纤维悬浮液搅拌流动的数值模拟

由于缺乏适当的本构方程，对纤维悬浮液流动的研究一直局限于纤维的牛顿流体悬浮液。本文采用ＭＵＣＭ模型对作者最近提出的纤维Ｏｌｄｒｏｙｄ－Ｂ流体悬浮液的本构方程作了改进，并对锚式桨搅拌槽的二维Ｏｌｄｒｏｙｄ－Ｂ流体和牛顿流体纤维悬浮液搅拌流动作了数值模拟。模拟的结果表明，本文所用的模型和方法能有效地抑制过大局部应力的影响并合理地处理流体的记忆效应。     

Steady-state particle distributions in shear flows of colloids and finely dispersed suspensions

On the basis of an analysis of the pseudoturbulent motion of both the suspended particles and the carrier fluid, the normal stress components in the dispersed phase are obtained for the problem of inclined confined flows of finely dispersed suspensions and colloids. These hydrodynamic pulsations are due to the shear and the work done by the average relative flow of the fluid phase on random concentration fluctuations of the disperse system because of the substantial slip of the phases of the suspension under gravity. The momentum conservation equations for the particles are obtained with allowance for the angle of inclination of the flow to the vertical and on the basis of these equations the suspension capacity of the flow as a function of the angle of inclination, particle size, Galileo number and other parameters is illustrated.Ekaterinburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 78–84, January–February, 1996.     

Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical analysis of complex flow of fibre suspensions in a viscoelastic fluid

Research efforts made so far to simulate fibre suspension flows are limited to fibre suspensions in Newtonian fluids. Though short fibre composites are mostly made of polymers, lack of suitable constitutive equations for fibre suspensions in viscoelastic fluids frustrates attempts to simulate flows of these suspensions. A preliminary work done by the author led to a constitutive equation for semiconcentrated fibre suspensions in the Oldroyd-B fluid. This paper describes the mathematical formulation of the flow problems for such a suspension and the numerical procedure to solve them. Some numerical results of flow past a sphere in a tube are also presented. This paper is supported by the University of Melbourne, the National Natural Science Foundation of China and Zhejiang Province.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

Flow and blockage of highly concentrated granular suspensions in non-Newtonian fluid

Studying the flow of highly concentrated granular suspensions represents a great challenge since they are characterized by a rather complex rheological behavior. In addition, macroscopic heterogeneities may be induced by the flow during rheological measurements due to the eventual relative motion between the liquid and the granular phases that may take place under certain conditions. Solid–liquid separation may ultimately lead to flow blockage. In the present investigation we consider experimentally the influence of the rheological properties of the suspending fluid on the transition between the flow and blockage of a concentrated suspension in a squeeze set-up geometry. The suspending fluid consists of an aqueous Xanthan solution for which rheological properties can be tuned by changing the polymer concentration. For each polymer concentration, it is shown that there exist flow parameters (squeeze velocity and gap thickness) for which one has a transition between homogeneous flow of the suspension and its blockage. Blockage diagrams, delimiting flowability and blockage zones, are then determined. Physical arguments are given to relate the evolution of the blockage diagrams to the flow parameters and rheological properties of suspending fluid.     

Flow-driven orientation dynamics of semiflexible fiber systems

We study the flow-induced orientation dynamics of semiflexible fibers in dilute fiber suspensions. Starting from the equations of motion for a two-rod model of flexible fibers in Stokes flow, the Smoluchowski equation for a connected monomer orientation distribution function is derived. We then obtain a set of equations for the time dependence of the first and second moments of the orientation distribution function, thus extending the Folgar Tucker equations for short rigid fiber suspensions to flexible fiber suspensions. The resulting generalized equations for the orientation dynamics of a suspension of flexible fibers are solved for simple channel flow. It is shown that all qualitative effects of bending and straightening of fibers and their influence on the orientation of flexible fibers are captured within our model. A scalar measure for the distribution of bending in a flow is introduced, which allows to detect the degree of bending of fibers. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

On co-rotational and other rate type constitutive equations

We derive expressions for the dilatational properties of suspensions of gas bubbles in incompressible fluids, using a cell model for the suspension. A cell, consisting of a gas bubble centered in a spherical shell of incompressible fluid, is subjected to a purely dilatational boundary motion and the resulting stress at the cell boundary is obtained. The same dilatational boundary motion is prescribed at the boundary of an “equivalent” cell composed of a one-phase, uniformly compressible fluid with unknown dilatational properties. By specifying that the stress at the boundary of the one-phase cell is equal to the stress at the boundary of the two-phase suspension cell, we obtain expressions for the unknown dilatational properties as a function of observable properties of the suspension. The dilatational viscosity of a suspension with a Newtonian continuous phase and the analogous properties for suspensions with non-Newtonian continuous phases are obtained as functions of the boundary motion, volume fraction of gas, and properties of the incompressible continuous phase. Results are presented for continuous phases which are Newtonian fluids, second-order fluids, and Goddard—Miller model fluids.     

The Dilute Rheology of Swimming Suspensions: A Simple Kinetic Model

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equation including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (J Fluid Mech 52(4):683–712, 1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.     

Granulation and bistability in non-Brownian suspensions

In granulation, a dense colloidal suspension is converted into pasty lumps by application of flow. Often, such lumps are bistable: each can exist either as a fluid droplet (with a shiny surface) or as a jammed granule, whose rough surface creates a bulk stress via capillary action. Such bistability can be explained if the bulk steady-state flow curve is sufficiently nonmonotonic that, above some threshold of stress, flow ceases entirely. This is a stronger condition than the one required to see discontinuous shear thickening, but closely related to it. For instance, inertia can play a role in shear thickening, but not in static bistability. Suitable flow curves were previously found in a phenomenological model of the colloidal glass transition, in which Brownian motion is arrested at high stresses. However, granulation often involves particles too large for Brownian motion to be significant, so that another nonmonotonicity mechanism is needed. A very recent theory, in which the proportion of frictional rather than lubricated contacts increases with stress, provides just such a mechanism, and we use it here to give a simple explanation of granular bistability in non-Brownian suspensions, which requires knowledge only of the steady-state flow curve. However, jamming is in general a history-dependent phenomenon which allows bistability to arise under broader conditions than those just described, possibly including systems that do not shear-thicken at all. In this paper, we focus on explanations of granular bistability based on steady-state shear-thickening, but also discuss alternative explanations based on flow history effects.     

A plane steady-state free-boundary problem for the navier-stokes equations

The model problem of the plane slow steady-state motion of a viscous incompressible fluid with a free boundary is investigated. It is assumed that the free boundary does not have any points in common with the solid surfaces confining the fluid. By the solution of the auxiliary fixed-boundary problem for the Navier-Stokes equations the problem is reduced to an operator equation describing the form of the free surface. The existence and uniqueness problems for the solution and its qualitative behavior are analyzed.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 91–102, May–June, 1972.In conclusion, the authors would like to thank R. M. Garipov and V. Kh. Izakson for affording an opportunity to become acquainted with the results of their unpublished work.     

Shallow waves in a two-layer vortex fluid under a lid

A mathematical model of the vortex motion of an ideal two-layer fluid in a narrow straight channel is considered. The fluid motion in the Eulerian-Lagrangian coordinate system is described by quasilinear integrodifferential equations. Transformations of a set of the equations of motion which make it possible to apply the general method of studying integrodifferential equations of shallow-water theory, which is based on the generalization of the concepts of characteristics and the hyperbolicity for systems with operator functionals, are found. A characteristic equation is derived and analyzed. The necessary hyperbolicity conditions for a set of equations of motion of flows with a monotone-in-depth velocity profile are formulated. It is shown that the problem of sufficient hyperbolicity conditions is equivalent to the solution of a certain singular integral equation. In addition, the case of a strong jump in density (a heavy fluid in the lower layer and a quite lightweight fluid in the upper layer) is considered. A modeling that results in simplification of the system of equations of motion with its physical meaning preserved is carried out. For this system, the necessary and sufficient hyperbolicity conditions are given. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 68–80, May–June, 1999.     

An elastohydrodynamic theory for the rheology of concentrated suspensions of deformable particles

The equations which govern thin films of a Newtonian liquid confined between deformable solid surfaces are applied to the regions of near contact in a concentrated suspension of deformable particles.For the case of slightly deformable elastic particles, one obtains the socalled “elastohydrodynamic” equations of lubrication theory.The appropriate asymptotic solution of these equations yields estimates for the viscosity, of a form proposed earlier by Frankel and Acrivos [1] for rigid particles, as well as a relaxation time for a suspension of near spheres. The present method, which goes beyond the dissipation calculation of Frankel and Acrivos to a derivation of the full stress tensor, yields the same form of dependence of viscosity on particle concentration. However, there is an as yet unexplained difference between the methods in the value of a numerical coefficient determined by the assumed packing of the spheres.While further work is needed on the kinetic theory for fluid suspensions, the methods employed here for the derivation of the stress tensor should have direct utility for certain solid dispersions, where it is possible to specify a priori the particle-packing in the system.     

Thermo-Gravitational Convection in a Near-Critical Fluid in a Side-Heated Enclosed Cavity

Convection near a thermodynamic critical point in a square cavity with lateral heating is investigated numerically on the basis of the Navier-Stokes equations for a compressible gas with a Van-der-Waals equation of state. Comparison of a near-critical fluid and a perfect gas with parameters equal to those of the real fluid near the critical point shows that, with the development of convection, the dynamics of these two media are qualitatively different; however, a certain similarity is observed for the steady-state regime. The dependence of the steady-state flow and heat transfer characteristics on the nondimensional governing parameters is investigated.     

Simulations of fibre orientation in dilute suspensions with front moving in the filling process of a rectangular channel using level-set method

The simulation of fibre orientation in dilute suspension with front moving is carried out using the projection and level-set methods. The motion of fibres is described using the Jeffery equation, and the contribution of fibres to the flow is accounted for by the configuration-field method. The dilute suspension of short fibres in Newtonian fluids is considered. The governing Navier–Stokes equation for the fluid flow is solved using the projection method with finite difference scheme, while the fibre-related equations are directly solved with the Runge–Kutta method. In the present study for fibres in dilute suspension flow for injection molding, the effects of various flow and material parameters on the fibre orientation, the velocity distributions and the shapes of the leading flow front are found and discussed. Our findings indicate that the presence of fibre motion has little influence on the front shape in the ranges of fibre parameters studied at the fixed Reynolds number. Influence of changing fibre parameters only causes variation of front shape in the region near the wall, and the front shape in the central core area does not vary much with the fibre parameters. On the other hand, the fibre motion has strong influence on the distributions of the streamwise and transverse velocities in the fountain flow. Fibre motion produces strong normal stress near the wall which leads to the reduction of transversal velocity as compared to the Newtonian flow without fibres, which in turn, leads to the increased streamwise velocity near the wall. Thus, the fibre addition to the flow weakens the strength of the fountain flow. The Reynolds number has also displayed significant influence on the distribution of the streamwise velocity behind the flow front for a given fibre concentration. It is also found that the fibre orientation is not always along the direction of the velocity vector in the process of mold filling. In the region of the fountain flow, the fibre near the centreline is more oriented across the streamwise direction compared to that in the region far behind the flow front. This leads to the fact that the fibre near the centreline in the region of fountain flow is more extended along the transverse direction. As the fibre orientation in the suspension flow and the shape of the flow front have great bearing on the quality of the product made from injection molding, this study has much implications for engineering applications. These results can also be useful in other fields dealing with fibre suspensions.     

The dynamic equations of motion for a two-component system

The motion of solid particles in a fluid flow is represented as a random process with independent increments. The resulting kinetic equation for the particle distribution has the form previously proposed [1]. The solution to this equation provides a system of equations for the hydrodynamics of the assembly of solid particles. These equations differ from ones previously proposed [2, 3] in having additional terms related to relative motion of the components, whose presence is due to anisotropy in the distribution of the normal stresses in the pseudogas.I am indebted to V. G. Levich for valuable discussions and for constant interest in the work.     

Review of chaos in the dynamics and rheology of suspensions of orientable particles in simple shear flow subject to an external periodic force

We review results obtained over a period of about a decade on a class of technologically and fundamentally important problems in suspension rheology viz., the dynamics and rheology of dipolar suspensions of orientable particles in simple shear flow. The areas explored in this review include effects such as the fluid flow field, external forcing, Brownian diffusion, hydrodynamic interactions and their impact on the rheological properties of the suspension. The main feature of the presentation is the use of a uniform framework in which one or more of the above effects can be studied, based on Langevin type equations for particle orientations combined with a brute-force technique for computing orientational averages. These models are capable of capturing complex dynamical behaviour in the system such as the presence of subharmonics or chaos, both in the dynamics and rheology. The tools developed allow for investigating how chaos in the system is affected by Brownian diffusion and hydrodynamic interactions. The presence of chaos opens up a number of novel possibilities for dynamical and rheological behaviour of the system, which can be put to efficient use in many ways, e.g. in separating particles by aspect ratio and possibly developing computer controlled intelligent rheology. The results also have implications for certain areas of chaos theory, such as a new intermittency route to chaos and the possibility of non-trivial collective behaviour in spatially extended systems. These studies highlight certain deficiencies in current techniques in the literature for handling the rheology of dilute and semi-dilute suspensions. In the presence of Brownian motion the proposed method computes the averages by simulating a set of deterministic ordinary differential equations rather than stochastic differential equations. The systems considered may also serve as a paradigm for analysing how microscopic chaotic fluctuations in spatially extended systems affect macroscopic averages. We also attempt to put our results into context with respect to recent work on rheochaos in complex fluids such as liquid crystals and nematic polymers.     

Fluid Flow-Induced Nonlinear Vibration of Suspended Cables

The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.