Oscillating sedimentation of spheres in viscoplastic fluids

In applications of chemical engineering often the sedimentation is used to separate disperse particles from liquid phases. Some real liquids, e.g., polymer fluids, paints, and skin creams show viscoplastic flow behavior, i.e., they have a yield stress. In such fluids it is possible that suspended particles do not move under action of gravity although the density of the particles is greater than the fluid density. A possibility to sediment stuck spherical particles is shown. The fluid is set in sinusoidal vibration so that the particles undergo forced oscillations. This effect is investigated for single spheres. A model is given and several theoretical results are discussed. A criterion is presented that allows one to predict the combinations of the vibration parameters (amplitude and frequency) which are needed to sediment the spheres. The theoretical investigations are confirmed by experiments. The motion of several glass and steel spheres in an oscillating tube filled with aqueous carbopol solutions are detected. The comparison between theory and experiment shows good agreement.Nomenclature C Stokes drag coefficient - D strain rate tensor - E unit tensor - F, F _w external resp. drag force - f body force vector - G Green deformation tensor - G dimensionless (shear) modulus - g acceleration of gravity - K abbreviation (Eq. (15)) - p pressure - R, spherical coordinates - R ₀, R _a sphere resp. body radius - T extra stress tensor - t time - S stress tensor - U ₀, W ₀ displacement amplitude - u displacement vector - velocity vector - V viscoelastic number - V _k sphere volume - V steady sink velocity - Y yield stress parameter - Y _g limiting value - Z Stokes number - density ratio - second invariant of D - radii ratio - y differential viscosity - , Lamé constants - ^* = + i complex (shear) modulus - f, _k fluid resp. sphere density - second invariant of T - _f yield stress - phase angle - frequency     

On the spin coating of viscoplastic fluids

The spin coating of a viscoplastic material is studied using a continuous viscosity function. Thus, the transient model requires the calculation of only velocity, pressure and the moving-free surface of the liquid film, but not the calculation of the yield surface within the liquid. A Finite Element/Newton-Raphson method is presented for solving this moving boundary problem after mapping the deforming domain onto a fixed one. Assuming axial symmetry, the effect of the Bingham, Reynolds, Capillary and gravitational Bond numbers is examined. The magnitude of the first two parameters affects significantly the flow field and the shape of the film as well as the required spinning time in order to produce a film of uniform thickness. Depending on their values, large departures from the corresponding Newtonian solution may be obtained. In these cases the film does not thin out uniformly, but a maximum in its profile is created at the center of the disk. Then, the magnitude of the Capillary number also affects the size of this maximum. The gravitational Bond number affects the film thickness and its profile to a lesser extent.Dedicated to the memory of Professor Tasos C. Papanastasiou     

Creeping flow of viscoplastic materials through a planar expansion followed by a contraction

In this work, the creeping flow of a viscoplastic fluid through a planar channel with an expansion followed by a contraction is analyzed numerically. The solution of the conservation equations of mass and momentum is obtained via the finite volume method. In order to model the non-Newtonian behavior of the fluid, it was used the generalized Newtonian fluid constitutive equation. The viscosity function was the one proposed by Souza Mendes and Dutra [Souza Mendes, P.R., Dutra, E.S.S., 2004. Viscosity function for yield-stress liquids. Appl. Rheol. 14, 296–302]. The yielded and unyielded regions are obtained for several combinations of rheological parameters. The influence of these parameters on pressure drop through the cavity is also obtained and analyzed.     

The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading

We have examined the problem of the dynamic growth of a single spherical void in an elastic-viscoplastic medium, with a view towards addressing a number of problems that arise during the dynamic failure of metals. Particular attention is paid to inertial, thermal and rate-dependent effects, which have not previously been thoroughly studied in a combined setting. It is shown that the critical stress for unstable growth of the void in the quasistatic case is strongly affected by the thermal softening of the material (in adiabatic calculations). Thermal softening has the effect of lowering the critical stress, and has a stronger influence at high strain hardening exponents. It is shown that the thermally diffusive case for quasistatic void growth in rate-dependent materials is strongly affected by the initial void size, because of the length scale introduced by the thermal diffusion. The effects of inertia are quantified, and it is demonstrated that inertial effects are small in the early stages of void growth and are strongly dependent on the initial size of the void and the rate of loading. Under supercritical loading for the inertial problem, voids of all sizes achieve a constant absolute void growth rate in the long term. Inertia first impedes, but finally promotes dynamic void growth under a subcritical loading. For dynamic void growth, the effect of rate-hardening is to reduce the rate of void growth in comparison to the rate-independent case, and to reduce the final relative void growth achieved.     

A thermodynamically consistent model for cavitating flows of compressible fluids

This work presents a logically consistent thermodynamic model to describe the isothermal cavitation phenomenon in compressible fluid flows. The fluid is regarded as a continuum mixture of liquid and vapor phases (both having the same velocity and temperature), which can or cannot coexist at a same material point and time. The volume fraction is considered as an internal variable and its constraint is treated as a material property, being part of the constitutive relations. Dissipative effects associated with the liquid-vapor phase change transformation and with the vapor volume fraction evolution are taken into account in such a way the Second Law of Thermodynamics is always satisfied. It is shown that the dissipative mechanisms are responsible for a cavitation threshold rule that leads to cavity formation under completely different situations from that characterized by the traditional (reversible) theory. The potentiality of the model as well as its basic features are illustrated and highlighted through a simple numerical example. It is demonstrated that the irreversibility associated with the phase change transformation may be seen as an intermediate case between two physically different non-dissipative situations. One in which the phase change takes place at a constant pressure (the saturated vapor pressure) and the other in which the vapor expands and contracts in the mixture without transforming into liquid.     

Compressible gas flow between closely spaced plates

A theoretical analysis is presented to solve numerically the steady state Navier–Stokes equations, continuity equation and energy equation for a compressible ideal gas flow between two closely spaced, in general nonparallel, infinitely wide plates (siider bearing). The analysis includes the gas inertia effect and covers both non-choked and choked flows. The results of the present analysis compare very well with both analytical and experimental results of compressible flow in a slider bearing comprised of two parallel and stationary plates. It was found that for choked flow the gas inertia effect is important, while the consideration of the energy equation does not affect the accuracy of the calculated flow substantially. Finally, the stiffness of a slider bearing is presented for different geometrical characteristics of the bearing.     

Effect of Compressible Foam Properties on Pressure Amplification During Shock Wave Impact

A comprehensive study is made of the influence of the physical properties of compressible open-cell foam blocks exposed to shock-wave loading, and particularly on the pressure distribution on the shock tube walls. Seven different foams are used, with three different shock Mach numbers, and three different slab lengths. Foam properties examined include permeability, density, stiffness, tortuosity and cell characteristics. The investigations concentrate on both side-wall and back-wall pressures, and the peak pressures achieved, as well as collapse velocities of the front face and the strength and nature of the reflected shock wave. The consequences of deviations from one-dimensionality are identified; primarily those due to wall friction and side-wall leakage. The results presented are the most comprehensive and wide ranging series conducted in a single facility and are thus a significant resource for comparison with theoretical and numerical studies. The different foams show significant differences in behavior, both in terms of peak pressure and duration, depending primarily on their density and permeability.This paper was based on work presented at the 2nd International Symposium on Interdisciplinary Shock Wave Research, Sendai, Japan on March 1–3, 2005.     

A universal cubic interpolation solver for compressible and incompressible fluids

A universal numerical solver commonly usable for compressible and incompressible fluids is proposed. The method approaches the MAC algorithm at very high sound speed and continuously approaches the algorithm for compressible fluid with decreasing sound speed. The advection term is treated by the CIP algorithm which was previously proposed. A single program is applied to one- and two-dimensional shock-tube problems, and two-dimensional liquid flow inside a cavity at high Reynolds number.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

Flow of viscoelastic fluids through a porous channel—I

The flow of viscoelastic fluids through a porous channel with one impermeable wall is computed. The flow is characterized by a boundary value problem in which the order of the differential equation exceeds the number of boundary conditions. Three solutions are developed: (i) an exact numerical solution, (ii) a perturbation solution for small R, the cross-flow Reynold's number and (iii) an asymptotic solution for large R. The results from exact numerical integration reveal that the solutions for a non-Newtonian fluid are possible only up to a critical value of the viscoelastic fluid parameter, which decreases with an increase in R. It is further demonstrated that the perturbation solution gives acceptable results only if the viscoelastic fluid parameter is also small. Two more related problems are considered: fluid dynamics of a long porous slider, and injection of fluid through one side of a long vertical porous channel. For both the problems, exact numerical and other solutions are derived and appropriate conclusions drawn.     

Elastic and viscous effects on flow pattern of elasto-viscoplastic fluids in a cavity

Inertialess flows of elasto-viscoplastic fluids inside a leaky cavity are numerically analyzed using the finite element technique, with the goal of understanding the influence of both the elastic and viscous effects on the topology of the yield surfaces of an elasto-viscoplastic material. Assuming that the collapse of the material microstructure is instantaneous, a mechanical model is composed of the governing equations of mass and momentum for incompressible fluids, and associated with a hyperbolic equation for the extra-stress tensor based on the Oldroyd-B model (Nassar et al., 2011). The main feature of the model is the consideration of the viscosity and relaxation time as functions of the strain rate to allow the shear-thinning of the viscosity and to restrict the elastic effects to the unyielded regions of the material. The numerical simulations are performed through a three-field Galerkin least-squares-type method in terms of the extra-stress tensor and the pressure and velocity fields. The results indicate that the material yield surfaces are strongly influenced by the interplay between the elastic and viscous effects, in accordance with recent experimental visualization of elasto-viscoplastic flows.     

Inhomogeneous flows in sheared complex fluids

Using dynamic light scattering in heterodyne mode, we measure velocity profiles in two complex fluids known to exhibit stress plateau behaviour under shear: a wormlike micelle solution and a lamellar phase. In both cases, our data provide evidence for the simplest shear-banding scenario, according to which the effective viscosity drop in the system is due to the nucleation and growth of a highly sheared band in the gap. We point out that the position of the interface between the two structures is stable at a fixed local shear stress *.This paper was presented at the first Annual European Rheology Conference (AERC) held in Guimarães, Portugal, September 11-13, 2003.     

三维有限变形弹性问题的分析方法

至今还未见到用通常的应力函数和位移函数分析三维有限变形弹性问题的报导。利用Hasegawa的工作和Adkins的摄动法,本文将位移函数用于求解表面力或体积力作用下的有限变形轴对称弹性问题,提出一个分析可压缩和不可压缩材料三维弹性问题的新的解析法,并用两个简单例子验证了这种分析方法。     

Bubble collapse in compressible fluids using a spectral element marker particle method. Part 1. Newtonian fluids

This paper is concerned with the development of a high‐order numerical scheme for the modelling of two‐phase Newtonian flows. The companion paper, herein referred to as Part 2, extends the scheme to two‐phase viscoelastic flows. The particular problem of the collapse of a two‐dimensional bubble in the vicinity of a rigid boundary is considered. The governing equations are discretized using the spectral element method, and the two phases are modelled using a marker particle method. The marker particle scheme is validated using the Zalesak slotted disk rotation test problem. A comprehensive set of results is presented for the problem of bubble collapse near a rigid wall, and qualitative agreement is obtained with other numerical studies and experimental observations. Viscous effects are shown to inhibit bubble collapse and prevent jet formation and are therefore likely to have a mitigating effect on cavitation damage.Copyright © 2011 John Wiley & Sons, Ltd.     

基于刚（粘）塑性流动理论的自然单元法研究

将自然单元法与刚（粘）塑性流动理论相结合,对自然单元法在金属塑性成形过程数值模拟中的应用进行了研究。采用基于Voronoi图和Delaunay三角化结构的Non-Sibsonian插值方法构造近似速度场向量,实现无网格方法中速度边界条件的直接精确施加,提出了基于刚（粘）塑性流动理论的无网格自然单元法。运用不完全广义变分...     

Detecting instabilities in flows of viscoelastic fluids

There is a growing interest in developing numerical tools to investigate the onset of physical instabilities observed in experiments involving viscoelastic flows, which is a difficult and challenging task as the simulations are very sensitive to numerical instabilities. Following a recent linear stability analysis carried out in order to better understand qualitatively the origin of numerical instabilities occurring in the simulation of flows viscoelastic fluids, the present paper considers a possible extension for more complex flows. This promising method could be applied to track instabilities in complex (i.e. essentially non‐parallel) flows. In addition, results related to transient growth mechanism indicate that it might be responsible for the development of numerical instabilities in the simulation of viscoelastic fluids. Copyright © 2003 John Wiley & Sons, Ltd.     

A coupled solid/fluids mixture theory that suffices for diffusion problems

In this paper, I begin with the general formulation of mixture theory by Bowen and present the derivation of a minimal set of field equations, constitutive relations, and material parameters suitable for the solutions of meaningful diffusion problems. The specific results are for a single solid and two fluids, and they may be extended to any number of fluids. I allude to the results of three problems, viz. (1) the injection of a fluid into a geological formation saturated with another fluid, (2) the drainage of two dissimilar fluids from a geological formation due to in-situ fluid pore pressures, and (3) the process of squeezing a sponge dry, in order to illustrate the general applicability of the derived theory.     

非结构动网格在多介质流体数值模拟中的应用

采用非结构动网格方法对含多介质的流场进行数值模拟.采用改进的弹簧方法来处理由于边界运动而产生的网格变形.采用基于格心的有限体积方法求解守恒型的ALE(Arbitrary Lagrangiall-Eulerian)方程,控制面通量的计算采用HLLC(Hartem,Lax,van Leer,Contact)方法,采用几何构造的方法使空间达到二阶精度,时间离散采用四阶Runge-Kutta方法.物质界面的处理采用虚拟流体方法.本文对含动边界的激波管、水下爆炸等流场进行数值模拟,取得较好的结果,不同时刻界面的位置和整个扩张过程被准确模拟.     

Effect of fluids properties on non-equilibrium capillarity effects: Dynamic pore-network modeling

We have developed a Dynamic Pore-network model for Simulating Two-phase flow in porous media (DYPOSIT). The model is applicable to both drainage and imbibition processes. Employing improved numerical and geometrical features in the model facilitate a physically-based pore-scale simulator. This computational tool is employed to perform several numerical experiments (primary and main drainage, main imbibition) to investigate the current capillarity theory. Traditional two-phase flow formulations state that the pressure difference between the two phase is equal to the capillary pressure, which is assumed to be a function of saturation only. Many theoretical and experimental studies have shown that this assumption is invalid and the pressure difference between the two fluids is not only equal to the capillary pressure but is also related to the variation of saturation with time in the domain; this is referred to as the non-equilibrium capillarity effect. To date, non-equilibrium capillarity effect has been investigated mainly under drainage. In this study, we analyze the non-equilibrium capillarity theory under drainage and imbibition as a function of saturation, viscosity ratio, and effective viscosity. Other aspects of the dynamics of two-phase flow such as trapping and saturation profile are also studied.