EFFECTS OF TENSOR CLOSURE MODELS AND 3-D ORIENTATION ON THE STABILITY OF FIBER SUSPENSIONS IN A CHANNEL FLOW

IntroductionFlowoffibresuspensionshasbeenveryfamiliarinmanyindustrialfields.Fibreadditivesplayanimportantroleindragreductioninmanytypesofflow[1- 3].Inthesuspensions,somebehavioroftheflowmaybealteredbythefibres.Oneoftheimportantexamplesisthehydrodynamicsta…     

The large deflection on problem of circular plate on elastic foundation and in conjunction with linear elastic structure

This paper deals with the axisymmetrical deformation of the circular plate in large deflection, which is on elastic foundation and in conjunction with a certain linear elastic structure. The governing integral equations are established by the method of mixed boundary condition 1 and the simplified form is given. The pertrubation method is used to obtain the solutions and an example of the composite structure made up of a circular plate and a cylindrical shell is presented.     

On the non-linear instability of fiber suspensions in a Poiseuille flow

The non-linear instability characteristics of fiber suspensions in a plane Poiseuille flow are investigated. The evolution equation of the perturbation amplitude analogous to Landau equation is formulated and solved numerically for different fiber parameters. It is found that the equilibrium amplitude decreases with the increase of the fiber aspect ratio and volume fraction, i.e. the addition of the fibers reduces the amplitude of the perturbation, and leads to the reduction of the flow instability. This phenomenon becomes significant for large volume concentration and aspect ratio. The mechanism of the reduction of the flow instability is also analyzed by taking into account of the modification of the mean flow and the energy transfer from the mean flow to the perturbation flow.     

Stability of some shear flows for concentrated suspensions

In this paper we investigate the stability of some viscometric flows for a concentrated suspension model which allows for the effects of shear-induced migration, including plane and circular Couette and Poiseulle flows, and unbounded and bounded torsional flows. In the bounded torsional flow, where its radial outer boundary is assumed frictionless, an exact closeform solution is given. With the exception of torsional flows, we find that a limit point for all the steady-state solutions can exist for certain range in the parameter values. In all cases, disturbances can persist for a long time, O (H ²/a ²), where H is a dimension of the flow field, and a is the particles' radius.     

Numerical prediction of fiber motion in a branching channel flow of fiber suspensions

Fiber orientation and dispersion in the dilute fiber suspension that flows through a T-shaped branching channel are simulated numerically based on the slender-body theory. The simulated results are consistent qualitatively with the experimental data available in the literature. The results show that the spatial distribution of fibers is dependent on the fiber aspect ratio, but has no relation with the volume fraction of fiber. The content ratio of fibers near the upper wall increases monotonically with an increasing Re number, and the situation is reverse for the region near the bottom wall. The orientation of fibers depends on Re number, however, the function of fiber volume fraction and aspect ratio is negligible. The fibers near the wall and in the central region of the channel align along the flow direction at all times, but the fibers in the other parts of the channel tend to align along the flow direction only in the downstream region.The project supported by the National Natural Science Foundation of China (10372090) and Doctoral Program of Higher Education in China (20030335001)The English text was polished by Ron Marshall     

Sidewall flow stability in the MHD channel flows

The velocity distribution between two sidewalls is M-shaped for the MHD channel, flows with rectangular cross section and thin conducting walls in a strong transverse magnetic field. Assume that the dimensionless numbersR _m ?1,M, N? 1, and σ* and that the distance between two perpendicular walls is very long in comparison with the distance between two sidewalls. First, the equation for steady flow is established, and the solution of M-shaped velocity distribution is given. Then, an equation for stability of small disturbances is derived based on the velocity distribution obtained. Finally, it is proved that the stability equation for sidewall flow can be transformed into the famous Orr-Sommerfeld equation, in addition, the following theorems are also proved, namely, the analogy theorem, the generalized Rayleigh's theorem, the generalized Fjørtoft's theorem and the generalized Joseph's theorems.     

A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions

A phase-transitional flow takes place during the filling stage by injection molding of short-fiber reinforced thermoplastics. The mechanical properties of the final product are highly dependent on the flow-induced distribution and orientation of particles. Therefore, modelling of the flow which allows to predict the formation of fiber microstructure is of particular importance for analysis and design of load bearing components. The aim of this paper is a discussion of existing models which characterize the behavior of fiber suspensions as well as the derivation of a model which treats the filling process as a phase-transitional flow of a binary medium consisting of fluid particles (liquid constituent) and immersed particles-fibers (solid-liquid constituent). The particle density and the mass density are considered as independent functions in order to account for the phenomenon of sticking of fluid particles to fibers. The liquid constituent is treated as a non-polar viscous fluid, but with a non-symmetric stress tensor. The state of the solid-liquid constituent is described by the antisymmetric stress tensor and the antisymmetric moment stress tensor. The forces of viscous friction between the constituents are taken into account. The equations of motion are formulated for open physical systems in order to consider the phenomenon of sticking. The chemical potential is introduced based on the reduced energy balance equation. The second law of thermodynamics is formulated by means of two inequalities under the assumption that the constituents may have different temperatures. In order to take into account the phase transitions of the liquid-solid type which take place during the flow process a model of compressible fluid and a constitutive equation for the pressure are proposed. Finally, the set of governing equations which should be solved numerically in order to simulate the filling process are summarized. The special cases of these equations are discussed by introduction of restricting assumptions.Received: 6 May 2002, Accepted: 16 December 2002, Published online: 29 July 2003PACS: 83.10.Ff, 83.70.Hg, 83.50.Cz Correspondence to: H. Altenbach     

An experimental study of flow-induced fiber orientation and concentration distributions in a concentrated suspension flow through a slit channel containing a cylinder

Flow-induced fiber orientation and concentration distributions were measured in a concentrated fiber suspension (CFS) and a dilute one (DFS). The channel has a thin slit geometry containing a circular cylinder. In the previous work, many researchers have qualitatively studied fiber orientation and concentration distributions in injection-molded products of fiber-reinforced plastics. In the present work, however, they are quantitatively estimated by direct observation of fibers in the concentrated suspension flow. For the CFS, some fibers rotate in an expansion part between the channel wall and the circular cylinder, and the fiber orientation becomes almost random state. On the other hand, fibers are perfectly aligned along the flow direction owing to the elongational flow near the centerline downstream of the cylinder. The fiber concentration has a flat distribution except near the channel wall and the centerline. For the DFS a minimum in the fiber concentration distribution was clearly observed on the centerline, and two peaks beside the centerline and near the channel wall. This characteristic distribution is caused by the fiber-wall and fiber-cylinder interactions. It is found that the obstacle such as the circular cylinder in the channel significantly affects the fiber orientation downstream of the obstacle for the CFD, while it affects the fiber concentration distribution for the DFS.     

Constitutive equations for fiber suspensions in viscoelastic media

The kinetic theory of elastic dumbells with a friction factor that depends on the fiber orientation is used to obtain constitutive equations for fiber suspensions in a polymer matrix. We followed the approach of Fan (X.J. Fan, in P. Moldenaers and R. Keunings (Eds.), Theoretical and Applied Rheology, Proceedings XIth International Congress on Rheology, Brussels, Elsevier, Amsterdam, 1992, pp. 850–852), and derived equations for polymer solutions based on the FENE-P, FENE-CR, and Giesekus models. Start-up and steady-state free shear flows are studied to explore the effects of the fiber-polymer coupling as well as the fiber volume fraction. Predictions based on different types of closure approximations for the fourth-order fiber orientation tensor are also discussed.     

Investigation of fiber motion near solid boundaries in simple shear flow

In this paper, fiber motion near a planar wall was investigated using a planar shear flow apparatus. Fibers were placed (one at a time) perpendicular to the flow direction at various locations throughout the flow field. The location and orientation of each fiber versus time was measured, using an image processing system, until the fiber aligned with the flow direction. When the centroid of the fiber was located at distances greater than a fiber length from the wall, Jeffery's equations governing particle motion were verified. For distances less than a fiber length and greater than a fiber diameter from the wall, the fiber experienced an increased rate of rotation. In this regime, the motion of the fiber could be described by Jeffery's equations if an increased effective shear rate was used. The effective shear rate was found to increase logarithmically with decreasing separation distance. The wall effect was higher for longer aspect ratio fibers and was also a function of orientation; fibers oriented perpendicular to the wall rotated faster than those oriented parallel to the wall at the same separation distance. Once the fiber aligned with the flow direction, it ceased to rotate within the field of view. In this orientation, the wall had a stabilizing effect on the fiber. In efforts to relate the increase in shear rate to the aspect ratio of the fiber and the separation distance between the fiber and a solid wall, a translation model based on the work of De Mestre and Russel was explored. This model allows one to quantify the increase in shear rate experienced by the fiber due to the presence of a wall or obstruction in the flow field. However, the model has its limitations and care should be taken when applying this model outside its realm of validity. When compared to experimental data, the translation model provides a very good estimate of the increased shear rate experienced by the fiber when it is located less than 2/3 of a fiber length from a planar wall. Received: 20 April 2000 Accepted: 28 September 2000     

Research on the motion of particles in the turbulent pipe flow of fiber suspensions

The motion of fibers in turbulent pipe flow was simulated by 3-D integral method based on the slender body theory and simplified model of turbulence. The orientation distribution of fibers in the computational area for different Re numbers was computed. The results which were consistent with the experimental ones show that the fluctuation velocity of turbulence cause fibers to orient randomly. The orientation distributions become broader as the Re number increases. Then the fluctuation velocity and angular velocity of fibers were obtained. Both are affected by the fluctuation velocity of turbulence. The fluctuation velocity intensity of fiber is stronger at longitudinal than at lateral, while it was opposite for the fluctuation angular velocity intensity of fibers. Finally, the spatial distribution of fiber was given. It is obvious that the fiber dispersion is strenghened with the increase of Re numbers.     

A new constitutive model for fibre suspensions: flow past a sphere

A new phenomenological constitutive equation for homogeneous suspensions of macrosized fibres is proposed. In the model, the local averaged orientation of the fibres is represented by a director field, which evolves in time in a manner similar to the rotation of a prolate spheroid. The stress is linear in the strain rate, but the viscosity is a fourth-order tensor that is directly related to the director field. In the limit of low-volume fractions of fibres, the model reduces properly to the leading terms of the constitutive equation for dilute suspensions of spheroids. The model has three parameters: the aspect ratio R of the fibres, the volume fraction , and A, which plays the role of the maximum-volume fraction of the fibres. Experimental shear data are used to estimate the parameter A, and the resulting model is used in a boundary-element program to study the flow past a sphere placed at the centre line of a cylinder for the whole range of volume fractions from 0.01 to near maximum volume fraction. The agreement with experimental data from Milliken et al. [1] is good.     

槽道纤维悬浮流稳定性的实验研究

对槽道纤维悬浮流进行染色线流动显示和流场PIV实验测量,实验中选用的是直径为20μm、长径比为20～100的尼龙纤维。PIV2100处理器被用来加工处理采集的实验数据。槽道长度1.5m,横截面为矩形,尺寸为105×19mm。实验结果说明在Reynolds数相同的情况下,纤维悬浮流比对应的牛顿流更不容易失稳,悬浮流中的纤维起着抑制流场失稳的作用,而且随着纤维体积分数和长径比的增大,抑制失稳的程度也提高。扰动衰减率的最小值随纤维体积分数和长径比的增加而增大,这一效果在大Re数时更明显。     

Linear stability analysis of flow in a periodically grooved channel

We have conducted the linear stability analysis of flow in a channel with periodically grooved parts by using the spectral element method. The channel is composed of parallel plates with rectangular grooves on one side in a streamwise direction. The flow field is assumed to be two‐dimensional and fully developed. At a relatively small Reynolds number, the flow is in a steady‐state, whereas a self‐sustained oscillatory flow occurs at a critical Reynolds number as a result of Hopf bifurcation due to an oscillatory instability mode. In order to evaluate the critical Reynolds number, the linear stability theory is applied to the complex laminar flow in the periodically grooved channel by constituting the generalized eigenvalue problem of matrix form using a penalty‐function method. The critical Reynolds number can be determined by the sign of a linear growth rate of the eigenvalues. It is found that the bifurcation occurs due to the oscillatory instability mode which has a period two times as long as the channel period. Copyright © 2003 John Wiley & Sons, Ltd.     

Orientation distribution of fibers and rheological property in fiber suspensions flowing in a turbulent boundary layer

A model relating the translational and rotational transport of orientation distribution function （ODF） of fibers to the gradient of mean ODF and the dispersion coefficients is proposed to derive the mean equation for the ODE Then the ODF of fibers is predicted by numerically solving the mean equation for the ODF together with the equations of turbulent boundary layer flow. Finally the shear stress and first normal stress difference of fiber suspensions are obtained. The results, some of which agree with the available relevant experimental data, show that the most fibers tend to orient to the flow direction. The fiber aspect ratio and Reynolds number have significant and negligible effects on the orientation dis- tribution of fibers, respectively. The additional normal stress due to the presence of fibers is anisotropic. The shear stress of fiber suspension is larger than that of Newtonian solvent, and the first normal stress difference is much less than the shear stress. Both the additional shear stress and the first normal stress difference increase with increasing the fiber concentration and decreasing fiber aspect ratio.     

Dynamics of short fiber suspensions in bounded shear flow

We have studied the dynamics of non-colloidal short fiber suspensions in bounded shear flow using the Stokesian dynamics simulation. Such particles make up the microstructure of many suspensions for which the macroscopic dynamics are not well understood. The effect of wall on the fiber dynamics is the main focus of this work. For a single fiber undergoing simple shear flow between plane parallel walls the period of rotation was compared with the Jeffrey’s orbit. A fiber placed close to the wall shows significant deviation from Jeffrey’s orbit. The fiber moving near a solid wall in bounded shear flow follows a pole-vaulting motion, and its centroid location from the wall is also periodic. Simulations were also carried out to study the effect of fiber–fiber interactions on the viscosity of concentrated suspensions.     

On the fiber orientation in steady recirculating flows involving short fibers suspensions

In this work we present a new numerical strategy to treat the 3D Fokker–Planck equation in steady recirculating flows. This strategy combines some ideas of the method of particles, with a more original treatment of the periodicity condition, which characterizes the steady solution of the FP equation in steady recirculating flows, as usually encountered in some rheometric devices. Using this numerical technique the fiber orientation distribution can be computed accurately in any steady recirculating flow. The simulation results can be used to identify some rheological parameters of the suspension, using an inverse technique, as well as to analyze the validity of some simplified models widely used, which require a closure relation. Thus, in this paper several closure relations of the fourth-order orientation tensor will be discussed in the context of a numerical example involving a steady recirculating flow.     

Peculiar features in the rheological behavior of electrorheological suspensions

Studies have been made of concentrated (up to 60%) diatomite suspensions in transformer oil, the structure and theological properties of which depend on an applied electric field. Studies have been conducted of steady-state and transient regimes of straining involving continuous and periodic shear. The structure in such suspensions is formed in the presence of an electric field of 10^–3 –10² duration. The suspensions under continuous stationary strain behave as non-Newtonian fluids with a yield stress dependent on electric intensity. Under periodic deformation conditions the test suspensions exhibit elasticity which abruptly diminishes with increasing deformation amplitude.     

Spatio-temporal analysis of the turbulent flow in a ribbed channel

The turbulent flow in a channel with transverse ribs over one wall is studied experimentally. The height of the obstacles is about one tenth of the channel height, and the spacing is 10 times their height. The Reynolds number based on the channel hydraulic diameter and bulk flow velocity is 15,000. Velocity fields are obtained with high spatial and temporal resolution along the streamwise/wall-normal plane by means of time-resolved particle image velocimetry. Beside mean velocity and Reynolds stresses, the flow is investigated through two-point correlations, distributions of spanwise-swirling events, space–time velocity diagrams and power spectral density. Although the separated flow reattaches before the following obstacle is approached, a strong rib-to-rib interaction occurs. Spanwise vortices, 0.2 rib heights in size, are generated in the free shear layer, travel across the whole pitch, and may impact on the next rib. The large scale motions triggered by the separation grow in size until they reach the following obstacle. Flapping of the separated shear layer is observed at frequencies consistent with previous studies, causing the instantaneous reattachment point to fluctuate. The flapping initiates at the downstream edge of the obstacle tip, rather than at the upstream edge where the instantaneous separation occurs.     

An accurate computation for rapidly varied flow in an open channel

A new method combining the Preissmann four-point scheme and the Holly–Preissmann reach-back scheme is introduced to solve the rapidly varied flow problem in an open channel. The Preissmann four-point scheme is well known for the computation of one-dimensional unsteady flow. The Holly–Preissmann reach-back scheme integrates the Holly-Preissmann two-point scheme with the concept of reach-back characteristics, which allows the characteristics to project several time steps beyond the current time level. A spontaneous surge formation case is used to demonstrate and evaluate the applicability of the new method. It has been found that the results from this method are quite compatible with those of Preissmann four-point scheme. In addition, with the appropriate choice of the number of reach-back time steps, this new method can always avoid the numerical oscillation which usually exists when one uses the Preissmann four-point scheme for the condition of Courant number not close to unity.