On the surface equations in two-phase flows and reacting single-phase flows

This paper summarizes the mathematical surface equations which are useful in two-phase flows and single-phase reacting flows. The connection between the interfacial area concentration transport equation for two-phase flows and the flame surface density transport equation for turbulent reacting flows is established. Several analytical examples are given to clarify the physical significance of the different quantities involved in the different transport equations. An introduction to the mathematical treatment of anisotropic interfaces is also given. This theory is illustrated on two different numerical examples: a single inclusion in a simple shear and a single inclusion in an uni-axial elongation.     

Averaged topological equations for dispersed two-phase flows

A general relationship between the volume fraction and the specific interfacial area for averaged dispersed two-phase flows is proposed. This relationship, expressed as a basic set of two scalar evolution equations and two vectorial non-uniformity state equations, is an analytical result obtained by a systematic approach using the derivatives of some generalized functions and a local volume-averaging technique. The proposed set of equations was expressed for measurable macroscopic parameters of the system and has the same generality as the averaged transport equations of two-phase flows. By combination of the basic set of equations, called the averaged topological equations (ATEs), second-order ATEs for the volume fraction were found. The second-order ATEs were expressed both by a Lagrangian formulation and by a Eulerian formulation. The importance and physical meaning of the ATEs developed in this study were clarified within the framework of the theory of kinematic waves.     

Particle motion and turbulence in dense two-phase flows

Measurements of particle mean and r.m.s. velocity were obtained by laser-Doppler anemometry in a descending solid-liquid turbulent flow in a vertical pipe with volumetric concentrations of suspended spherical particles of 270 μm mean diameter in the range 0.1–14%. Similar measurements were obtained in the flow downstream of an axisymmetric baffle of 50% area blockage placed in the pipe with volumetric concentrations of 310 μm particles up to 8% and of 665 μm particles up to 2%. In order to enable measurements in high particle concentrations without blockage of the laser beams the refractive index of the particles was matched to that of the carrier fluid.

The results show that the particle mean velocity profiles become more uniform and the particle r.m.s. velocity decreases with increasing concentration in both flow cases. The particle mean velocity in the pipe flow also decreases with concentration and the relative velocity, the difference between the particle velocity and the fluid velocity in single-phase flow, decreases with increasing Reynolds number. The length of the recirculation region downstream of the baffle was shorter than in single-phase flow by 11 and 24% for particle concentrations of 4 and 8%, respectively. The particle mean velocities were hardly affected by size for concentrations up fo 2%, but the r.m.s. velocities were lower with the larger particles.     

Determining equations of two-phase flows through anisotropic porous media

A new interpretation of the concept of relative phase permeability is given. Relative phase permeabilities are represented in the form of fourth-rank tensors. It is shown that in the case of anisotropic porous media functions depending not only on the saturation but also on the anisotropy parameters represented in the form of ratios of the principal values of the absolute permeability coefficient tensor correspond to the classical representation of the relative phase permeabilities. For a two-phase flow in anisotropic porous media with orthotropic and transversely-isotropic symmetry a generalized two-term Darcy’s law is analyzed. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 87–94, March–April, 1998. The work was carried out with support from the Russian Foundation for Fundamental Research (project No. 96-01-00623).     

The setting up of the equations for two-phase flows by the kinetic theory method

The fundamental equations for two-phase flows are deduced from the Boltzmann's equation. The collision terms are treated with a method similar to what is used in the classical kinetic theory for handling the transport properties of dense gases. It is shown that collision pressure and collision thermal flux exist in gas-particle flows in addition to the general partial pressure and partial thermal flux. Their physical natures are quite different from those of the general partial pressure and partial thermal flux. The applicability of the binary collision assumption and the molecular chaos assumption to gas-particle flows is also discussed. Finally, the equations for two-phase flows obtained by the method of the kinetic theory are compared with those obtained by average continuum models and by the model of particle clouds. The results from the kinetic theory show clearly the physical significance of various parameters and clarify some confusing concepts. Institute of Mechanics, Academia Sinica     

The effect of virtual mass on the basic equations for unsteady one-dimensional heterogeneous flows

A separated two-component flow model is presented which includes virtual mass forces coupling the momentum equations of the two components. It is shown that for physically realistic situations four real roots of the characteristics determinant can exist. These are associated with the acoustic propagation velocities and the fiow velocities of the constitutive phases. Direct analytical solution of the full characteristic determinant is difficult. However, for low Mach number flows an acoustic propagation velocity is obtained which falls between the well-known true separated and homogeneous wave speeds, and compares favorably with experimental data for glass/water and air/water mixtures.     

Two-phase flows: Constitutive equations for lift and Brownian motion and some basic flows

Constitutive relations for the lift force on the particulate phase and the effect of Brownian motion are presented. These constitutive relations are derived subject to three new principles of constitutive equations. The effects of lift and Brownian motion in basic parallel flows are considered in order to determine the importance and the consequences of these effects. The relation of the Brownian motion model involving momentum balance to the diffusive model of particle motions is studied. Dimensional and scaling arguments are given.     

On the equations of motion for accelerated frames

In this communication we present the equations of Euler generalized for the motion of a body in an accelerated reference frame using the generalized work-energy principle. The equivalence among the generalized Euler equation, the generalized Lagrange equation, and the generalized Kane equation are shown when applied to the motion of a body of a holonomic system that depend onn generalized coordinates. Therefore when the generalized coordinates can be reduced to two sets of independent coordinates, the generalized Euler equation can be split into two uncoupled equations that are not independent of each other.Universidade da Beira Interior, Covilhã, Portugal. Published in Prikladnaya Mekhanika, Vol. 31, No. 9, pp. 79–89, September, 1995.     

Rigorous expunction of poisson's ratio from the reissner-meissner equations

Solutions of the two Reissner-Meissner linear equations for the torsionless, axisymetric deformation of elastically isotropic shells of revolution of constant thickness subject to edge conditions and variable normal pressure are compared with the solutions of a simplified version of these equations obtained by neglecting terms containing Poisson's ratio. The relative pointwise differences in the predicted values for the change in the meridional angle and a stress function are shown to be of the same order of magnitude as the inherent errors in classical, first-approximation shell theory. These results do not depend on physical arguments or asymptotic integration techniques, but rather follow from the structure of the Reissner-Meissner equations themselves. The advantage of the simplified equations is that they may be combined into a single comple-valued equation containing no conjugates of the dependent variable.     

On the Squire's transformation for stratified two-phase flows in inclined channels

The applicability of the Squire's transformation for stability analysis of stratified two-phase flow in horizontal and inclined channels is examined. It is shown that for the considered flow such a transformation requires some additional constraints on the change of the inclination angle and flow rates of each of the phases. While the Squire's theorem (on the two-dimensionality of the critical disturbances) rigorously holds for the horizontal two-phase flow, for the inclined flow an exact mathematical theorem cannot be formulated. Nevertheless, it has been proven that 2D perturbations are the critical ones also for the case of inclined channel, since the transformation of a 3D stability problem to its 2D analog is associated with a stabilizing effect of reducing the system inclination, in addition to the reduction of the phases flow rates as in the case of horizontal flows.     

Eulerian prediction of the fluid/particle correlated motion in turbulent two-phase flows

An approximate equation governing the turbulent fluid velocity encountered along discrete particle path is used to derive the fluid/particle turbulent moments required for dispersed two-phase flows modelling. Then, closure model predictions are compared with results obtained from large-eddy simulation of particle fluctuating motion in forced isotropic fluid turbulence.