Mechanical Equilibrium Conditions Research Articles

Тепловая конвекция в ячейке Хеле–Шоу при наличии у жидкости зависимости температуропроводности от температуры

The investigation of thermal convection of a fluid with the dependence of thermal diffusivity on temperature in a vertical Hele Shaw cell heated from below has been fulfilled theoretically.The expression for equilibrium temperature distribution in a cavity has been derived analytically. It has been found that the dependence of temperature on the vertical coordinate looks like a square rootlaw.The linear stability of mechanical equilibrium state against small normal perturbations has been investigated by means of Galerkin method. It has been shown that the most dangerous perturbation in a cavity under considerationis described by the mode which corresponds to the two vortexsteady flow. The numerical simulation of overcritical steady and oscillatory flows has been carried out in the approximation of plane trajectories.This simplification of theoretical modelis consistent with all experimental data on thermal convection in similar cavities. It has been shown that the inclusion of the dependence of thermal diffusivity on temperature into the mathematical model leads to the up down symmetry breakdown for the small values of over criticality.

Ultrasensitive fiber tilt sensor based on a mobile inscribed microbubble along the arc-shaped inwall of the microcavity.

We report a novel fiber tilt sensor based on a mobile microbubble inscribed in an ellipsoidal liquid-filled microcavity. When the tilt angle changes, the microbubble in the liquid can drift along the upper arc-shaped inwall of the microcavity and stay at a vertex to get a state of mechanical equilibrium. The drifting microbubble induces a drastic change on the optical path difference between the light beams in the microcavity and, consequently, on the reflection spectrum, which makes it a tilt angle sensitive element. The experimental results show that the tilt angle sensitivity is as high as 160.21nm/deg, and the detection resolution of such a sensor can reach up to 2.3×10-3 deg. Moreover, such a fiber tilt sensor is capable of distinguishing the tilt direction, since the movement of the microbubble is tilt direction dependent.

Thermodynamic Investigation of the Effect of Interface Curvature on the Solid-Liquid Equilibrium and Eutectic Point of Binary Mixtures.

Thermodynamic phase behavior is affected by curved interfaces in micro- and nanoscale systems. For example, capillary freezing point depression is associated with the pressure difference between the solid and liquid phases caused by interface curvature. In this study, the thermal, mechanical, and chemical equilibrium conditions are derived for binary solid-liquid equilibrium with a curved solid-liquid interface due to confinement in a capillary. This derivation shows the equivalence of the most general forms of the Gibbs-Thomson and Ostwald-Freundlich equations. As an example, the effect of curvature on solid-liquid equilibrium is explained quantitatively for the water/glycerol system. Considering the effect of a curved solid-liquid interface, a complete solid-liquid phase diagram is developed over a range of concentrations for the water/glycerol system (including the freezing of pure water or precipitation of pure glycerol depending on the concentration of the solution). This phase diagram is compared with the traditional phase diagram in which the assumption of a flat solid-liquid interface is made. We show the extent to which nanoscale interface curvature can affect the composition-dependent freezing and precipitating processes, as well as the change in the eutectic point temperature and concentration with interface curvature. Understanding the effect of curvature on solid-liquid equilibrium in nanoscale capillaries has applications in the food industry, soil science, cryobiology, nanoporous materials, and various nanoscience fields.

Phase equilibrium pressure of dielectric system under influence of electrostatic field

Abstract Electrostatic field can change the pressure of dielectric system, and this effect is related to the exerted direction of the electric field. With this characteristic, enhancing mass and heat transfer can be achieved by applying external electric field. In this paper, based on the laws of thermodynamics, the fundamental equation of the dielectric system with the influences of electrostatic field was established. The mechanical equilibrium condition was deduced from this fundamental equation by the free energy criterion for thermodynamic equilibrium state, which is the effective pressure of each phases must be equal to each other. This effective pressure includes the mechanical effect of the electrostatic field on the dielectric system. It consists of two parts, real pressure within the system and the electric tension on the interface between phases, which leads to the difference of the real pressure within different phases. For a two-phase system with a planar interface, when the electric field is parallel to the phase interface, the real pressure within the phase with larger dielectric constant is larger; when the electric field is perpendicular to the phase interface, real pressure within the phase with larger dielectric constant is smaller; and while the field acts on part of the system, the real pressure in the region with electric field is higher than that without electric field. With this characteristic, the vapor pressure of gas-liquid system could be reduced with exerting electrostatic field along the direction parallel to the interface, or it could be increased when the exerting direction is perpendicular to the interface. Our conclusions derived from equilibrium thermodynamics are consistent with the electrohydrodynamic results.

Condition of Mechanical Equilibrium at the Phase Interface with Arbitrary Geometry

The authors produced an expression for the mechanical equilibrium condition at the phase interface within the force definition of surface tension. This equilibrium condition is the most general one from the mathematical standpoint and takes into account the three-dimensional aspect of surface tension. Furthermore, the formula produced allows describing equilibrium on the fractal surface of the interface. The authors used the fractional integral model of fractal distribution and took the fractional order integrals over Euclidean space instead of integrating over the fractal set.

Efficient and robust relaxation procedures for multi-component mixtures including phase transition

We consider a thermodynamic consistent multi-component model in multi-dimensions that is a generalization of the classical two-phase flow model of Baer and Nunziato. The exchange of mass, momentum and energy between the phases is described by additional source terms. Typically these terms are handled by relaxation procedures. Available relaxation procedures suffer from efficiency and robustness resulting in very costly computations that in general only allow for one-dimensional computations. Therefore we focus on the development of new efficient and robust numerical methods for relaxation processes. We derive exact procedures to determine mechanical and thermal equilibrium states. Further we introduce a novel iterative method to treat the mass transfer for a three component mixture. All new procedures can be extended to an arbitrary number of inert ideal gases. We prove existence, uniqueness and physical admissibility of the resulting states and convergence of our new procedures. Efficiency and robustness of the procedures are verified by means of numerical computations in one and two space dimensions.

Nonmonotonic Casimir interaction: The role of amplifying dielectrics

The normal and the lateral Casimir interactions between corrugated ideal metallic plates in the presence of an amplifying or an absorptive dielectric slab has been studied by the path-integral quantization technique. The effect of the amplifying slab, which is located between corrugated conductors, is to increase the normal and lateral Casimir interactions, while the presence of the absorptive slab diminishes the interactions. These effects are more pronounced if the thickness of the slab increases, and also if the slab comes closer to one of the bounding conductors. When both bounding ideal conductors are flat, the normal Casimir force is non-monotonic in the presence of the amplifying slab and the system has a stable mechanical equilibrium state, while the force is attractive and is weakened by intervening the absorptive dielectric slab in the cavity. By replacing one of the flat conductors by a flat ideal permeable plate the force becomes non-monotonic and the system has an unstable mechanical equilibrium state in the presence of either an amplifying or an absorptive slab. When the left side plate is conductor and the right one is permeable, the force is non-monotonic in the presence of a double-layer DA (dissipative-amplifying) dielectric slab with a stable mechanical equilibrium state, while it is purely repulsive in the presence of a double-layer AD (amplifying-dissipative) dielectric slab.

An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows

Numerical solutions of the Euler equations using real gas equations of state (EOS) often exhibit serious inaccuracies. The focus here is the van der Waals EOS and its variants (often used in supercritical fluid computations). The problems are not related to a lack of convexity of the EOS since the EOS are considered in their domain of convexity at any mesh point and at any time. The difficulties appear as soon as a density discontinuity is present with the rest of the fluid in mechanical equilibrium and typically result in spurious pressure and velocity oscillations. This is reminiscent of well-known pressure oscillations occurring with ideal gas mixtures when a mass fraction discontinuity is present, which can be interpreted as a discontinuity in the EOS parameters. We are concerned with pressure oscillations that appear just for a single fluid each time a density discontinuity is present. The combination of density in a nonlinear fashion in the EOS with diffusion by the numerical method results in violation of mechanical equilibrium conditions which are not easy to eliminate, even under grid refinement.A cure to this problem is developed in the present paper for the van der Waals EOS based on previous ideas. A special extra field and its corresponding evolution equation is added to the flow model. This new field separates the evolution of the nonlinear part of the density in the EOS and produce oscillation free solutions. The extra equation being nonconservative the behavior of two established numerical schemes on shocks computation is studied and compared to exact reference solutions that are available in the present context. The analysis shows that shock conditions of the nonconservative equation have important consequence on the results. Last, multidimensional computations of a supercritical gas jet is performed to illustrate the benefits of the present method, compared to conventional flow solvers.

Thermostability analysis of line-tension-associated nucleation at a gas-liquid interface.

The influence of line tension on the thermostability of a droplet nucleated from an oversaturated vapor at the interface of the vapor and another immiscible liquid is investigated. Along with the condition of mechanical equilibrium, the notion of extremization of the reversible work of formation is considered to obtain the critical parameters related to heterogeneous nucleation. From the energetic formulation, the critical reversible work of formation is found to be greater than that of homogeneous nucleation for high value of the positive line tension. On the other hand, for high value of the negative line tension, the critical reversible work of formation becomes negative. Therefore, these thermodynamic instabilities under certain substrate wettability situations necessitate a free-energetics-based stability of the nucleated droplet, because the system energy is not minimized under these conditions. This thermostability is analogous to the transition-based stability proposed by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)]10.1021/j100009a041 in the case of partial wetting phenomena along with the positive line tension. The thermostability analysis limits the domain of the solution space of the present critical-value problem as the thermodynamic transformation in connection with homogeneous and workless nucleation is considered. Within the stability range of the geometry-based wetting parameters, three limiting modes of nucleation, i.e., total-dewetting-related homogeneous nucleation, and total-wetting-associated and total-submergence-associated workless nucleation scenarios, are identified. Either of the two related limiting wetting scenarios of workless nucleation, namely, total wetting and total submergence, is found to be favorable depending on the geometry-based wetting conditions. The line-tension-associated nucleation on a liquid surface can be differentiated from that on a rigid substrate, as in the former, the stability based on mechanical equilibrium and a typical case of workless nucleation with complete submergence are observed.

Multipole analysis of the strain-mediated coupling between proteins adsorbed at tubular lipid membrane surface.

The tubular lipid membranes (TLMs) pulled out from vesicles are often used in in vitro studies of the interactions between curvature-inducing proteins and highly curved membranes. The protein molecules adsorbed at the membrane surface deform the TLM and couple with each other due to the induced strain. Here we propose an approach which models the single curvature-inducing protein action on the lipid bilayer by the multipole, the superposition of the point forces applied to the membrane in the region of the protein adsorption. We show that to be localized in the area of the protein size at the TLM surface, the force multipoles satisfying the mechanical equilibrium conditions should be composed of three or more point forces. The protein coupling energy mediated by the membrane strain is studied in detail. In the region of the tubular membrane stability the maximal distance between two neighboring interacting protein-induced force multipoles is estimated to be of the order of the TLM cross section perimeter. In the vicinity of the TLM instability in the region of the vanishing stretching force applied to the TLM, the interaction radius increases drastically. The high affinity of the single curvature-inducing protein molecule to the regions in the vicinity of the TLM ends is explained and related to the boundary conditions in the experimental set-ups. The reasons for the aggregate formation on the membrane surface are also discussed.

A container loading algorithm with static mechanical equilibrium stability constraints

The Container Loading Problem (CLP) literature has traditionally guaranteed cargo static stability by imposing the full support constraint for the base of the box. Used as a proxy for real-world static stability, this constraint excessively restricts the container space utilization and has conditioned the algorithms developed for this problem. In this paper we propose a container loading algorithm with static stability constraints based on the static mechanical equilibrium conditions applied to rigid bodies, which derive from Newton’s laws of motion. The algorithm is a multi-population biased random-key genetic algorithm, with a new placement procedure that uses the maximal-spaces representation to manage empty spaces, and a layer building strategy to fill the maximal-spaces. The new static stability criterion is embedded in the placement procedure and in the evaluation function of the algorithm. The new algorithm is extensively tested on well-known literature benchmark instances using three variants: no stability constraint, the classical full base support constraint and with the new static stability constraint—a comparison is then made with the state-of-the-art algorithms for the CLP. The computational experiments show that by using the new stability criterion it is always possible to achieve a higher percentage of space utilization than with the classical full base support constraint, for all classes of problems, while still guaranteeing static stability. Moreover, for highly heterogeneous cargo the new algorithm with full base support constraint outperforms the other literature approaches, improving the best solutions known for these classes of problems.

Reflection of cylindrical converging shock wave over a plane wedge

The cylindrical converging shock reflection over a plane wedge is investigated experimentally and numerically in a specially designed shock tube which converts a planar shock into a cylindrical one. When the converging shock is moving along the wedge, both the shock strength and the incident angle are changing, which provides the possibility for the wave transition. The results show that both regular reflection (RR) and Mach reflection (MR) are found on the wedge with different initial incident angles. The wave transitions from direct Mach reflection (DiMR) to inverse Mach reflection (InMR) and further to transitioned regular reflection (TRR) are observed with appropriate initial incident angles. The instability development in the shear layer and strong vortices formation near the wall are evident, which are ascribed not only to the interaction of two shear layers but also to the shock impact and the shock converging effect. Because of the flow unsteadiness after the converging shock, the detachment criterion provides a good estimation for the RR → MR transition, but fails to predict the DiMR → InMR transition, and MR is found to persist slightly below the mechanical equilibrium condition. A hysteresis process is found in the MR → TRR transition and becomes more apparent as the increase of the initial incident angle due to the shock converging effect.

Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach

A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects.

A simplified coupled thermo-mechanical model for the transient analysis of temperature-sensitive hydrogels

The present work investigates the deformation characteristics of a temperature sensitive hydrogel. The state of mechanical and chemical equilibrium is first formulated via a variational approach, upon which a finite element model is developed and implemented through user-defined material subroutine UMAT in the commercial software ABAQUS. We will show that this UMAT implementation allows for more versatility in the imposition of initial conditions over existing models developed using UHYPER subroutine. Furthermore, we propose an approach to simulate the transient swelling process of a temperature sensitive hydrogel. This is achieved through the simultaneous application of three user-defined subroutines, which model the constitutive properties of the gel, as well as the diffusion of solvent molecules within the gel. Several numerical case studies are presented to verify the present model developed with experimental data, as well as to illustrate its capabilities in simulating a wide array of complex gel phenomena, including surface creasing, bifurcation and buckling of gels. Through these numerical examples, we are able to gain deeper insights, and explain some of the new interesting physical phenomena observed in reported experiments.

Stable equilibrium configurations of an oblate capsule in simple shear flow

The objective of the paper is to determine the stable mechanical equilibrium states of an oblate capsule subjected to a simple shear flow, by positioning its revolution axis initially off the shear plane. We consider an oblate capsule with a strain-hardening membrane and investigate the influence of the initial orientation, capsule aspect ratio$a/b$, viscosity ratio${\it\lambda}$between the internal and external fluids and the capillary number$Ca$which compares the viscous to the elastic forces. A numerical model coupling the finite element and boundary integral methods is used to solve the three-dimensional fluid–structure interaction problem. For any initial orientation, the capsule converges towards the same mechanical equilibrium state, which is only a function of the capillary number and viscosity ratio. For$a/b=0.5$, only four regimes are stable when${\it\lambda}=1$: tumbling and swinging in the low and medium$Ca$range ($Ca\lesssim 1$), regimes for which the capsule revolution axis is contained within the shear plane; then wobbling during which the capsule experiences precession around the vorticity axis; and finally rolling along the vorticity axis at high capillary numbers. When${\it\lambda}$is increased, the tumbling-to-swinging transition occurs for higher$Ca$; the wobbling regime takes place at lower$Ca$values and within a narrower$Ca$range. For${\it\lambda}\gtrsim 3$, the swinging regime completely disappears, which indicates that the stable equilibrium states are mainly the tumbling and rolling regimes at higher viscosity ratios. We finally show that the$Ca$–${\it\lambda}$phase diagram is qualitatively similar for higher aspect ratio. Only the$Ca$-range over which wobbling is stable increases with$a/b$, restricting the stability ranges of in- and out-of-plane motions, although this phenomenon is mainly visible for viscosity ratios larger than 1.

Soret-driven convection and separation of binary mixtures in a horizontal porous cavity submitted to cross heat fluxes

An analytical and numerical study of Soret-driven convection in a horizontal porous layer saturated by a binary fluid and subjected to uniform cross heat fluxes is presented. The flow is driven by the combined buoyancy effect due to temperature and induced mass fraction variations through a binary water ethanol mixture. In the first part of the study, a closed-form analytical solution in the limit of a large aspect ratio of the cell (A >> 1) is developed. We are mainly concerned with the determination of the mass fraction gradient of the component of interest along the horizontal direction, which determines the species separation. In the second part, numerous numerical simulations are carried out in order to validate the analytical results and extend heat and mass transfer to an area not covered by the analytical study. Good agreement is found between analytical and numerical results concerning the species separation obtained for a unicellular flow. In this configuration, the Soret separation process is improved by two control parameters: the heat flux density imposed on the horizontal walls of the cell and the ratio, a, of heat flux density imposed on vertical walls to that on horizontal walls. The influence of the heat flux density ratio, a, on the transient regime (relaxation time) is also investigated numerically. We observe that an increase in the parameter a leads to a decrease in the relaxation time. Thus, for a cell heated from below without lateral heating, the onset of convection from the mechanical equilibrium state is analyzed. The linear stability analysis shows that the equilibrium solution loses its stability via a stationary bifurcation or a Hopf bifurcation depending on the separation ratio and the normalized porosity of the medium. The linear stability results are widely corroborated by direct 2D numerical simulations. The thresholds of various multicellular solutions are determined in terms of the governing parameters of the problem using nonlinear direct numerical simulations.

О конвективной устойчивости коллоидной суспензии в ячейке Хеле–Шоу

The article is devoted to the influence of sedimentation and thermal diffusion on threshold of convective instability of a suspension in the Hele-Shaw cell heated from below. Impurity distribution adjustment in colloid in a state of mechanical equilibrium is not uniform, due to the gravitational separation of the mixture. In the channels with the height much less than the sedimentation length, concentration gradient is almost constant. In this case, the ratio of the average impurity concentration to the sedimentation length determines the instability threshold. Mathematical model that describes the convective motion of the colloid, has been based on the Boussinesq approximation. The solution of the linearized system of equations has been performed using the Galerkin method with a large number of basic functions. During research, neutral instability curves are obtained for different values of the Boltzmann number. The influence of gravity stratification on the structure of near-threshold flows has been established. The dependence of the threshold of convective instability and the wave number are determined as a function of parameters of the problem, such as the Boltzmann number and the parameter of thermal diffusion. Received 31 .10.20 16; accepted 05 . 12 .20 16

Fundamentals of numerical analysis of linear tension of two-dimensional drops

Fundamentals of numerical analysis of linear tension of two-dimensional drops of molecules adsorbed on uniform crystal faces have been developed. A lattice-gas model has been applied for description of adsorption. Molecular distributions have been computed in a quasi-chemical approximation, allowing for the effects of direct correlations of interacting particles. Differences in the algorithms of calculation of linear tension on a two-dimensional vapor–liquid interface connected with the effect of the transition region structure, taking into account metastability of coexisting small phases, and a variation in methods of determining line tension at a curved interface according to the material balance condition (equimolecular line) or mechanical equilibrium condition. This has allowed us to compute the equilibrium molecular distribution in small two-dimensional drops and the linear tension value at the interface as a function of the drop size and temperature.

Repulsive Casimir interaction: Boyer oscillators at nanoscale

We study the effect of temperature on the repulsive Casimir interaction between an ideally permeable and an ideally polarizable plate in vacuo. At small separations or for low temperatures the quantum fluctuations of the electromagnetic field give the main contribution to the interaction, while at large separations or for high temperatures the interaction is dominated by the classical thermal fluctuations of the field. At intermediate separations or finite temperatures both the quantum and thermal fluctuations contribute. For a system composed of one infinitely permeable plate between two ideal conductors at a finite temperature, we identify a stable mechanical equilibrium state, if the infinitely permeable plate is located in the middle of the cavity. For small displacements the restoring force of this Boyer oscillator is linear in the deviation from the equilibrium position, with a spring constant that depends inversely on the separation between the two conducting plates and linearly on temperature. Furthermore, an array of such oscillators presents an ideal Einsteinian crystal that displays a fluctuation force between its outer boundaries stemming from the displacement fluctuations of the Boyer oscillators.

Application of a pseudomorphous layer on a vicinal substrate as a test sample for high-resolution X-ray diffractometry

A procedure whereby a pseudomorphous layer on a vicinal substrate is used as a test sample for high-resolution X-ray diffractometry is discussed. An algorithm for calculating the angles between the layer- and substrate-induced peaks according to the substrate-cut deviation and the sample composition, which are known under mechanical-equilibrium conditions of the system, is proposed and illustrated by examples. It is demonstrated that the given sample enables us to control the accuracy of the measured angles of reflection from the layer to within 0.0005°. Two examples are investigated in detail. The first is an AlGaAs/GaAs(001) + 2.5° layer for which the calculations are compared with published experimental data. The second is a GaAs:Te/GaAs(001) + 4° structure measured by means of a Bruker D8 Discover diffractometer.