Evaluation of drag correction factor for spheres settling in associative polymers

Drag correction factors are calculated for the creeping motion of spheres descending in various associative polymers of different concentration with various sphere-container ratios and Weissenberg numbers. The simple-shear rheology and linear viscoelasticity of these polymeric fluids have been previously presented and modeled with the BMP (Bautista–Manero–Puig) equation of state (Mendoza-Fuentes et al., Phys Fluids 21:033104, 2009). The drag on the sphere is initially kept nearly constant for small Weissenberg numbers, We < 0.1. As the Weissenberg number increases, We < 0.1, a reduction in drag is found. Experimental results show the presence of a critical Weissenberg number at which a drag reduction occurs. The reduction in the drag correction factor is associated to the onset of extension-thinning, which coincides with the formation of a negative wake. No increase in the drag correction factor was observed, due to the simultaneous opposing effects of extension-thickening and shear-thinning viscosity. The shape of the drag correction factor curve may be predicted considering the extensional properties of the solutions, as suggested elsewhere (Chen and Rothstein, J Non-Newton Fluid Mech 116:205–215, 2004).     

Computational analysis of techniques to determine extensional viscosity from entrance flows

Recent computational analysis of entrance flows (Mitsoulis et al. 1998) suggests that the entrance pressure drop is insensitive to large changes in steady extensional viscosity-a result that directly contradicts a large body of experimental work in this area. A re-examination of entrance flows using numerical simulations is presented in this work which shows that entrance pressure drops do depend on the steady extensional viscosity, provided the extension rate in the entrance flow is large enough. Numerical simulations are presented using both the strain thinning and thickening versions of the Phan-Thien–Tanner (PTT) constitutive model. Several techniques for extracting extensional viscosity from entrance pressure are applied to the results of these simulations. The resulting predictions of extensional viscosity are compared to the steady extensional viscosity curves predicted by the PTT constitutive model used to generate the simulated pressure drop curves. The analytical techniques examined here are shown to provide reasonably accurate estimates of the steady extensional viscosity. This work also clearly demonstrates the advantage of using variable power-law coefficients for the rheological properties, used as inputs to the analyses, to capture the extensional behavior at deformation rates below the power law region more accurately. Received: 23 July 1999/Accepted: 24 November 1999     

Experimental studies on drag reduction and rheology of mixed cationic surfactants with different alkyl chain lengths

Experimental studies of the effects of mixtures of cationic surfactants on their drag reduction and rheological behaviors are reported. Cationic alkyl trimethyl quaternary ammonium surfactants with alkyl chain lengths of C₁₂ and C₂₂ were mixed at different molar ratios (total surfactant concentrations were kept at 5 mM with 12.5 mM sodium salicylate (NaSal) as counterion). Drag reduction tests showed that by adding 10% (mol) of C₁₂, the effective drag reduction range expanded to 4–120 °C, compared with 80–130 °C with only the C₂₂ surfactant. Thus mixing cationic surfactants with different alkyl chain lengths is an effective way of tuning the drag reduction temperature range. Cryo-TEM micrographs revealed thread-like micellar networks for surfactant solutions in the drag reducing temperature range, while vesicles were the dominant microstructures at non-drag reducing temperatures. High extensional viscosity was the main rheological feature for all solutions except 50% C₁₂ (mol) solution, which also does not show strong viscoelasticity. It is not clear why this low extensional viscosity solution with relatively weak viscoelasticity is a good drag reducer. Received: 3 November 1999/Accepted: 5 January 2000     

The modified imbedded disc retraction method for measurement of interfacial tension in polymer melts

The imbedded disc retraction method is used to estimate interfacial tension in LLDPE/PS system with PS as the imbedded disc. Shape evolution of a disc of one material (PS) imbedded into the matrix of another material is observed (LLDPE). Three to five repetitions at three different temperature levels are observed. The Newtonian model of Rundqvist et al. (1996) for the imbedded disc retraction is modified to include elastic effects. The modified model is derived assuming uniaxial extension, starting with the lower convected Maxwell model. Both the original model and modified imbedded disc retraction model are used in data analysis. The mean values of interfacial tension at 190 °C, 200 °C, and 210 °C are 6.8 ± 0.7 mN/m, 3.9 ± 0.3 mN/m, and 3.7 ± 0.2 mN/m, respectively. A method of estimating whether elastic effects will significantly affect the estimated interfacial tension value during retraction for the given polymer pair is provided. Received: 6 August 1999 Accepted: 2 January 2001     

A new particle tracking algorithm based on deterministic annealing and alternative distance measures

We describe a new particle tracking algorithm for the interrogation of double frame single exposure data, which is obtained with particle image velocimetry. The new procedure is based on an algorithm which has recently been proposed by Gold et al. (Gold et al., 1998) for solving point matching problems in statistical pattern recognition. For a given interrogation window, the algorithm simultaneously extracts: (i) the correct correspondences between particles in both frames and (ii) an estimate of the local flow-field parameters. Contrary to previous methods, the algorithm determines not only the local velocity, but other local components of the flow field, for example rotation and shear. This makes the new interrogation method superior to standard methods in particular in regions with high velocity gradients (e.g. vortices or shear flows). We perform benchmarks with three standard particle image velocimetry (PIV) and particle tracking velocimetry (PTV) methods: cross-correlation, nearest neighbour search, and image relaxation. We show that the new algorithm requires less particles per interrogation window than cross-correlation and allows for much higher particle densities than the other PTV methods. Consequently, one may obtain the velocity field at high spatial resolution even in regions of very fast flows. Finally, we find that the new algorithm is more robust against out-of-plane noise than previously proposed methods. Received: 1 March 1999 / Accepted: 29 July 1999     

Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow

An algorithm is derived for calculating flow-induced birefringence using a bead-spring model with and without excluded volume effects. The simulation results for the bead-spring model compare well with experimental results for stress and birefringence in extensional flows of dilute solutions of polystyrene molecular weight 2 million in a filament-stretching device in both “theta” and “good” solvents (Orr and Sridhar 1999; Sridhar et al. 2000). In a “good” solvent, both stress and birefringence rise much more rapidly with strain than in a “theta” solvent, making extensional rheology a very sensitive indicator of solvent quality. Received: 7 December 1999 Accepted: 23 May 2000     

Relaxation time spectra of star polymers

 The linear viscoelastic data for model star polymer melts with varying functionality and arm molecular weight were represented by means of a modified Baumgaertel-Schausberger-Winter (BSW) relaxation time spectrum, based on data analysis with the parsimonius model of Baumgaertel et al., reported in 1990. In the case of high arm functionality, the second slow terminal relaxation observed by Vlassopoulos et al. in 1997, was captured with a straightforward extension of the BSW model using broad cut-off functions. This study represents a potentially promising attempt to extend the applicability of this representation of viscoelastic data to more complex architectures, beyond simple linear chains which are characterized by self-similarity. The casting of linear viscoelastic data into spectra allows the exploration of star polymer behavior. It is a necessary step in preparation for large scale complex flow calculations in conjunction with constitutive models and for material databases. Received: 18 November 1998/Accepted: 12 August 1999     

Steady shear rheology of dilute polystyrene particle gels

This paper describes an experimental study on dispersions of monodisperse polystyrene (PS) spheres with a typical radius of 1 μm, dispersed in an electrolyte at high ionic strength, screening the electrostatic repulsion. These suspensions gelate at rest even at low volume fractions of PS particles. The density of the particles is matched with the solvent by using deuterium oxide for volume fractions φ≤0.117. Steady-state flow curves, viscosity as a function of shear rate, are measured and reported for 0.014<φ<0.322. The measured flow curves are analyzed on the basis of two models: 1. In the giant floc model (van Diemen and Stein 1983, 1984; Schreuder et al. 1986, 1987; Laven et al. 1988), at low shear rates, the shear is not distributed homogeneously but is limited to certain shear planes; the energy dissipation during steady flow is due primarily to overcoming the viscous drag on the suspended particles during motion caused by encounters of particles in the shear planes. Though this model was developed for higher solid volume fractions (0.35–0.425), we found that it also describes the rheology of dilute particle gels for 0.15≤φ≤0.3, using the same values for the parameters in the model as in the high solid volume fraction region. For φ<0.15, the model also describes the data if the fraction of distance by which a moving particle entrains its neighbors, is assumed to increase in this φ region. 2. The model of de Rooij (de Rooij et al. 1993, 1994) considers aggregates in shear flow to be monodisperse impermeable spheres with a fractal structure. The permeability is taken into account by considering a hydrodynamic radius smaller than the gyration radius in the Krieger-Dougherty expression for the hydrodynamic contribution to the viscosity. Through the use of a yield criterion the aggregate radius is modeled as a function of shear rate. We found that the model describes our experimental results, with a combination of parameter values used already by de Rooij, but only for φ<0.15. Received: 7 May 1998 Accepted: 22 December 1998     

Investigation of the flow around a circular cylinder under the influence of an electromagnetic force

An investigation of the effect of the local electromagnetic body force on the flow behavior around a circular cylinder is conducted. The electromagnetic force is applied locally on the cylinder surface in the range of 70–130° from the stagnation point along the cylinder circumference in both clockwise and counterclockwise directions. The numerical results predict that the Lorentz force applied in the circumferential direction on the cylinder moves the separation point rearward, and reduces the drag. To validate the numerical results, an experiment is conducted with a circular cylinder of 5 cm diameter. The electrodes and permanent magnets are flush mounted on the cylinder in such a way that the Lorentz force is generated in the circumferential direction. Flow visualization with polystyrene particles and direct drag measurement using strain gages are made. The fluid used is natural sea water of electric conductivity of about 4 (Ω m)^-1. Induction effect can be neglected in the present investigation due to the low flow speed and the Lorentz force is proportional to E×B where E is an applied electric field and B is a magnetic field. Received: 7 June 1998/Accepted: 28 April 1999     

Dynamics of shear aligning of nematic liquid crystal monodomains

The equations of linear and angular momentum for nematic liquid crystals have been described with Ericksen's transversely isotropic fluid [TIF] model and solved for start-up of shear flow at constant rate and varying initial alignment conditions. An analytical solution for the rotation provides predictions of the nematic director which closely agree with experimental results of Boudreau et al. (1999), supporting the validity of Ericksen's TIF model. The solution is limited to flows where the effects of director gradients are negligible. Received: 13 September 1999/Accepted: 24 January 2000     

Squeezing flow of semi liquid foods between parallel Teflon coated plates

Samples of commercial tomato paste, low fat mayonnaise and mustard about 6–8 mm thick were squeezed to 0.8 mm at various speeds between 5 mm min⁻¹ and 25 mm min⁻¹ between Teflon-coated parallel plates 127 mm in diameter using an Instron UTM Model 5542. All the log force vs log height relationships had a clearly identified linear region. This indicated that a dominant squeezing flow regime was achieved at about 3 mm height, and that the machine has the proper stiffness to perform the tests. The stress level at a pre-selected height in this region is a measure of consistency, sensitive enough to distinguish between products of different brands. The residual stress after relaxation for about 2 min was on order of 10–50% of the initial stress, an indication that all three foods have a considerable structural integrity. In all three products there was a considerable discrepancy between the observed rate effects and predictions based on a pseudoplastic (power law) model. It could be described by the empirical relation (F_v1 − F_R)/(F_v2 − F_R)=(V₁/V₂)^m where F_v1 and F_v2 are the forces at the given displacement reached at speeds v₁ and v₂ respectively, F_R is the residual force after relaxation (found to be practically rate independent), and m is a constant of the order of 0.15–0.33, independent of the compression velocities ratio but characteristic of the food and brand. The calculated elongational viscosity was not a unique function of biaxial strain rate. To a certain extent, this was probably due to imperfect lubrication. But it was also a manifestation of these products considerable structural integrity which cannot be accounted for by models developed for ideal liquids. Received: 1 November 1999 Accepted: 2 May 2000     

Squeeze flow between plane and spherical surfaces

Experiments are described in which a constant force F squeezed a fluid, either between two parallel circular plates, or between a plate and convex spherical lens. Newtonian fluids obeyed the relation of Stefan (1874) for plates, and the relation of Adams et al. (1994) for plate and lens. The non-Newtonian yield stress fluids Brylcreem, Laponite and Sephadex were squeezed between plates of various diameter D to attain a stationary separation h. Only for separations greater than h ^* (which depended on the fluid) did Brylcreem and Laponite obey the relation F/D ³ ∝ h ⁻¹ of Scott (1931) and give a yield stress in agreement with the vane method. For Sephadex the dependence of F/D ³ on h disagreed with Scott's relation, but varied as h ^−5/2 for h > 0.6 mm and h ^−3/2 for h < 0.6 mm. On rotating one plate in its plane the yield stress fluids at a fixed F suffered a marked decrease of h. This, and the existence of h ^*, are discussed in terms of the soft glassy material model of Sollich et al. (1997) and Sollich (1998). Brylcreem and Laponite were squeezed between a plate and lenses of various curvature and their yield stress obtained using the relation of Adams et al. (1994) was compared with measurements by plate-plate squeeze-flow and vane methods. Received: 12 April 2000 Accepted: 26 October 2000     

Viscoelastic flow past confined objects using a micro–macro approach

This paper is concerned with the numerical prediction of viscoelastic flow past a cylinder in a channel and a sphere in a cylinder using molecular-based models. The basis of the numerical method employed is a micro–macro model in which the polymer dynamics is described by the evolution of an ensemble of Brownian configuration fields. The spectral element method is used to discretize the equations in space. Comparisons are made between the macroscopic simulations based on the Oldroyd B constitutive model and microscopic simulations based on Hookean dumbbells, and excellent agreement is found. The micro–macro approach can be used to simulate models, such as the finitely extensible nonlinear elastic (FENE) dumbbell model, which do not possess a closed-form constitutive equation. Numerical simulations are performed for the FENE model. The influence of the model parameters on the flow is described and, in particular, the dependence of the drag as a function of the Weissenberg number.     

Experimental study of flow past a square cylinder at high Reynolds numbers

 Measurements of two-components of velocity in the wake of a square cylinder using a hot-wire anemometer are reported. Two Reynolds numbers, namely 8700 and 17,625, have been considered. The measurements were carried out in a low-speed, low-turbulence wind tunnel. Benchmark experiments at much lower Reynolds numbers show good agreement between the present experiments and published results. At higher Reynolds numbers, the experimental data reveal anticipated trends in terms of wake recovery and turbulence decay. Both velocity and velocity fluctuations show symmetry about the wake axis. The experimental data have been compared with the large eddy simulation (LES) calculation reported by Wang et al. [University of Illinois at Urbana – Champaign (1996) Report CFD 96-03] and LDV measurements of Lyn et al. [J Fluid Mech (1995) 304: 285–319]. The agreement among the three sets is generally acceptable in terms of the time-averaged velocity components, but not the velocity fluctuations. The turbulence fluctuations in the present experiments are seen to be lower than in the referred work. The differences have been traced to factors such as the aspect ratio, blockage ratio and upstream turbulence. Experiments with increased upstream turbulence did show a reduction in the discrepancy between the present experiments and the published data. An assessment of the experimental data in terms of physical mechanisms revealed that (a) streamwise normal stresses were correlated with the vortex centers, and (b) the turbulence kinetic energy profiles are similar to the turbulence shear stress. Spectral analysis of the velocity signals was carried out in the present work. Energy transfer from the mean flow to the streamwise velocity fluctuation was confirmed in the near wake. A redistribution of the kinetic energy between the streamwise and transverse components of velocity over a longer distance downstream was subsequently observed. Received: 17 May 1999/Accepted: 29 December 1999     

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。     

基于涡粘性模型的翼型湍流流场熵产率计算

利用有限体积法实现了基于非正交同位网格的SIMPLE算法。基于熵分析方法,采用涡粘性模型求解湍流熵产方程,系统研究了湍流模型对二维翼型绕流流场熵产率的影响。通过计算NACA0012翼型在来流雷诺数为2.88×10⁶时,0°攻角~16.5°攻角范围内的翼型表面压力系数分布和升阻力特性,验证了算法及程序的正确性。结果表明,选择不同湍流模型时,翼型流场熵产的计算结果存在差异,湍流耗散是引起流场熵产的主要原因;翼型流场的熵产主要发生在翼型前缘区、壁面边界层和翼型尾流区域,流场熵产率与翼型阻力系数线性相关;当产生分离涡时,粘性耗散引起的熵产下降。     

Lift force on rotating sphere at low Reynolds numbers and high rotational speeds

The lift force on an isolated rotating sphere in a uniform flow was investigated by means of a three-dimensional numerical simulation for low Reynolds numbers (based on the sphere diameter) (Re&lt;68.4) and high dimensionless rotational speeds (Г5). The Navier-Stokes equations in Cartesian coordinate system were solved using a finite volume formulation based on SIMPLE procedure. The accuracy of the numerical simulation was tested through a comparison with available theoretical, numerical and experimental results at low Reynolds numbers, and it was found that they were in close agreement under the above mentioned ranges of the Reynolds number and rotational speed. From a detailed computation of the flow field around a rotational sphere in extended ranges of the Reynolds number and rotational speed, the results show that, with increasing the rotational speed or decreasing the Reynolds number, the lift coefficient increases. An empirical equation more accurate than those obtained by previous studies was obtained to describe both effects of the rotational speed and Reynolds number on the lift force on a sphere. It was found in calcttlations that the drag coefficient is not significantly affected by the rotation of the sphere. The ratio of the lift force to the drag force, both of which act on a sphere in a uniform flow at the same time, was investigated. For a small spherical particle such as one of about 100μm in diameter, even if the rotational speed reaches about 10^6 revolutions per minute, the lift force can be neglected as compared with the drag force.     

Three-dimensional viscoelastic simulation of coextrusion process: comparison with experimental data

 Three-dimensional numerical simulation of viscoelastic coextrusion process has been performed and numerical results were compared with the experimental data of Karagiannis et al. (1990). By varying the magnitude of the second normal stress difference and its ratio of Fluid I and Fluid II, we were able to control the interface profile and the degree of encapsulation along the downstream direction. By increasing the parameter α (α_Fluid I=α_Fluid II) from 0.1 to 0.4 in the Giesekus model and increasing the α ratio (α_Fluid Iα_Fluid II) between Fluid I and Fluid II from 2.0 to 4.0 in the permissible range of realistic polymeric systems, the interface profile and the degree of encapsulation along the downstream direction were fitted with the experimental results. There was little difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction when the α ratio was set to 3.0 (0.3:0.1). Fluid I with larger magnitude of the second normal stress difference protrudes into Fluid II with smaller magnitude of the second normal stress difference around the symmetric plane, while Fluid II wraps around Fluid I near the side walls. As the ξ ₁ ratio (ξ _{1 ,Fluid I}ξ _{1 ,Fluid II}) increases from 1.0 to 3.0 for the two-mode Phan-Thien and Tanner model, it was found that the curvature of the interface profile increased, and the difference between the numerical results and the experimental data in the interface profile and the degree of encapsulation along the downstream direction was almost negligible when the ξ ₁ ratio was set to 3.0 (0.54:0.18). Although the parameters of viscoelastic models were fitted by using the shear viscosity data only, quantitative agreements between the numerical results and the experimental coextrusion data were quite satisfactory. Received: 24 April 2001 Accepted: 5 June 2001     

Experiments with three-dimensional riblets as an idealized model of shark skin

The skin of fast sharks exhibits a rather intriguing three-dimensional rib pattern. Therefore, the question arises whether or not such three-dimensional riblet surfaces may produce an equivalent or even higher drag reduction than straight two-dimensional riblets. Previously, the latter have been shown to reduce turbulent wall shear stress by up to 10%. Hence, the drag reduction by three-dimensional riblet surfaces is investigated experimentally. Our idealized 3D-surface consists of sharp-edged fin-shaped elements arranged in an interlocking array. The turbulent wall shear stress on this surface is measured using direct force balances. In a first attempt, wind tunnel experiments with about 365,000 tiny fin elements per test surface have been carried out. Due to the complexity of the surface manufacturing process, a comprehensive parametric study was not possible. These initial wind tunnel data, however, hinted at an appreciable drag reduction. Subsequently, in order to have a better judgement on the potential of these 3D-surfaces, oil channel experiments are carried out. In our new oil channel, the geometrical dimensions of the fins can be magnified 10 times in size as compared to the initial wind tunnel experiments, i.e., from typically 0.5 mm to 5 mm. For these latter oil channel experiments, novel test plates with variable fin configuration have been manufactured, with 1,920–4,000 fins. This enhanced variability permits measurements with a comparatively large parameter range. As a result of our measurements, it can be concluded, that 3D-riblet surfaces do indeed produce an appreciable drag reduction. We found as much as 7.3% decreased turbulent shear stress, as compared to a smooth reference plate. However, in direct comparison with 2D riblets, the performance of 3D-riblets is still inferior by about 1.7%. On the other hand, it appears conceivable, with a careful design of the fin shape (possibly supported by theory), that this inferiority in performance might be reduced. Nevertheless, at present, it seems to be rather unlikely, that 3D-riblets can significantly outperform 2D-riblets. Finally, one interesting finding remains to be mentioned: The optimum drag reduction for short 3D-riblets occurs at a lower rib height than for longer 3D-riblets or for infinitely long 2D-riblets. The same observation had been made previously on shark scales of different species with differing rib lengths, but no explanation could be given. Received: 1 March 1999/Accepted: 16 July 1999     

Shear-banding structure orientated in the vorticity direction observed for equimolar micellar solution

 The non-monotonic shear flow of a viscoelastic equimolar aqueous surfactant solution (cetylpyridinium chloride-sodium salicylate) is investigated rheologically and optically in a transparent strain-controlled Taylor Couette flow cell. As reported before, this particular wormlike micellar solution exhibits first a shear thinning and then a pronounced shear-thickening behavior. Once this shear-thickening regime is reached, a transient phase separation/shear banding of the solution into turbid and clear ring-like patterns orientated perpendicular to the vorticity axis, i.e., stacked like pancakes, is observed (Wheeler et al. 1998; Fischer 2000). The solution exhibit several unique features as no induction period of the shear induced phase, no structural build-up at the inner rotating cylinder, jumping pancake structure of clear and turbid ringlike phases, and oscillating shear stresses appear once the pancake structure is present. According to our analysis this flow phenomenon is not purely a mechanical or rheological driven hydrodynamic instability but one has to take into account structural changes of the oriented micellar aggregates (flow induced non-equilibrium phase transition) as proposed by several authors. Although this particular flow behavior and the underlying mixture of shear induced phases and mechanical instabilities is not fully understood yet, some classification characteristics based on a recent theoretical approach by Schmitt et al. (1995) and Porte et al. (1997) where a strong coupling between the flow instability (non-homogeneous flow profile due to the bands) and the structural changes causes the observed transient phenomena can be derived. In reference to the presented model the observed orientation of the rings is typical for complex fluids that undergo a spinodal phase separation coupled with a thermodynamic flow instability. In contrast to other shear banding phenomena, this one is observed in parallel plate, cone-plate, and Couette flow cell as well as under controlled stress and controlled rate conditions. Therefore, it adds an additional aspect to the present discussion on shear banding phenomena, i.e., the coupling of hydrodynamics and phase transition of rheological complex fluids. Received: 8 January 2001 Accepted: 15 May 2001