A two-fluid model for reactive dilute solid–liquid mixtures with phase changes

Based on the Eulerian spatial averaging theory and the Müller–Liu entropy principle, a two-fluid model for reactive dilute solid–liquid mixtures is presented. Initially, some averaging theorems and properties of average quantities are discussed and, then, averaged balance equations including interfacial source terms are postulated. Moreover, constitutive equations are proposed for a reactive dilute solid–liquid mixture, where the formation of the solid phase is due to a precipitation chemical reaction that involves ions dissolved in the liquid phase. To this end, principles of constitutive theory are used to propose linearized constitutive equations that account for diffusion, heat conduction, viscous and drag effects, and interfacial deformations. A particularity of the model is that the mass interfacial source term is regarded as an independent constitutive variable. The obtained results show that the inclusion of the mass interfacial source term into the set of independent constitutive variables permits to easily describe the phase changes associated with precipitation chemical reactions.     

The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows

In a recent study, Ganpule and Khomami (submitted to J. Non-Newtonian Fluid Mech.) have shown that in order to accurately describe the experimentally observed interfacial instability phenomenon in superposed channel flow of viscoelastic fluids, a constitutive equation that can accurately depict not only the steady viscometric properties of the experimental test fluids, but also their transient viscoelastic properties must be used in the analysis. In the present study, the effect of differences in transient viscoelastic properties which can arise either due to the differences in the predictive capabilities of various constitutive models or from the presence of multiple modes of relaxation on the interfacial instabilities of the superposed pressure driven channel flows has been investigated. Specifically, a linear stability analysis is performed using nonlinear constitutive equations which predict identical steady viscometric properties but different transient viscoelastic properties. It is shown that different nonlinear constitutive equations give rise to the same mechanism of interfacial instability, but the boundaries of the neutral stability contours and the magnitudes of the growth/decay rates, especially at intermediate and shortwaves, are shifted due to the overshoots in the transient viscoelastic responses predicted by the constitutive equations. In addition, the effect of the presence of multiple modes of relaxation on interfacial stability is studied using single and multiple mode upper convected Maxwell (UCM) fluids and it is shown that pronounced differences in the intermediate and shortwave linear stability predictions arise due to the fact that the increase in the number of modes gives rise to additional fast as well as slow relaxation modes and the presence of these additional relaxation modes gives rise to differences in the transient viscoelastic response of the fluids in the absence of any overshoots. The effect of fluid inertia on the interfacial stability of viscoelastic liquids is examined and it is shown that at longwaves, inertia has a pronounced effect on the stability of the interface, whereas at shortwaves, elastic and viscous effects dominate. Furthermore, the mechanism of viscoelastic interfacial instabilities is studied by a careful examination of disturbance eigenfunctions as well as performing a disturbance energy analysis. The results indicate that the mechanism of viscoelastic interfacial instabilities can be described in terms of interaction of mechanisms of purely viscous and purely elastic instabilities. However, since more than one mechanism for the instability is at work, the disturbance energy analysis can not clearly distinguish between them due to the fact that the eigenfunctions used in the energy analysis contain the information regarding both viscous and elastic effects. Hence, the mechanism of the instability must be determined by a careful examination of disturbance eigenfunctions.     

The Burnett equations for a relativistic gas

The Burnett constitutive equations for the dynamic pressure, heat flux and pressure deviator are obtained from the relativistic Boltzmann equation by using fourteen moment equations and an iteration method akin to the Maxwellian iteration procedure of the non-relativistic theory. The non-relativistic and ultra-relativistic limits of all Burnett coefficients are given and it is shown that the non-relativistic limit of the Burnett constitutive equations agree with those obtained from the non-relativistic Boltzmann equation. Received July 21, 2000     

On the dynamic bending of layered plates

The dynamic bending of layered metallic plates is studied. Their layers are reinforced with wires, and there are interfacial layers between the contact surfaces. A general approach to solving the corresponding dynamic problems is proposed. It is the principle of virtual work. A system of ordinary differential constitutive equations is obtained and used to determine the amplitude coefficients of the series of coordinate functions that approximates the deflections. The constitutive equations are reduced to Volterra equations of the second kind. Algorithms are developed to calculate the coefficients of the constitutive equations depending on the coordinate functions and on the reinforcement pattern. Possible methods to set up coordinate functions and solve integral equations are indicated     

Numerical study on the two-phase flow distribution in a T-junction

The main objective of this paper is to investigate the ability of a two-dimensional two-fluid computer code to predict the phase separation in a T-junction. A new semi-implicit numerical scheme is developed for solving the two-fluid model equations. Special attention is directed to the modelling of the constitutive for the interfacial friction term. Detailed distribution of void fraction, pressure and velocities are obtained for an air–water mixture in a vertical tee. Good agreement was obtained between the computer code results and the experimental data for the phase separation in the T-junction.     

A generalized multilength scale nonlinear composite plate theory with delamination

A new type of plate theory for the nonlinear analysis of laminated plates in the presence of delaminations and other history-dependent effects is presented. The formulation is based on a generalized two length scale displacement field obtained from a superposition of global and local displacement effects. The functional forms of global and local displacement fields are arbitrary. The theoretical framework introduces a unique coupling between the length scales and represents a novel two length scale or local-global approach to plate analysis. Appropriate specialization of the displacement field can be used to reduce the theory to any currently available, variationally derived, displacement based (discrete layer, smeared, or zig-zag) plate theory.The theory incorporates delamination and/or nonlinear elastic or inelastic interfacial behavior in a unified fashion through the use of interfacial constitutive (cohesive) relations. Arbitrary interfacial constitutive relations can be incorporated into the theory without the need for reformulation of the governing equations. The theory is sufficiently general that any material constitutive model can be implemented within the theoretical framework. The theory accounts for geometric nonlinearities to allow for the analysis of buckling behavior.The theory represents a comprehensive framework for developing any order and type of displacement based plate theory in the presence of delamination, buckling, and/or nonlinear material behavior as well as the interactions between these effects.The linear form of the theory is validated by comparison with exact solutions for the behavior of perfectly bonded and delaminated laminates in cylindrical bending. The theory shows excellent correlation with the exact solutions for both the inplane and out-of-plane effects and the displacement jumps due to delamination. The theory can accurately predict the through-the-thickness distributions of the transverse stresses without the need to integrate the pointwise equilibrium equations. The use of a low order of the general theory, i.e. use of both global and local displacement fields, reduces the computational expense compared to a purely discrete layer approach to the analysis of laminated plates without loss of accuracy. The increased efficiency, compared to a solely discrete layer theory, is due to the coupling introduced in the theory between the global and local displacement fields.     

Poromechanics of freezing materials

When subjected to a uniform cooling below the freezing point a water-infiltrated porous material undergoes a cryo-deformation resulting from various combined actions: (i) the difference of density between the liquid water and the ice crystal, which results in the initial build-up of an in-pore pressure at the onset of crystallization; (ii) the interfacial effects arising between the different constituents, which eventually govern the crystallization process in connection with the pore access radius distribution; (iii) the drainage of the liquid water expelled from the freezing sites towards the air voids; (iv) the cryo-suction process, which drives liquid water towards the already frozen pores as the temperature further decreases; (v) the thermomechanical coupling between the solid matrix, the liquid water and the ice crystal. We work out a comprehensive theory able to encompass this whole set of actions. A macroscopic approach first provides the constitutive equations of freezing poroelastic materials, including the interfacial energy effects. This approach reveals the existence of a thermodynamic state function—namely the liquid saturation degree as a function of the temperature only. The macroscopic ice-dependent poroelastic properties are then upscaled from the knowledge of the elastic properties of the solid matrix, of the pore access radius distribution, and of the capillary curve. The theory is finally illustrated by analysing quantitatively the effects of the cooling rate and of the pore radius distribution upon the cryo-deformation of water-infiltrated porous materials. The theory succeeds in accounting for the experimentally observed shrinkage of embedded air voids, while predicting the partial melting of the ice already formed when the cooling suddenly stops.     

Research note on application of a constitutive inequality in nonassociated plasticity

A constitutive inequality proposed by several authors is applied, apparently for the first time, to a nonassociated elastic/plastic constitutive model. The main mathematical difficulty ensues from the presence of a nonsymmetric dyadic matrix in the constitutive equations. A general property of such products is used to derive a simple sufficient condition for the constitutive inequality. This condition involves the hardening modulus and the spectral condition number of a matrix arising from the constitutive equations. Several examples are given.     

Numerical Flow Simulation for Bingham Plastics in a Single-Screw Extruder

Numerical simulations have been performed concerning the operation of a single-screw extruder, pumping a Bingham plastic under isothermal, developed flow conditions. Under the assumption of sufficiently low Reynolds numbers, inertia effects are neglected. The singular rheological behavior of the Bingham plastic is considered as the limiting case within a class of generalized Newtonian liquids with smooth constitutive equations. The validation of this regularization process is shown for a related flow problem where the Bingham solution is known analytically. A mixed finite-element method is applied to the flow in the screw-extruder to reduce the equations of motion, the continuity equation, and the regularized constitutive equation to a set of nonlinear algebraic equations, which are solved using a Newton method. In particular, the pumping characteristics of a given screw geometry are extracted from the finite-element calculations, i.e., the dependence of the volumetric flow rate and of the power requirement on the axial pressure drop, on the screw speed, and on the rheological parameters. Calculated flow fields clearly show the size and position of regions in the extruder channel where the Bingham plastic behaves like a solid. Received: 12 December 1995 and accepted 12 November 1996     

Irreversible thermodynamics of liquid crystal interfaces

A macroscopic theory for the dynamics of isothermal compressible interfaces between nematic liquid crystalline polymers and isotropic viscous fluids has been formulated using classical irreversible thermodynamics. The theory is based on the derivation of the interfacial rate of entropy production for ordered interfaces, that takes into account interfacial anisotropic viscous dissipation as well as interfacial anisotropic elastic storage. The symmetry breaking of the interface provides a natural decomposition of the forces and fluxes appearing in the entropy production, and singles out the symmetry properties and tensorial dimensionality of the forces and fluxes. Constitutive equations for the surface extra stress tensor and for surface molecular field are derived, and their use in interfacial balance equations for ordered interfaces is identified. It is found that the surface extra stress tensor is asymmetric, since the anisotropic viscoelasticity of the nematic phase is imprinted onto the surface. Consistency of the proposed surface extra stress tensor with the classical Boussinesq constitutive equation appropriate to Newtonian interfaces is demonstrated. The anisotropic viscoelastic nature of the interface between nematic polymers (NPs) and isotropic viscous fluids is demonstrated by deriving and characterizing the dynamic interfacial tension. The theory provides for the necessary theoretical tools needed to describe the interfacial dynamics of NP interfaces, such as capillary instabilities, Marangoni flows, wetting and spreading phenomena.     

Energy release rates for delamination of composite beams

This paper deals with the formulation of a mechanical/numerical model for analyzing the delamination effects of layered composite beams. The laminate is modelled through a multiple-beam model and interfacial constitutive laws are obtained by introducing interlaminar bilateral and unilateral springs. Delamination growth is described by employing the classical energy release rate criterion. A path-following procedure with delamination growth control is presented for the numerical analysis of the given model. Numerical results on delamination buckling and growth in compressed beams are given and comparisons with simplified theories are established.     

Simulation and transport phenomena of a ternary two-phase flow

A chemical flood model for a three-component (petroleum, water, injected chemical) two-phase (aqueous, oleic) system is presented. It is ruled by a system of nonlinear partial differential equations: the continuity equation for the transport of each of its components and Darcy's equation for the two-phase flow. The transport mechanisms considered are ultralow interfacial tension, capillary pressure, dispersion, adsorption, and partition of the components between the fluid phases (including solubilization and swelling).The mathematical model is numerically solved in the one-dimensional case by finite differences using an explicit and direct iterative procedure for the discretization of the conservation equations. Numerical results are compared with Yortsos and Fokas' exact solution for the linear waterflood case including capillary pressure effects and with Larson's model for surfactant flooding. The effects of the above-mentioned transport mechanisms on concentration profiles and on oil recovery are also analyzed.     

Irreversible dilatancy or compressibility of viscoplastic rock-like materials and some applications

An elastic/viscoplastic constitutive equation for rock-like materials which is able to describe, besides creep, an irreversible compressibility and/or dilatancy of the volume is formulated. Mathematical definitions are used for the concepts of dilatancy, compressibility, compressibility/dilatancy boundary, and failure. Two examples of such constitutive equations are given for a hard rock and a softer rock.Two applications of this constitutive equation are presented. First the creep dilatancy and/or compressibility of the rocks surrounding an oil well or a pressure shaft are studied for various initial primary stresses. The location and the conditions under which the rock surrounding the well may become dilatant are then shown. The same problem is studied for the case of horizontal tunnels. Various depths and primary stresses are considered and it is shown where possible failure may occur. For both applications numerical examples are given.     

Theoretical analysis of non-uniform extensional flows in relation to processing rheology and predictions of some constitutive equations

In this paper, a characteristic equation involving the stream function, already given by one of the authors in a previous work for classifying axisymmetric incompressible flows, is re-considered. Non-uniform nearly extensional flows are derived as particular solutions from this equation. Using experimental data in the literature for polymer solutions and melts, it is proved that particular solutions of the characteristic equation lead to kinematics very close to those encountered in the fiber-spinning process. The kinematic equations satisfactorily correlating the fiber-spinning data are used in order to determine the ability of constitutive equations to predict realistic stresses in the flow domain. The rheological parameters of the fluids, obtained from experiments, are used for computation of differential and integral constitutive equations in the spinning conditions. Comparisons with the stress response of adequate constitutive equations are given and discussed.Also affiliated to: Université Joseph Fourier Grenoble I and Institut National Polytechnique de Grenoble, Associé au CNRS (URA 1510)     

Macroscopic constitutive equations of thermo-poroviscoelasticity derived using eigenstrains

Macroscopic constitutive equations for thermo-viscoelastic processes in a fully saturated porous medium are re-derived from basic principles of micromechanics applicable to solid multi-phase materials such as composites. Simple derivations of the constitutive relations and the void occupancy relationship are presented. The derivations use the notion of eigenstrain or, equivalently, eigenstress applied to the separate phases of a porous medium. Governing coupled equations for the displacement components and the fluid pressure are also obtained.     

DECAY RATE OF SAINT-VENANT END EFFECTS FOR PLANE DEFORMATIONS OF PIEZOELECTRIC-PIEZOMAGNETIC SANDWICH STRUCTURES

This paper is concerned with the decay of Saint-Venant end effects for plane deformations of piezoelectric （PE）-piezomagnetic （PM） sandwich structures, where a PM layer is located between two PE layers with the same material properties or reversely. The end of the sandwich structure is subjected to a set of self-equilibrated magneto-electro-elastic loads. The upper and lower surfaces of the sandwich structure axe mechanically free, electrically open or shorted as well as magnetically open or shorted. Firstly the constitutive equations of PE mate- rials and PM materials for plane strain are given and normalized. Secondly, the simplified state space approach is employed to arrange the constitutive equations into differential equations in a matrix form. Finally, by using the transfer matrix method, the characteristic equations for eigen- values or decay rates axe derived. Based on the obtained characteristic equations, the decay rates for the PE-PM-PE and PM-PE-PM sandwich structures are calculated. The influences of the electromagnetic boundary conditions, material properties of PE layers and volume fraction on the decay rates are discussed in detail.     

Separation phenomena at the interface of a finitely deformed composite sphere

The problem of the finite deformation of a composite sphere subjected to a spherically symmetric dead load traction is revisited focusing on the formation of a cavity at the interface between a hyperelastic, incompressible matrix shell and a rigid inhomogeneity. Separation phenomena are assumed to be governed by a vanishingly thin interfacial cohesive zone characterized by uniform normal and tangential interface force–separation constitutive relations. Spherically symmetric cavity shapes (spheres) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation, the dead loading and the volume concentration of inhomogeneity. Spherically symmetric and non-symmetric bifurcations initiating from spherically symmetric equilibrium states are analyzed within the framework of infinitesimal strain superimposed on a given finite deformation. A simple formula for the dead load required to initiate the non-symmetrical rigid body mode is obtained and a detailed examination of a few special cases is provided. Explicit results are presented for the Mooney–Rivlin strain energy density and for an interface force–separation relation which allows for complete decohesion in normal separation.     

填隙二阶流体下圆球平行于壁平移的粘性阻力

离散元法是分析散体力学行为的数值方法。存在填隙流体时，颗粒之间或颗粒与壁之间产生的法向挤压力和切向阻力、阻力矩，是湿颗粒离散元法的理论基础。二阶流体是以微小偏离牛顿流体本构而考虑时间影响的一种流体。它具有常粘度，并且第一和第二法向应力差正比于剪切率的平方。根据Reynolds润滑理论，采用小参数法，导出了存在填隙二阶流体时，圆球沿平行于平壁缓慢移动时流体的速度场和压力方程，进而求出切向阻力和阻力矩的解析解。有趣的是在推导时所得的速度场和压力方程形式比牛顿流体要复杂得多，但最终结果表明圆球沿平行于平壁移动时因填隙二阶流体引起的切向阻力和阻力矩与牛顿流体时的结果相同。