The flow behavior of fiber suspensions in Newtonian fluids and polymer solutions.

This is the second part of a study examining the mechanical properties and capillary flow of fiber suspensions in Newtonian fluids and in polymer solutions. In part I results for the viscous and elastic properties of the fiber suspensions were presented and it was shown that the fiber suspensions exhibited normal stresses in Newtonian as well as in viscoelastic suspending media. It was thus expected that circulating secondary flows would occur near the entrance to a capillary. Four types of fillers (glass, carbon, nylon and vinylon fibers) suspended in glycerin, HEC solutions and Separan solutions were investigated. The entrance flow patterns were visualized and the pressure fluctuations measured. The visualization enabled the eddies occurring in the fiber suspensions in Newtonian fluids to be analysed and classified into two tpyes. The results from the flow visualization were correlated with the pressure fluctuations. Empirical equations for the tube length correction factor due to the elasticity were obtained.     

An analysis on entrance region effect of the laminar radial flow between two parallel disks

In this paper, B. B. Golubef method^[1] is used for calculating the radial diffuse flow between two parallel disks for the first step. The momentum integral equation together with the energy integral equation is derived from the boundary layer momentum equation, and the expression of secondary approximation explicit function in which the channel length of entrance region varies with the boundary layer thickness can be obtained by using Picard iteration[2] in the solution of the energy integral equation. Therefore, this has made it possible to analyze directly and analytically the coefficients of the entrance region effect. In particular, when the outer diameter of disk is smaller than the entrance region length, the advantage of this method can be prominently manifest.Only because the energy integral equation is employed, the terms in the pressure loss coefficient can be independently derived theoretically. The computable value of the pressure loss coefficient presented in this paper is nearer to the testing value than that in ref. [3] when the entrance correction Reynolds number Re<100. Therefore the results in this paper within Re<100 are both reliable and simple.     

A remark on similarity analysis of boundary layer equations of a class of non-Newtonian fluids

A similarity analysis of three-dimensional boundary layer equations of a class of non-Newtonian fluid in which the stress, an arbitrary function of rates of strain, is studied. It is shown that under any group of transformation, for an arbitrary stress function, not all non-Newtonian fluids possess a similarity solution for the flow past a wedge inclined at arbitrary angle except Ostwald-de-Waele power-law fluid. Further it is observed, for non-Newtonian fluids of any model only $90°$ of wedge flow leads to similarity solutions. Our results contain a correction to some flaws in Pakdemirli׳s [14] (1994) paper on similarity analysis of boundary layer equations of a class of non-Newtonian fluids.     

An SPH multi‐fluid model based on quasi buoyancy for interface stabilization up to high density ratios and realistic wave speed ratios

We introduce a smoothed particle hydrodynamics (SPH) concept for the stabilization of the interface between 2 fluids. It is demonstrated that the change in the pressure gradient across the interface leads to a force imbalance. This force imbalance is attributed to the particle approximation implicit to SPH. To stabilize the interface, a pressure gradient correction is proposed. In this approach, the multi‐fluid pressure gradients are related to the (gravitational and fluid) accelerations. This leads to a quasi‐buoyancy correction for hydrostatic (stratified) flows, which is extended to nonhydrostatic flows. The result is a simple density correction that involves no parameters or coefficients. This correction is included as an extra term in the SPH momentum equation. The new concept for the stabilization of the interface is explored in 5 case studies and compared with other multi‐fluid models. The first case is the stagnant flow in a tank: The interface remains stable up to density ratios of 1:1000 (typical for water and air), in combination with artificial wave speed ratios up to 1:4. The second and third cases are the Rayleigh‐Taylor instability and the rising bubble, where a reasonable agreement between SPH and level‐set models is achieved. The fourth case is an air flow across a water surface up to density ratios of 1:100, artificial wave speed ratios of 1:4, and high air velocities. The fifth case is about the propagation of internal gravity waves up to density ratios of 1:100 and artificial wave speed ratios of 1:4. It is demonstrated that the quasi‐buoyancy model may be used to stabilize the interface between 2 fluids up to high density ratios, with real (low) viscosities and more realistic wave speed ratios than achieved by other weakly compressible SPH multi‐fluid models. Real wave speed ratios can be achieved as long as the fluid velocities are not very high. Although the wave speeds may be artificial in many cases, correct and realistic wave speed ratios are essential in the modelling of heat transfer between 2 fluids (eg, in engineering applications such as gas turbines).     

The squeeze flow of a bi-viscosity fluid between two rigid spheres with wall slip

In this paper, the squeeze flow between two rigid spheres with a bi-viscosity fluid is examined. Based on lubrication theory, the squeeze force is calculated by deriving the pressure and velocity expressions. The results of the normal squeeze force are discussed, and fitting functions of the squeeze and correction coefficients are given. The squeeze force between the rigid spheres increases linearly or logarithmically with the velocity when most or part of the boundary fluid reaches the yield state, respectively. Furthermore, the slip correction coefficient decreases with the increase in the velocity. The investigation may contribute to the further study of bi-viscosity fluids between rigid spheres with wall slip.     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.     

非牛顿流体力学研究的若干进展

简要介绍微极性流体,应力偶流体,非整数阶Maxwell流体和智能流体的基本概念,通过简单的例子阐述它们与牛顿流体的本质差别,并介绍非牛顿流体在以上热点研究领域的若干进展．     

圆柱管道内运动的小球受到的黏滞阻力的数值分析

为了更好地运用落球法测量研究流体的黏滞系数,研究小球在黏性流体中下落的受力情况,本文对小球在充满黏性流体的圆柱管道的下落过程进行分析。利用COMSOL4.4仿真模拟,建立了合理的仿真模型,并分析了小球受到的黏滞阻力与小球的大小、下落位置的关系。结果表明:选择速度项二阶近似、压强项一阶近似的离散化方法,可以得到和理论值非常相符的仿真结果;当下落过程中小球球心始终在圆柱轴线上时,小球受到的黏滞阻力相对于Stokes力的修正系数,是小球半径与圆柱管道半径的比例函数,本文得到了更大范围的符合理论解的修正系数;当下落过程中小球的球心偏离圆柱轴线时,对于同样大小的小球,黏滞阻力、压强力、黏性力均随着球心到轴线的距离先减小后增大,且具有不同的极小值点。     

Flow measurement on an oscillating pipe flow near the entrance using the UVP method

 The authors have carried out a study to investigate and clarify the characteristics of purely oscillating pipe flows over the developing region. The main objective of this study is to establish the method of time-dependent velocity profiles obtained by the ultrasonic velocity profile (UVP) measurement method. First, the relationship between the test fluids and the microparticles, as reflectors of ultrasonic pulses, was investigated. In addition, the relationship between the sound speeds of the test fluids and the wall materials was studied. Second, the UVP was used to obtain the instantaneous velocity profiles in oscillating pipe flows, and the developing characteristics of the flows were analyzed. Finally, the “entrance length” (by analogy with a unidirectional pipe flow) required for oscillating pipe flows was analyzed by examining the amplitude of the harmonic spectral components of the oscillating frequency. A fast Fourier transform (FFT) is proposed as the applicable method to estimate the “entrance length”. From the Fourier transform of the velocity on the centerline, nonlinear oscillation of fluid occurs in the “entrance length” of the oscillating flows, and the viscous dissipation of the higher-order velocity harmoncis determines the entrance region. The “entrance length” can be obtained from the dissipation length of the third-order harmonic. These results prove that the UVP method is highly applicable to carry out the flow measurement in the “entrance length” of oscillating pipe flow. Received: 20 March 2000 / Accepted: 10 August 2001     

A projection scheme for Navier-Stokes with variable viscosity and natural boundary condition

Combiningthe Navier-Stokes systems with Neumann (or natural) boundary condition to characterize a fluid flow is frequent. The popular projection (or pressure correction) methods inspired by Chorin and Temam are not well adapted to such boundary condition, which translate in loss of accuracy. If some alternative projection methods have been proposed to reduce the accuracy loss due to the Neumann condition in case of Newtonian fluids, little has been proposed for generalized Newtonian fluids. In this work, we propose two methods derived from the incremental pressure correction projection that can be used for fluids with inhomogeneous or variable viscosity with natural boundary condition. Both time and space accuracy of the methods will be illustrated using a manufactured solution.     

非牛顿流体入口收敛流动分析

本文讨论了非牛顿流体的入口收敛流动问题,考虑到粘弹性流体在流动中的粘滑行为,应用最小能原理,导出了扩展的入口收敛流边界流线方程和流体自然收敛锥角方程,并与前人的工作进行了比较和分析。     

Effect of variable slip boundary conditions on flows of pressure driven non-Newtonian fluids

In microfluidic devices it has been suggested a scheme for enhancing the mixing of two fluids is to use patterned, slip boundary conditions. This has been shown to induce significant transverse flow for Newtonian fluids [S.C. Hendy, M. Jasperse, J. Burnell, Effect of patterned slip on micro- and nanofluidic flows, Phys. Rev. E 72 (2005) 016303]. Here we study the effect of patterned slip on non-Newtonian fluids. Using a power-law model it is shown for shear-thickening fluids patterned slip can induce significant transverse flows comparable in size to those produced for Newtonian fluids. However, for shear-thinning fluids this transverse flow is suppressed. We predict a convenient way to increase the transverse flow for shear-thinning fluids is to use a patterned slip boundary condition coupled to a sinusoidally time-varying pressure gradient. This system is studied using a simple linearized White–Metzner model which has a power-law viscosity function [R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, Volume 1: Fluid Mechanics, John Wiley & Sons, New York, 1987]. In this case it is shown the two variations combine to produce transverse flow, which can be increased by increasing the frequency of the sinusoidal time-dependent fluctuation.     

分层流体中混合流体团运动生成内波的研究现状

本文综述了分层流体中混合流体团运动生成内波的研究进展状况,系统地介绍了它的应用背景、实验和理论结果以及有待进一步研究的问题。      

Turbulent heat transfer in circular tube flows of viscoelastic fluids

The effects of thermal entrance length, polymer degradation and solvent chemistry were found to be critically important in the determination of the drag and heat transfer behavior of viscoelastic fluids in turbulent pipe flow. The minimum heat transfer asymptotic values in the thermally developing and in the fully developed regions were experimentally determined for relatively high concentration solutions of heat transfer resulting in the following correlations: $$\begin{gathered} j_H = 0.13\left( {\frac{x}{d}} \right)^{ - 0.24} \operatorname{Re} _a^{ - 0.45} thermally developing region \hfill \\ x/d< 450 \hfill \\ j_H = 0.03 \operatorname{Re} _a^{ - 0.45} thermally developed region \hfill \\ x/d< 450 \hfill \\ \end{gathered} $$ For dilute polymer solutions the heat transfer is a function ofx/d, the Reynolds number and the polymer concentration. The Reynolds analogy between momentum and heat transfer which has been widely used in the literature for Newtonian fluids is found not to apply in the case of drag-reducing viscoelastic fluids.     

Velocity measurement of start-up flow of polymeric solutions in a planar contraction channel

The start-up flow of polymeric solutions in an abrupt contraction channel with a rectangular cross-section was experimentally studied. Aqueous solutions with 0.2 wt% and 1.0 wt% of polyacrylamide were used as test fluids. Temporal changes in velocity were measured with a laser Doppler velocimeter. The velocity overshot just after the onset of the flow. The changes in velocity caused by the rearrangement of the velocity distribution were observed. A three-dimensional flow structure and the development of a vortex region were found near the entrance to the contraction. A decrease in the axial velocity just upstream from the contraction was observed. The velocity profile is related to the three-dimensional nature of the flow and the elongational rheological properties of the test fluid. The experimental results indicate that three-dimensional analysis of viscoelastic flows is required especially for the flow in rectangular channels. Received: 6 July 1998 Accepted: 1 December 1998     

Exact solution for compressive flow of viscoplastic fluids under perfect slip wall boundary conditions

Based on the perfect slip condition between rigid walls and fluids, the compressive flow of Herschel-Bulkley fluids and biviscous fluids was studied. The explicit expressions of stresses and fluid velocity were given. To move the rigid walls for a Herschel-Bulkley fluid with the yield stress (τ₀), the mean pressure applied onto the rigid wall should be larger than 2τ₀/. No yield surface exists in the interior of the fluids when flow occurs. For a biviscous fluid, a critical load was given. The fluid behaves like the Bingham fluid when the external applied load onto the wall is larger than the critical load, otherwise the fluid is Newtonian. Received: 10 June 1997 Accepted: 22 September 1997     

Velocity field of convergent flow into a rectangular slit for a polymeric fluid

Three-dimensional velocity distributions in the entry region of a rectangular slit contraction were investigated using a dual-beam laser Doppler velocimeter. The flow of a silicone oil (a Newtonian fluid) and a solution of silicone rubber in the same silicone oil (a viscoelastic fluid) was studied at low Reynolds numbers (Re < 0.5). In contrast to the usual velocity distribution of a Newtonian fluid, the viscoelastic fluid showed the following characteristic features: (1) a pronounced axial velocity overshoot immediately after the slit entrance and a maximum before the slit exit; (2) appearance of an axial flow deceleration region just before the sharp acceleration near the slit entrance. Even more remarkably, a saddle form of velocity profile was found in the entrance region. This flow pattern is completely different from that found for Newtonian fluids and has not yet been explained using existing rheological analysis.Parts of this paper were presented at the IX. Intern. Congress on Rheology at Acapulco (Mexico), October 8–13, 1984     

Stability of plane-parallel vibrational flow in a two-layer system

The stability of the interface separating two immiscible incompressible fluids of different densities and viscosities is considered in the case of fluids filling a cavity which performs horizontal harmonic oscillations. There exists a simple basic state which corresponds to the unperturbed interface and plane-parallel unsteady counter flows; the properties of this state are examined. A linear stability problem for the interface is formulated and solved for both (a) inviscid and (b) viscous fluids. A transformation is found which reduces the linear stability problem under the inviscid approximation to the Mathieu equation. The parametric resonant regions of instability associated with the intensification of capillary-gravity waves at the interface are examined and the results are compared to those found in the viscous case in a fully numerical investigation.     

GPU accelerated simulations of three-dimensional flow of power-law fluids in a driven cube

Newtonian fluid flow in two- and three-dimensional cavities with a moving wall has been studied extensively in a number of previous works. However, relatively a fewer number of studies have considered the motion of non-Newtonian fluids such as shear thinning and shear thickening power law fluids. In this paper, we have simulated the three-dimensional, non-Newtonian flow of a power law fluid in a cubic cavity driven by shear from the top wall. We have used an in-house developed fractional step code, implemented on a Graphics Processor Unit. Three Reynolds numbers have been studied with power law index set to 0.5, 1.0 and 1.5. The flow patterns, viscosity distributions and velocity profiles are presented for Reynolds numbers of 100, 400 and 1000. All three Reynolds numbers are found to yield steady state flows. Tabulated values of velocity are given for the nine cases studied, including the Newtonian cases.     

Derivation of the Forchheimer Law via Homogenization

In this paper we derive the Forchheimer law via the theory of homogenization. In particular, we study the nonlinear correction to Darcy's law due to inertial effects on the flow of a Newtonian fluid in rigid porous media. A general formula for this correction term is derived directly from the Navier–Stokes equation via homogenization. Unlike other studies based on the same approach that concluded for the nonlinear correction to be cubic in velocity for isotropic media, the present work shows that the nonlinear correction is quadratic. An example is constructed to illustrate our theory. In this example, the analytic solution to the Navier–Stokes equation is obtained and is utilized to show the validity of the quadratic correction. Both incompressible and compressible fluids are considered.