0.29ms. PACS 47.40.Nm; 82.40.FpPart of this paper was presented at the 5th International Workshop on Shock/Vortex Interaction, Kaohsiung, October 27–31, 2003.     

Slow Dynamics for the Cahn‐Hilliard Equation in Higher Space Dimensions: The Motion of Bubbles

It is known that the Van der Waals‐Cahn‐Hilliard (W‐C‐H) dynamics can be approximated by a Quasi‐static Stefan problem with surface tension. It turns out that the Stefan problem has a manifold of equilibria equal in dimension to that of the domain Ω: any sphere of fixed radius with interface contained in the domain is an equilibrium (indistinguishable from the point of view of the perimeter functional). We resolve this degeneracy by showing that at the W‐C‐H level this manifold is replaced by a quasi‐invariant stable manifold, on which the typical solution preserves its “bubble” like shape until it reaches the boundary. Moreover, we show that the “bubble” moves superslowly. We also obtain an equation that determines those special spheres that correspond to equilibria at the W‐C‐H level. Our work establishes the phenomenon of superslow motion in higher space dimensions in the class of single interface solutions. (Accepted February 12, 1996)     

Nonlinearly Elastic Shell Models: A Formal Asymptotic Approach I. The Membrane Model

We consider a family of three‐dimensional shells with the same middle surface, all composed of the same nonlinearly elastic Saint Venant‐Kirchhoff material. Using the method of asymptotic expansions with the thickness as the “small” parameter, and making specific assumptions on the applied forces, the geometry of the middle surface, and the kinematic boundary conditions, we show how a “limiting”, “large‐deformation” two‐dimensional model can be identified in this fashion. By linearization, this nonlinear membrane model reduces to the linear membrane model. (Accepted January 13, 1997)     

Mixed convection from upward flow of hot air to a cooled vertical pipe

An experimental study had been carried out to investigate the buoyancy-opposed mixed convection from an upward flow of hot air to a vertical pipe with a cooled surface. The investigation covered a wide range of flow regime, ranging from the “free convection significant” to the “forced convection significant” conditions. Reynolds number of the flow extended from 966 to 14780, whereas the Buoyancy parameter, Ω [=Gr_d/(Re_d)²], varied from 0.008 to 2.77. A steady stream of hot air at a moderate pressure and a Prandtl number of 0.707 was arranged to flow upward through a vertical steel pipe, whose external wall was cooled uniformly by ambient air at 20°C. Test section of the vertical pipe was 1625 mm long with an internal diameter of 156 mm and an external diameter of 166 mm. Air temperature at inlet of the test section was varied from 40°C to 70°C. Both radial temperature and velocity profiles of the airflow were measured at inlet and exit of the test section respectively. Temperatures along the pipe wall were also measured. Non-dimensional expression for the prediction of the average heat transfer coefficient of the mixed convection from an upward flow of hot air to a vertical pipe with a cooled surface was developed from the experimental results. Convection heat transfer was found to impair when the flow is laminar and was enhanced for turbulent flow condition. Received on 20 July 1998     

Inertial-microfluidic radial migration in solid/liquid two-phase flow through a microcapillary: Particle equilibrium position

Although equilibrium of spherical particles under radial migration has been extensively investigated, mostly in macroscale flows with characteristic lengths on the order of centimeters, it is not fully characterized at relatively small Reynolds numbers, 1 ≤ Re ≤ 100. This paper experimentally studies “inertial microfluidic” radial migration of spherical particles in circular Poiseuille flow through a microcapillary. Microparticle tracking experiments are performed to obtain the spatial distribution of the particles by adopting a depth-resolved measurement technique. Through the analysis of the radial distribution of particles, inertial microfluidic circular Poiseuille flow is shown to induce a strong radial migration of particles at substantially small Re, which is quite in contrast to the pipe flows at large Re previously reported. This particle migration phenomenon is so prominent that particle equilibrium positions are formed even at small Re. However, it turns out that there exists a certain critical Re below which particle equilibrium position is almost fixed, but above which it seems to drift toward the channel wall.     

Stability of equilibrium for an inverted two-link mathematical pendulum with critical tracking forces

We investigate nonlinear stability for equilibrium of a pendulum with viscoelastic components. The tracking force is chosen so that the matrix of the linearized part of the perturbed motion has two purely imaginary roots or one zero and one negative root. The other two roots are complex with negative real part. The boundary of the domain of stability is divided into “dangerous” and “safe” (in the sense of Bautin) zones. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 100–105, September, 1999.     

Measurements of mean and fluctuating wall shear stress beneath spanwise-invariant separation bubbles

 Pulsed-wire measurements of wall shear stress have been made beneath two separation bubbles. In one a cross flow was generated by means of a (25°) swept separation line. Fluctuating stresses in orthogonal “streamwise” and cross-flow directions are very nearly equal and independent of at least moderate cross flow velocity. These fluctuations are largely determined by large-scale motions in the outer flow, whereas the mean shear stresses are not. The pdf of the “streamwise” fluctuations is unchanged by the cross flow. When a cross flow is present the pdf of the cross-flow stresses is similar to the “streamwise” pdf. Dependence on Reynolds number is the same in both flows. Received: 10 April 1998/Accepted: 17 July 1998     

Heat transfer characteristics of the algebraically decaying Glauert jet

The classical exponentially decaying wall jet considered independently by Tetervin (NACA TN 1644 40 pp, 1948), Akatnov (Leningrad Politek Inst Trudy 5:24–31, 1953) and Glauert (J Fluid Mech 1:625–643, 1956) as well as its algebraically decaying counterpart (which will be referred to hereafter as “algebraic Glauert Jet”, or AG-jet for short) belong to the same similarity class of solutions of the boundary layer equations. We investigate in this paper the thermal characteristics of a nonpreheated AG-jet over a permeable wall for prescribed constant wall temperature and prescribed constant heat flux. Their scaling behavior for small and large values of the Prandtl number is discussed in detail and compared to that of the classical Tetervin–Akatnov–Glauert wall jet.     

Stability of tribotechnical laminated coatings subjected to tracing loads

The problems of surface stability of laminated coatings subjected to the combined effect of tracing surface loads and a stationary temperature field is considered. The material of the layer and the base are considered linearly elastic and trasversally isotropic. Fulfillment of conditions of absolutely rigid force and ideal contacts are assumed between the individual elements of the laminar medium. The subcritical stress state of the laminar medium under consideration is piecewise-homogeneous. Interbranch Engineering Center “Tribotekhnika”, Dnepropetrovsk, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 6, pp 32–38, June, 1999.     

Fixed Points of Multi-valued Maps and Static Coulomb Friction Problems

The existence of solutions to the Signorini problem with Coulomb friction is a long standing open question. We prove the existence of generalized solutions that satisfy the pointwise Coulomb friction conditions on the entire interface and the normal nonpenetration condition on the complement of a subset with arbitrarily small but possibly positive measure. Furthermore, the penetration itself can also be made arbitrarily small. Although “measure zero” instead of “arbitrarily small measure” would be needed to fully resolve the issue, these generalized solutions seem to be the closest answer available to date. Their existence is proved by a suitable application of Ky Fan's fixed point theorem for multi-valued maps. The same method can be used with a number of variants involving contact of two or more elastic bodies and possible debonding phenomena. This revised version was published online in August 2006 with corrections to the Cover Date.