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Electro-elastic instabilities in electro-osmotic contraction/expansion flows (EOF) of viscoelastic fluids are experimentally investigated in this work. Several microchannel configurations are used, including the cases with hyperbolic-shaped contractions followed by an abrupt expansion, or abrupt contractions followed by a hyperbolic-shaped expansion. Such cases were selected to assist in the understanding of the EOF instability mechanisms and at which conditions they occur. A reference Newtonian fluid was also used in the experiments, and a wide range of electric fields were imposed to drive the flow. Flow visualizations for a Newtonian fluid allowed to assess the impact of dielectrophoresis on the velocity field of the seeding particles. For each geometry, depending on the polymer concentration, the flow was found to be quasi-Newtonian below a critical electric potential difference, characterized by smooth parallel steady flow patterns in the upstream and downstream channels. Above the critical voltage difference two types of electro-elastic instabilities were found to occur, leading to an unsteady symmetric flow, and to time-dependent irregular flow.

Three local self-similar solutions are obtained for the laminar planar jet flow of viscoelastic fluids, described by the FENE-P constitutive equation, through an order of magnitude simplification of the governing equations. The more general solution is shown to be more accurate than two further simplified solutions, here called the delta and the Olagunju-type solutions, at least for the profiles of conformation tensor components. In the limit of vanishing viscoelasticity, all conformation tensor components reduce to the same low elasticity asymptotic behavior, and the polymer stresses become Newtonian-like. The general solution is then used to obtain the laws of the decay of the centerline velocity and of the growth of the jet half-width and to ascertain the effects of the Weissenberg number, maximum polymer extensibility parameter and the ratio of polymer to total viscosities upon the flow characteristics. The general local self-similar solution compares well with results of numerical simulations obtained by the RheoFoam module of the freeware OpenFoam code.

This work presents a numerical study of the 4:1 planar contraction flow of a viscoelastic fluid described by the simplified Phan-Thien–Tanner model under the influence of slip boundary conditions at the channel walls. The linear Navier slip law was considered with the dimensionless slip coefficient varying in the range ½0; 4500. The simulations were carried out for a small constant Reynolds number of 0.04 and Deborah numbers (De) varying between 0 and 5. Convergence could not be achieved for higher values of the Deborah number, especially for large values of the slip coefficient, due to the large stress gradients near the singularity of the reentrant corner. Increasing the slip coefficient leads to the formation of two vortices, a corner and a lip vortex. The lip vortex grows with increasing slip until it absorbs the corner vortex, creating a single large vortex that continues to increase in size and intensity. In the range De = 3–5 no lip vortex was formed. The flow is characterized in detail for De ¼ 1 as function of the slip coefficient, while for the remaining De only the main features are shown for specific values of the slip coefficient., The authors gratefully acknowledge funding by COMPETE, FEDER and Fundacao para a Ciencia e a Tecnologia (FCT) through projects PEst-C/CTM/LA0025/2013 (Strategic Project - LA 25 - 2013-2014, PTDC/EME-MFE/113988/2009 and PTDC/EME-MFE/114322/2009. AMA would also like to thank FCT for the financial support through the scholarship SFRH/BPD/75436/2010.

A tensorially consistent near-wall four equation model is developed to model turbulent flow of dilute polymer solutions. The model is validated up to the maximum drag reduction limit, by utilizing the data obtained from direct numerical simulations using the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive model. Eight sets of direct numerical simulation (DNS) data are used to analyze budgets of relevant physical quantities, such as the nonlinear terms in the FENE-P constitutive equation, the turbulent kinetic energy, the wall normal Reynolds stress and dissipation transport. Closures were developed in the framework of the k - e - v 2 ¯ - f model for the viscoelastic stress work, the viscoelastic destruction of the rate of dissipation, the viscoelastic turbulent viscosity, and the interactions between the fluctuating components of the conformation tensor and of the velocity gradient tensor terms. Predicted polymer stress, velocity profiles and turbulent flow characteristics are all in good agreement with the literature, from which six independent DNS data sets were used covering a wide range of rheological and flow parameters, including high Reynolds number flows, and showing significant improvements over the corresponding predictions of other existing models.

This work presents a finite volume method used to solve the relevant coupled equations for general electro-osmotic flows of viscoelastic fluids, using the upper-convected Maxwell and the simplified Phan-Thien–Tanner models. Three different implementations of the electro-osmosis physical models were carried out, which depend on the required level of approximation. In the first implementation, the Poisson–Nernst–Planck equations were incorporated into the code and the electric charge distribution required to quantify the electric field forcing of the momentum equation is calculated from these fundamental equations. The second implementation is an approximation in which a stable Boltzmann distribution of ions is assumed to occur in the electric double layer, represented by the Poisson–Boltzmann equations. Finally, the so-called Debye–Huckel approximation was also implemented in the Poisson–Boltzmann–Debye–Huckel model, which is valid for cases with a stable Boltzmann distribution of ions characterized by a small ratio of electrical to thermal energies. Numerical simulations were undertaken in a cross-slot geometry, to investigate the possible appearance of purely electro-elastic flow instabilities, by considering the effect of the electric field. For pure electro-osmotic flow, i.e., in the absence of an imposed pressure gradient, we were able to capture the onset of an unsteady asymmetric flow above a critical Weissenberg number, which is lower than the corresponding value for pressure-driven viscoelastic flow. Additionally, we derive analytically the fully-developed electro-osmosis driven channel flow of polymer solutions described by the FENE-P and PTT models with a Newtonian solvent and assuming a thin electric double layer.

A new low-Reynolds-number k–e turbulence model is developed for flows of viscoelastic fluids described by the finitely extensible nonlinear elastic rheological constitutive equation with Peterlin approximation (FENE-P model). The model is validated against direct numerical simulations in the low and intermediate drag reduction (DR) regimes (DR up to 50%). The results obtained represent an improvement over the low DR model of Pinho et al. (2008) [A low Reynolds number k–e turbulence model for FENE-P viscoelastic fluids, Journal of Non-Newtonian Fluid Mechanics, 154, 89–108]. In extending the range of application to higher values of drag reduction, three main improvements were incorporated: a modified eddy viscosity closure, the inclusion of direct viscoelastic contributions into the transport equations for turbulent kinetic energy (k) and its dissipation rate, and a new closure for the cross-correlations between the fluctuating components of the polymer conformation and rate of strain tensors (NLTij). The NLTij appears in the Reynolds-averaged evolution equation for the conformation tensor (RACE), which is required to calculate the average polymer stress, and in the viscoelastic stress work in the transport equation of k. It is shown that the predictions of mean velocity, turbulent kinetic energy, its rate of dissipation by the Newtonian solvent, conformation tensor and polymer and Reynolds shear stresses are improved compared to those obtained from the earlier model.

In this work we propose a new type of microfluidic rectifier, which is able to operate efficiently under creeping flow conditions. The flow of Newtonian and non-Newtonian fluids was investigated experimentally in different microchannels with triangular (nozzle/diffuser) and hyperbolic shapes in order to achieve high anisotropic flow resistance between the two flow directions. The Newtonian fluid used was de-ionized water and the viscoelastic fluids were aqueous solutions of polyacrylamide and polyethylene oxide with different molecular weights. Pressure drop measurements were performed in addition to visualizations of the flow patterns by streak line photography for a wide range of flow rates. For the Newtonian flows, inertia leads to the appearance of recirculations for both flow directions, but no significant rectification effects appear. For the viscoelastic fluids, two distinct behaviors are identified: at low flow rates, the pressure drops are similar in both flow directions; above a critical flow rate (or Deborah number), the flow patterns become quite different, leading to different flow rates in the forward and backward flow directions for the same pressure drop, i.e., rectification effects emerge. In particular, the viscoelastic fluid flow becomes unsteady in the forward direction, due to the presence of elastic instabilities, which leads to a significant increase in the flow resistance. Flow resistance ratios greater than three were achieved for the hyperbolic rectifier, clearly in excess of the value for the triangular-shaped rectifier and for other geometries proposed in the literature for operation in creeping flow conditions. This high diodicity is associated with the distinct nature of the extensional flows in the forward and backward directions of the hyperbolic-type microgeometry.

This work deals with the development of a numerical technique for simulating three-dimensional viscoelastic free surface flows using the PTT (Phan-Thien–Tanner) nonlinear constitutive equation. In particular, we are interested in flows possessing moving free surfaces. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. The fluid is modelled by a Marker-and-Cell type method and an accurate representation of the fluid surface is employed. The full free surface stress conditions are considered. The PTT equation is solved by a high order method, which requires the calculation of the extra-stress tensor on the mesh contours. To validate the numerical technique developed in this work flow predictions for fully developed pipe flow are compared with an analytic solution from the literature. Then, results of complex free surface flows using the PTT equation such as the transient extrudate swell problem and a jet flowing onto a rigid plate are presented. An investigation of the effects of the parameters ɛ and ξ on the extrudate swell and jet buckling problems is reported.

A systematic numerical investigation on creeping flows in planar sudden expansions of viscoelastic fluids obeying the upper-convected Maxwell model is carried out to assess the combined effects of viscoelasticity, through the Deborah number, and expansion ratio (ER), which was varied between 1.25 and 32. At large expansion ratios (ER ≥ 4) the flow becomes dominated by the downstream duct size and appropriately normalized quantities tend to be independent of ER. The recirculation size and strength become decreasing functions of De, whereas the Couette correction (the normalized entry pressure drop due to the presence of the expansion) increases. At small ER (ER ≤ 3), however, no simple scaling laws are found and there is a complex interaction between De and ER leading to non-monotonic variations, with an initial decrease in the recirculation length at low Deborah numbers, followed by an enhancement as De increases.

The flow of a viscoelastic fluid through a microfluidic flow-focusing device is investigated numerically with a finite-volume code using the upper-convected Maxwell (UCM) and Phan-Thien–Tanner (PTT) models. The conceived device is shaped much like a conventional planar “cross-slot” except for comprising three inlets and one exit arm. Strong viscoelastic effects are observed as a consequence of the high deformation rates. In fact, purely elastic instabilities that are entirely absent in the corresponding Newtonian fluid flow are seen to occur as the Deborah number (De) is increased above a critical threshold. From two-dimensional numerical simulations we are able to distinguish two types of instability, one in which the flow becomes asymmetric but remains steady, and a subsequent instability at higher De in which the flow becomes unsteady, oscillating in time. For the UCM model, the effects of the geometric parameters of the device (e.g. the relative width of the entrance branches, WR) and of the ratio of inlet average velocities (VR) on the onset of asymmetry are systematically examined. We observe that for high velocity ratios, the critical Deborah number is independent of VR (e.g. Dec ≈ 0.33 for WR = 1), but depends non-monotonically on the relative width of the entrance branches. Using the PTT model we are able to demonstrate that the extensional viscosity and the corresponding very large stresses are decisive for the onset of the steady-flow asymmetry.

The log-conformation formulation, proposed by Fattal and Kupferman [J. Non-Newt. Fluid Mech. 123 (2004) 281], has helped to provide further insights into the High-Weissenberg Number Problem. In this work, we investigate the performance of the log-conformation formulation in the Finite Volume Method (FVM) framework for creeping flows of viscoelastic fluids in steady and unsteady flows around a confined cylinder. The Oldroyd-B and Phan-Thien-Tanner (PTT) constitutive equations were used to assess the effect of different theological behaviour on the flow patterns and solution stability. The calculation of the polymer stress contribution is carried out with both the standard technique and with the log-conformation methodology. For all test cases, up to the critical conditions when both methods converge to a steady solution, the use of the log-conformation technique provides solutions with similar accuracy as the standard approach. In terms of stability the log-conformation formulation is found to be significantly more robust, and Solutions could be obtained at higher Deborah number flows.

An analytical solution is derived for the time-dependent flow of an infinite pool of fluid described by the viscoelastic upper convected Maxwell (UCM) model driven by an oscillating porous plate in the presence of cross flow. Whereas for a Newtonian fluid there is a solution regardless of the amount of suction or blowing, for viscoelastic fluids the solution breaks down under certain conditions. When the suction velocity exceeds the elastic shear wave speed there is no solution. For sub-critical blowing through the plate the stream-wise velocity profiles are periodic, but when the blowing speed exceeds the elastic shear wave speed non-periodic chaotic-like waves appear under certain conditions, which are characterized. The flow characteristics are properly scaled by the reciprocal square root of the Reynolds number with and without cross flow. Generally speaking, the flow properties are controlled by the cross flow and by the fluid elasticity at low and high Deborah numbers, respectively.

When viscous dissipation effects are important in duct flows the Brinkman number is widely used to quantify the relationship between the heat generated by dissipation and the heat exchanged at the wall. For Newtonian laminar fully developed pipe flow the use of the classical expression for this dimensionless group is appropriate, but under different conditions it can lead to misleading conclusions, such as when comparing flows through different cross-section ducts, flow regimes and mainly non-Newtonian flows. In this work a generalized Brinkman number is proposed, based on an energy balance for the power dissipated by friction, that allows proper quantification of viscous heating effects and reduces to the classical definition in laminar Newtonian pipe flow. The advantages of the new definition are shown and expressions are given for generalized Brinkman numbers in the most common cases.

A Reynolds-averaged Navier–Stokes equation framework is developed for modeling polymer-induced turbulent drag reduction by utilizing the finitely extensible nonlinear elastic-Peterlin (FENE-P) constitutive relationship to describe fluid rheology. A self-consistent system of model equations is derived. The dominant correlations among the flow and polymer conformation variables are identified by analyzing results from recent direct numerical simulations (DNS) for dilute polymer solutions. Closures are developed for turbulent correlations that arise from viscoelasticity and incorporated into a single-point k − ɛ model. Comparison of model predictions with DNS data obtained for an identical set of rheological and geometric parameters in the low drag reduction regime allows for a critical evaluation of the model and closure approximations. Overall, the model predictions are encouraging. This approach can be extended to higher order turbulence models and cases with higher levels of drag reduction.

We report a systematic numerical investigation of the creeping flow of three different viscoelastic models, the UCM, Oldroyd-B and the linear form of the PTT model, through a 1:3 planar sudden expansion. Although the effect of elasticity is to reduce both the length and intensity of the recirculation region downstream of the expansion, we show that this reduction is much lower than previous studies have suggested and that, at high Deborah number, a significant region of recirculation still exists for all of the models studied. © 2006 Elsevier B.V. All rights reserved.

Two solutions are presented for fully-developed pipe and planar flows of multimode viscoelastic models. The fluids have a Newtonian solvent contribution and the polymer modes are described by the Phan-Thien—Tanner (PTT), the FENE-P or the Giesekus equation. The first solution is exact and can handle any number of modes, but is only semi-analytical. The second solution, which is presented only for the PTT model with a linear stress coefficient and the FENE-P model, can also handle any number of modes. It is based on a truncated series expansion and is completely analytical, but provides only an approximated solution. The complexity of the multimode solutions is investigated first with the exact semi-analytical method and it is shown that at high Deborah number flows the high-order stresses can become as important as the stress of the first mode. It is also under these conditions that the approximated analytical solution deviates from the exact semi-analytical solution. A criterion for the accurate use of the approximated solution is presented. Fortran codes are provided to obtain these solutions at the internet address at the end. © 2006 Elsevier B.V. All rights reserved.

Large velocity overshoots with very strong velocity gradients appear near the side walls of a smooth contraction (contraction ratio 2.86:1) following a square duct and preceding a plane sudden expansion of large expansion ratio (D/d = 1.43), when viscoelastic fluids flow in the laminar regime, as observed experimentally by Poole et al. [R.J. Poole, M.P. Escudier, A. Afonso, F.T. Pinho, Laminar viscoelastic flow over a backward-facing step, in: Proceedings of the XIV International Congress on Rheology, Paper NF-06, Seoul, South Korea, 2004; R.J. Poole, M.P. Escudier, P.J. Oliveira, Laminar flow of a viscoelastic shear-thinning liquid through a plane sudden expansion preceded by a gradual contraction, Proc. R. Soc. London Series A 461 (2005) 3827–3845; R.J. Poole, M.P. Escudier, A. Afonso, F.T. Pinho, Laminar flow of a viscoelastic shear-thinning liquid over a backward-facing step preceded by a gradual contraction, Phys. Fluids, submitted for publication]. To help understand this phenomenon an extensive numerical investigation was undertaken in the five parameter space of a fluid described by the Phan-Thien–Tanner (PTT) constitutive equation with non-zero second normal-stress difference (N2 ≠ 0 due to ξ ≠ 0) and a solvent contribution. The numerical simulations were able to qualitatively capture the velocity overshoots on the XZ-centreline profiles, and observed also overshoots on the XY-mid-plane and the so far unreported enhanced velocity peak at the duct corners. For this particular geometry the overshoots took place for particular combinations of the independent parameters, in particular they required large Weissenberg numbers, large second normal-stress differences, strain-hardening of the extensional viscosity and intense shear-thinning, together with non-negligible inertia. The parametric investigation also reports the variations of the three-dimensional velocity profiles which progressed from being rather flat shaped at low elasticity and high Reynolds numbers (typical of purely viscous behaviour) towards a slim almost triangular shape at high elasticity numbers.

Analytical solutions are obtained for heat transfer in concentric annular flows of viscoelastic fluids modeled by the simplified Phan-Thien–Tanner constitutive equation. Solutions for thermal and dynamic fully developed flow are presented for both imposed constant wall heat fluxes and imposed constant wall temperatures, always taking into account viscous dissipation. Equations are presented for the normalized temperature profile, the bulk temperature, the inner and outer wall temperatures and, through their definitions for the inner and outer Nusselt numbers as a function of all relevant non-dimensional parameters. Some special results are discussed in detail. Given the complexity of the derived equations, for ease of use compact exact expressions are presented for the Nusselt numbers and programmes to calculate all quantities are made accessible on the internet. Generally speaking, fluid elasticity is found to increase the heat transfer for imposed heating at the wall, especially in combination with internal heat generation by viscous dissipation, whereas for imposed wall temperatures it reduces heat transfer when viscous dissipation is weak.

We present analytical solutions for fully developed pipe and channel flows of two viscoelastic fluids possessing a Newtonian solvent, where the polymer contribution is either described by the Phan-Thien–Tanner (PTT) or FENE-P models. We derive in detail the pipe flow solution for the PTT fluid, and present the final solutions for the remaining three cases. This constitutes an important addition to existing results where the presence of a solvent with Newtonian characteristics has been consistently overlooked, as it posed considerable difficulties to the task of obtaining a closed form solution. In addition, interesting aspects of the solutions are discussed. © 2005 Elsevier B.V. All rights reserved.

A model is proposed for the new stress term in the momentum equation of a modified generalised Newtonian fluid that is used to mimic viscoelastic effects of fluids exhibiting drag reduction in turbulent pipe flow. The new stress quantifies the cross-correlation between the fluctuating viscosity and the fluctuating rate of strain and had been neglected in the k-e low Reynolds number model originally developed by Cruz and Pinho [J. Non-Newt. Fluid Mechanics 114 (2003) 109]. With the inclusion of this new stress, the predictions of turbulent kinetic energy (k + ) in drag reducing pipe flow are significantly improved at the cost of a slight deterioration in the prediction of other quantities for some of the fluids. The value of the coefficient C in the original model of Cruz and Pinho was also modified to correct a mistake and this was shown to improve the predictions for all fluids except for the solution of 0.125% PAA used to calibrate the model. Comparison of predictions with DNS results for a FENE-P model showed large overprediction of drag reduction and emphasize the need for improvements in extensional rheometry. © 2004 Elsevier B.V. All rights reserved.

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (eDe 2 ) and the radius ratio. For axial dominated flows fRe and CM are found to depend only on eDe 2 and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers. © 2004 Elsevier B.V. All rights reserved.

Measurements of pressure on the cylinder surface for the flow of Newtonian and non-Newtonian fluids around a circular cylinder were carried out, from which several parameters were calculated: the form drag coefficient ( CD), the pressure rise coefficient ( Cpb − Cpm) and the wake angle (θw). The non-Newtonian fluids were aqueous solutions of CMC and tylose with varying degrees of shear-thinning and elasticity, at weight concentrations of 0.1–0.6% and the experiments encompassed the transition and shear-layer transition regimes. For low Reynolds numbers flows elasticity on the shear layers was responsible for an increase in drag reduction with polymer concentration. Within the shear-layer transition regime the increase of the wake angle and pressure rise coefficient for the more concentrated solutions reduced CD by narrowing the near wake. For the tylose solutions a good correlation was found between the elasticity number and the mean pressure rise coefficient. © 2004 Elsevier B.V. All rights reserved.

The solution of thermal entry problem for pipe and channel flow of highly viscous liquids having viscoelastic properties, here modeled by the FENE-P constitutive equation, is obtained semi-analytically. Two types of boundary condition are considered, either an imposed wall temperature or a prescribed wall heat flux, and the effects of viscous dissipation are also taken into account. The analysis leads to a Sturm-Liouville problem similar to that found for the simplified Phan-Thien–Tanner fluid with linear stress coefficient, provided an adequate transformation of constitutive variables is performed first. The way to do this is explained and representative results for thermal entry flow of the FENE-P fluid are given. © 2004 Elsevier B.V. All rights reserved.

Based on a generalised Newtonian fluid (GNF) model, modified to account for strain-thickening of the extensional viscosity, this paper derives transport equations for mass, momentum, Reynolds stresses, turbulent kinetic energy and its rate of dissipation. An analysis of order of magnitude identifies the relevant new terms and suggestions are made to model those terms needed to ensure closure in the perspective of a low Reynolds number k–e model. Specifically, a closed model for the time-average viscosity is proposed that takes into account its non-linearity and dependence on the second and third invariants of the fluctuating rate of deformation tensor. The turbulence model is qualitatively shown to increase the rate of decay of turbulent kinetic energy in isotropic grid turbulence for certain rheological conditions. The performance of the turbulence model in a pipe flow is assessed in a companion paper by Cruz and Pinho [J. Non-Newtonian Fluid Mech., in press]. © 2003 Elsevier B.V. All rights reserved.

A low Reynolds number k–e turbulence model is developed for predicting turbulent wall flows of viscoelastic fluids. The model uses a non-linear molecular viscosity that is affected by the turbulent fluctuations and a new damping function is introduced to account for near-wall effects. This new function was made equal to the eddy viscosity damping function which was derived taking into account viscometric and elastic effects. Flow predictions compare favourably with results from experiments with several viscoelastic fluids, especially the friction factor and the mean velocity. Comparisons of the turbulence kinetic energy are less good, but the model is able to capture the shift of the peak turbulence kinetic energy and rate of dissipation away from the wall, the decrease in those peak values and of the production of k, and the Reynolds shear stress deficit across the pipe. An advantage of the present single-point turbulence closure relative to previous attempts at modelling polymer drag reduction, is the fact that here the input is only the mean velocity and fluid properties and no other modifications of the turbulence model are required to deal with different fluids. Further developments of the turbulence model are suggested at the end.

A detailed experimental study on the flow characteristics of various vortex shedding regimes was carried out for the flow of non-Newtonian fluids around a cylinder. The fluids were aqueous solutions of carboxymethyl cellulose (CMC) and tylose at weight concentrations ranging from 0.1 to 0.6%, which had varying degrees of shear-thinning and elasticity. Two cylinders of 10 and 20 mm diameter were used in the experiments, defining an aspect ratio of 12 and 6 and producing blockages of 5 and 10%, respectively. The Reynolds number (Re) ranged from 50 to 9 × 10 3 . Shear-thinning gave rise to a decrease of the cylinder boundary-layer thickness and to a reduction of the diffusion length (ld), which raised the Strouhal number, St. In the laminar shedding regime, a modified Strouhal number was successful at overlapping the shedding frequency variation with the Reynolds number for the various solutions. In contrast, fluid elasticity was found to increase the formation length ( lf ), and this contributed to a decrease of the Strouhal number. The overall effect of shear-thinning and elasticity was an increase in the Strouhal number. The increase in polymer concentration and the corresponding increase in fluid elasticity were responsible for the reduction of the critical Reynolds number marking the sudden decrease of the formation length, Relf . In the shear layer transition regime, the formation length and Strouhal number data collapsed onto single curves as function of a Reynolds number difference, which confirmed Coelho and Pinho (J. Non-Newtonian Fluid Mech. (2003), accepted for publication) finding that an important effect of fluid rheology was in changing the demarcations of the various flow regimes. © 2003 Elsevier Science B.V. All rights reserved.

Analytical solutions are derived for the steady state channel and pipe flows of viscoelastic fluids obeying the complete single-mode Phan-Thien–Tanner (PTT) constitutive equation with a linear stress coefficient in the absence of a solvent viscosity contribution. The results include the profiles of all relevant stresses, the axial velocity and the viscosity across the gap. The three material functions of the single-mode PTT model in steady Couette flow are also given and it is shown that the conditions of the maximum point in the shear stress versus shear rate curve are related to the conditions for existence of steady state solutions in the channel and pipe flows. The range of model parameters for which a classical steady solution exists is established. © 2001 Elsevier Science B.V. All rights reserved.

An aqueous suspension of 1% by weight laponite was investigated in terms of its rheology and hydrodynamic behaviour in a sudden expansion flow. The fluid was shear-thinning, thixotropic and had an yield stress which was measured by direct and indirect methods. The oscillatory tests showed that the elasticity of the 1% Laponite suspension was very small. The high Reynolds number turbulent flow downstream of a sudden expansion, with fully-developed inlet conditions, showed no major difference in relation to the flow of water. There were no differences between the laponite and water mean and turbulent flow characteristics upstream and downstream of the expansion plane, except for a small anticipation of the loci of maximum Reynolds stresses with the suspension, but this had no further consequence. In conclusion, although the laponite suspension was thixotropic, shear-thinning and viscoplastic, its hydrodynamic behaviour in a sudden expansion was akin to that of water, a result which could not be anticipated.

Accurate solutions are obtained with the numerical method of Oliveira et al. [J. Non-Newtonian Fluid Mech. 79 (1998) 1] for the inertialess plane flow around a confined cylinder. This numerical procedure is based on the finite-volume method in non-orthogonal block-structured meshes with a collocated arrangement of the dependent variables, and makes use of a special interpolation practice to avoid stress‐velocity decoupling. Two high-resolution schemes (MINMOD and SMART) are implemented to represent the convective terms in the constitutive equations for the upper convected Maxwell and Oldroyd-B fluids, and the resulting predictions of the drag coefficient on the cylinder are shown to be as accurate as existing finite-element method predictions based on the supposedly very accurate h-p refinement technique. Numerical uncertainties are quantified with help of Richardson’s extrapolation technique and the orders of convergence of the differencing schemes are established and shown to be second-order accurate. Calculations performed with a wake-refined mesh predicted the variation of the maximum longitudinal normal stress in the wake as De 3 and De 5 depending on Deborah number. © 2001 Elsevier Science B.V. All rights reserved.

The independent analysis of flow measurements is frequently hampered by incomplete characterisation of the working fluid. This problem is particularly acute in situations which require working fluids with identical properties, such as the development of scaling laws for the turbulent flow of drag-reducing liquids. In this paper, we demonstrate that the viscometric viscosity, loss and storage moduli for two of the most common polymers used for flow experiments, carboxymethylcellulose (CMC) and xanthan gum (XG), are practically insensitive to the chemistry of the tap water used as a solvent, to the method of mixing, and to the biocide added. However, the properties of CMC from two different manufacturers were found to be significantly different, whereas there was no difference between XG solutions prepared from different batches from the same manufacturer. Our conclusion is that for a given concentration in water, the properties of certain non-Newtonian liquids, such as CMC and XG, are essentially fixed and reproducible. Although the situation is less than ideal, comparisons of fluid-flow data from entirely independent laboratories can thus be made even in the absence of direct rheological measurements.

An analytical solution is given for the kinematic and stress variations across the radial gap of a concentric annular flow in fully developed conditions. The fluid is viscoelastic and obeys the non-linear rheological constitutive equation proposed by Phan-Thien and Tanner [1]. This constitutive model simulates well the material functions of many polymer melts and solutions and therefore, the present results are useful in a number of practical situations. The ratio of pressure drop to flow rate drop is found to be a complex mathematical function of the radial position of zero shear stress and this, in turn, depends weakly on the elasticity, based on the product of an elongational parameter by a Deborah number defined with an averaged velocity. There is thus a non-linear coupling which could not be solved in an explicit way for the inverse problem of an imposed flow rate, but an iterative procedure gives a ready result. For the direct problem of a given pressure drop the present results represent an exact explicit solution to the axial annular flow problem. Representative profiles of the solution are given and discussed. It is found that, for a given flow rate, the pressure drop scaled with the corresponding Newtonian value is independent of the diameter ratio. © 2000 Elsevier Science B.V. All rights reserved.

Improved accuracy and enhanced convergence rate are achieved when a finite-volume method (FVM) is used in conjunction with a high-resolution scheme (MINMOD) to represent the stress derivatives in the constitutive equation, because it avoids oscillations of the solution field near sharp stress gradients. Calculations for the benchmark flow of an upper-convected Maxwell fluid through a 4:1 plane contraction were carried out at a constant Reynolds number of 0.01 and varying Deborah numbers in four consistently refined meshes, the finest of which had a normalised cell size of 0.005 in the vicinity of the re-entrant corner. The MINMOD scheme was able to provide converged solutions up to Deborah numbers well beyond those attained by other second-order accurate schemes. The asymptotic behaviour of velocity and stresses near the re-entrant corner was accurately predicted as compared with Hinch’s theory [1]. The simulations improved previous results for the same flow conditions obtained with less accurate schemes, and the present results can be used as benchmark values up to a Deborah value of 3 with quantified numerical uncertainties. © 2000 Elsevier Science B.V. All rights reserved.

Detailed mean velocity, normal Reynolds stress and pressure drop measurements were carried out with 0.4 to 0.6% by weight aqueous solutions of Tylose, a methylhydroxil cellulose (molecular weight 6000) from Hoechst after a selection process from a set of low molecular weight fluids. The viscosity measurements of the Tylose solutions showed shear-thinning behaviour, and the oscillatory and creep tests measured elastic components of the stress of the order of the minimal detectable values by the rheometer. These low molecular weight polymer solutions delayed transition from the laminar to the turbulent regime and showed drag reductions of half that reported to occur with other low elasticity shear-thinning high molecular aqueous polymer solutions. Near the wall the axial turbulent stress was higher than with water, whereas the two transverse components of turbulence were reduced. This near-wall behaviour is typical of drag reducing fluids based on high molecular weight polymers, but in the core of the pipe the three components of turbulence were higher than for the water flows, especially in the radial and tangential directions.

Measurements of mean axial velocity and of the three normal stresses have been obtained in fully developed pipe-flow with four concentrations of a polymer (sodium carboxymethyl cellulose) in aqueous solution and with water and viscous Newtonian fluids encompassing a range of Reynolds numbers from 240 to 111,000. The results quantify the delay in transition from laminar to turbulent flow caused by shear-thinning, the suppression of turbulent fluctuations particularly in the radial and tangential components of normal stress, and the drag reduction at the higher Reynolds numbers. They also confirm that the maximum drag reduction asymptote is appropriate to these shear-thinning solutions.

0377-0257/$ see front matter 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jnnfm.2013.09.003 DOI of original article: http://dx.doi.org/10.1016/j.jnnfm.2005.08.013 ⇑ Corresponding author. Tel.: +351 275329923; fax: +351 275329972. E-mail addresses: doac@mecanica.coppe.ufrj.br (D.O.A. Cruz), fpinho@fe.up.pt (F.T. Pinho), pjpo@ubi.pt (P.J. Oliveira). D.O.A. Cruz , F.T. Pinho , P.J. Oliveira c,⇑

Centro de Estudos de Fenomenos de Transporte, Faculdade de Engenharia, Universidade do Porto, rua Dr. Roberto Frias s/n, 4200-465 Porto, Portugal b School of Energy and Power Engineering, Jiangsu University, Zhenjiang, Jiangsu 212013, PR China Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, USA Department of Biomedical and Chemical Engineering, Department of Physics, Syracuse University, NY 13244, USA