Lip Vortex Research Articles

The importance of downstream events in microfluidic viscoelastic entry flows: Consequences of increasing the constriction length

Polymer solutions and melts can exhibit large upstream corner and lip vortices, unstable and diverging flow and an enhanced pressure drop when flowing through a geometry containing a constriction. In the present work, we use a planar microfluidic device to show that the length of the downstream constriction plays an important role in the upstream kinematics and the extra pressure drop. That is, the elastic flow phenomena observed upstream of a constriction during entry flows of polymer solutions are not exclusively a result of the stretching dynamics induced by the converging flow—the downstream relaxation events are, at least, equally important. Flow visualization experiments with semi-dilute solutions of a high molecular weight polymer showed that large stable symmetric vortices could be reduced to highly chaotic asymmetric flow, merely by increasing the length of the constriction—the Reynolds number and elasticity number were both held constant. This was accompanied by a higher extra pressure. These results support the hypothesis that elastic flow instabilities originate downstream of the constriction (at the expansion) and move progressively upstream with time and/or flowrate. These findings may also partly explain the discrepancies commonly observed between the results of entry flow experiments and numerical simulations, in which the downstream geometry is very rarely considered. Lastly, we illustrate how to minimize the occurrence of unstable flow upstream of a constriction, which is a necessary condition for closed microrheometry devices used to characterize low viscosity elastic fluids.

Modelling the three-dimensional flow of a semi-dilute polymer solution in microfluidics—on the effect of aspect ratio

The flow of polymer solutions in microfluidic devices is inherently three-dimensional, especially in the non-linear flow regime, and often results in flow phenomena that might not even be encountered in macro-devices. Using a multi-mode Phan-Thien–Tanner model, three-dimensional (3-D) simulations of a semi-dilute polyethylene oxide (PEO) solution through 8:1 planar contraction micro-channels with various depths have been carried out to systematically study the effect of the aspect ratio on the flow fields. Vortex dynamics in the upstream flow section and excess pressure drop are quantified in detail. A transition from a salient-corner vortex mechanism to a lip vortex mechanism is observed as the aspect ratio is varied from 1 to 1/4, which corresponds to the elasticity numbers El = 36.3 to 48.3. The numerical results show that varying the aspect ratio of microfluidic channels has similar effects to varying other parameters, such as fluid properties, which influence the elasticity number. Thus, our results support the view that vortex growth mechanism is determined by the elasticity number, which is fixed for a given fluid and geometry. The principle is of significance to the design of new microfluidic chips for a wide range of applications.

Anomalous pressure drop behaviour of mixed kinematics flows of viscoelastic polymer solutions: a multiscale simulation approach

A long-standing unresolved problem in non-Newtonian fluid mechanics, namely, the relationship between friction drag and flow rate in inertialess complex kinematics flows of dilute polymeric solutions is investigated via self-consistent multiscale flow simulations. Specifically, flow of a highly elastic dilute polymeric solution, described by first principles micromechanical models, through a 4:1:4 axisymmetric contraction and expansion geometry is examined utilizing our recently developed highly efficient multiscale flow simulation algorithm (Koppol, Sureshkumar & Khomami, J. Non-Newtonian Fluid Mech., vol. 141, 2007, p. 180). Comparison with experimental measurements (Rothstein & McKinley, J. Non-Newtonian Fluid Mech., vol. 86, 1999, p. 61) shows that the pressure drop evolution as a function of flow rate can be accurately predicted when the chain dynamics is described by multi-segment bead-spring micromechanical models that closely capture the transient extensional viscosity of the experimental fluid. Specifically, for the first time the experimentally observed doubling of the dimensionless excess pressure drop at intermediate flow rates is predicted. Moreover, based on an energy dissipation analysis it has been shown that the variation of the excess pressure drop with the flow rate is controlled by the flow-microstructure coupling in the extensional flow dominated region of the flow. Finally, the influence of the macromolecular chain extensibility on the vortex dynamics, i.e. growth of the upstream corner vortex at low chain extensibility or the shrinkage of the upstream corner vortex coupled with the formation of a lip vortex that eventually merges with the upstream corner vortex at high chain extensibility is elucidated.

The effects of chain conformation in the microfluidic entry flow of polymer–surfactant systems

An associative polymer–surfactant system has been used to observe the effects of chain conformation in the entry flow through a microfabricated planar 16:1:16 contraction–expansion geometry. The well-studied system of the flexible polymer poly(ethylene oxide) (PEO) and anionic surfactant sodium dodecyl sulfate (SDS) was used. Dilute polymer solutions with increasing SDS concentration were characterized in steady and dynamic shear, as well as capillary breakup extensional rheology. Based on this characterization, the primary quantitative difference is an increase in zero-shear viscosity as a result of the PEO chain expansion brought on by association of SDS surfactant micelles. However, these quantitatively similar solutions were observed to exhibit much more qualitatively different flow patterns via fluorescent streak imaging in the entry flow. In contrast to previous work on PEO solutions, the PEO–SDS systems were observed to transition to a steady viscoelastic flow regime characterized by stable lip vortices at much lower elasticity and Weissenberg numbers. The resulting insight gained regarding the utility of microfluidic flows in elucidating effects of subtle conformational changes further illustrates the potential for using microfabricated devices as rheometric tools for measuring the properties of dilute and weakly viscoelastic fluids.

Visualization of shear banding and entry Poiseuille flow oscillations in a micellar aqueous solution

A study of the plane Poiseuille flow of a micellar aqueous solution of cetylpyridinium chloride 100 mM and sodium salicylate 60 mM was performed in this work. The experiments were run at 27.5 °C under controlled pressure using a transparent flow cell, where simultaneous measurements of transmitted light intensity, pressure drop and flow rate were performed in order to asses the flow stability. Particle image velocimetry (PIV) was also used to analyze the flow kinematics upstream of the contraction. Different regimes were observed in the flow curve, including shear banding and spurt. In the high shear rate branch the flow became unstable and was composed by asymmetric shear bands of structured and isotropic fluid, which oscillated with respect to the zero-shear stress plane. Symmetric lip vortices were observed to grow upstream of the contraction, and then to oscillate and decrease their average length under unstable flow. The shear bands downstream of the contraction oscillated in the same way as upstream vortices, with a frequency that increased along with the flow rate. Finally, the oscillating flow upstream of the contraction produced jets or spurts of highly oriented material followed by recoiling, akin to those reported in the melt fracture regime of polymers.

Re-entrant corner flows of UCM fluids: The Cartesian stress basis

For a given re-entrant corner geometry, we describe a two parameter family of solutions for the local asymptotic behaviour of the flow and stress fields of UCM fluids. The two parameters used are the coefficients of the upstream wall shear rate and pressure gradient. In describing this parametric solution, the relationship between the Cartesian and natural stress basis is explained, reconciling these two equivalent formulations for the problem. The asymptotic solution structure investigated here comprises an outer (core) region together with inner regions (single wall boundary layers) located at the upstream and downstream walls. It is implicitly assumed that there are no regions of recirculation at the upstream wall, i.e. we consider flow in the absence of a lip vortex. The essential feature of the analysis is a full description of the matching between the outer and inner regions in both the Cartesian and natural stress bases, as well as the derivation of numerical estimates of important solution parameters such as the coefficients of the stream function and extra-stresses in the outer (core flow) region together with the downstream wall shear rate. This work is divided into two papers, the first one describing the solution structure in the Cartesian stress basis for the core and upstream boundary layer, and the second paper using the natural stress basis which allows the downstream boundary layer solution to be linked through the core to the upstream boundary layer solution. It is the latter formulation of this problem which allows a complete solution description.

Simulation of viscoelastic flows of polymer solutions in abrupt contractions using an arbitrary Lagrangian Eulerian (ALE) based finite element method

We present a method for simulation of viscoelastic flows using an Arbitrary Lagrangian Eulerian (ALE) technique based finite element formulation. The ALE technique provides advantages of both Lagrangian and Eulerian frameworks by allowing the computational mesh to move in an arbitrary manner, independent of the material motion. In the present method, a fractional step ALE technique is employed in which the Lagrangian phase of material motion and convection arising out of mesh motion are decoupled. In the first step the relevant flow and constitutive equations are solved in Lagrangian framework. The simpler representation of polymer constitutive equations in a Lagrangian framework avoids the difficulties associated with convective terms thereby resulting in a robust numerical formulation. In the second step the mesh is moved in ALE mode and the associated convection of the stress is performed using a Godunov type scheme. This ALE technique is easy to implement and can accurately simulate the complex viscoelastic behaviour of transient polymer flow through complex geometries. In the present study, steady flows through abrupt contractions of planar and axisymmetric geometries are studied by performing transient flow simulations until steady state is achieved. The proposed method is validated with previously published numerical and experimental studies for polymer solutions obeying the Oldroyd B and Phan Thien Tanner (PTT) models. The simulated corner and lip vortex enhancement mechanism and flow behavior are in good agreement with experimentally obtained flow visualization photographs. The strength of the proposed method lies in its ability to simulate free surface flows such as swell.

Re-entrant corner flows of upper convected Maxwell fluids: the small and high Weissenberg number limits

We discuss here the steady planar flow of the upper convected Maxwell fluid at re-entrant corners in the singular limits of small and large Weissenberg number. The Weissenberg number is a parameter representing the dimensionless relaxation time and hence the elasticity of the fluid. Its value determines the strength of the fluid memory and thus the influence of elastic effects over viscosity. The small Weissenberg limit is that in which the elastic effects are small and the fluid's memory is weak. It is an extremely singular limit in which the behaviour of a Newtonian fluid is obtained in a main core region away from the corner and walls. Elastic effects are confined to boundary layers at the walls and core regions nearer to the corner. The actual asymptotic structure comprises a complicated four-region structure. The other limit of interest is the large Weissenberg limit (or high Weissenberg number problem) in which the elastic effects now dominate in the main regions of the flow. We explain how the transition in solution from Weissenberg order 1 flows to high Weissenberg flows is achieved, with the singularity in the stress field at the corner remaining the same but its effects now extending over larger length-scales. Implicit in this analysis is the absence of a lip vortex. We also show (for the main core region) that there is a small reduction in the velocity field at the corner and walls where it becomes smoother. This high Weissenberg number limit has a six-region local asymptotic structure and comment is made on its relevance to the case in which a lip vortex is present.

Stabilised computations for viscoelastic flows under compressible implementations

We analyse and contrast different stabilisation methodologies embedded within a time-marching incremental pressure-correction formulation. Numerical solutions are presented for an Oldroyd-B model under compressible implementations, considering flow through a planar four-to-one abrupt contraction. Various alternative stabilisation strategies and their combinations are analysed to hone the response of the base hybrid finite element/volume implementation. To reflect the stabilised properties of each scheme, the study interrogates levels of stable Weissenberg number ( We) solution. Results indicate that most improvement has been encountered with a Strain-Rate Stabilisation scheme, where critical We-levels have more than doubled above neutral variants, while stress peaks levels have been constrained. Here, differed-correction characterises temporal error norm stress behaviour and the nature of the re-entrant corner stress singularity. At a selected We-level and under a specific flow setting, all scheme variants have produced similar salient-corner vortex behaviour, predicting vortex reduction under increasing We. In contrast, lip-vortex features are found to be significantly affected by the particular re-entrant corner treatment applied. When present, lip vortices grow with increasing We. Relaxation of the incompressible constraint points to important numerical anomalies, present under certain discretisations.

Re-entrant corner flows of UCM fluids: the initial formation of lip vortices

We discuss here a third type of boundary-layer structure that arises in steady planar flows at re-entrant corners for the upper convected Maxwell fluid. This structure extends the class of similarity solutions that are associated with the inviscid flow equations and hold in an outer core region local to the corner. In previous work ( Renardy 1995 J. Non-Newtonian Fluid Mech. 58 , 83–39 ; Evans 2005 Proc. R. Soc. A 461 , 117–142 ), single and double layer structures were used to the match outer core similarity solutions to wall boundary layers in which viscometric behaviour is obtained. Here a second double layer structure is discussed, which completes the range of validity for the core similarity solutions. This structure is fundamentally different to the single layer structure in that it only admits reverse flow solutions at the upstream wall, a situation of practical relevance to describe the initial formation of lip vortices in contraction flows.

Visualizations of Boger fluid flows in a 4:1 square–square contraction

AbstractVisualizations of the 3‐D flow in a 4:1 square–square sudden contraction for two viscoelastic Boger fluids and two Newtonian fluids were carried out at low Reynolds numbers. In these creeping flow conditions, the vortex length remained unchanged for Newtonian fluids, whereas a nonmonotonic variation with flow rate was observed for the Boger fluids. Initially, the corner vortex slightly increased with flow rate to a local peak at a Deborah number of De2 ≈ 6, before decreasing significantly to a minimum at De2 ≈ 15 (De2 is based on downstream characteristics). Finally, for Deborah numbers > 20 there was intense vortex enhancement until a periodic flow was established at higher flow rates (De2 ≈ 45–52). The strong elastic vortex enhancement was preceded by the appearance of diverging streamlines on the approach flow and, for the Boger fluid with higher polymer concentration, vortex enhancement took place through a lip vortex mechanism. AIChE J, 2005

Time-Weissenberg Number Superposition in 4:1 Planar Contraction Flow of a Viscoelastic Fluid

Vortex enhancement mechanism in 4:1 planar contraction flow of a viscoelastic fluid has been investigated, where the Oldroyd-B model was considered as a constitutive equation. Finite element method was used based on the fractional step method for time marching and DEVSS/DG as a stabilizing scheme. The prediction of vortex dynamics agreed well with the experimental results. At first, a lip vortex was generated at the reentrant corner at low Weissenberg number (We). As We increases, the size of the salient corner vortex becomes reduced, while the lip vortex becomes more pronounced and finally takes over the corner vortex. The trajectory of a lip vortex at high We was found to follow the steady state positions of the lip vortex at lower We, which means that the fluid has a memory and there exists a simple relationship between We and time. Accordingly, a simple universal behavior was found to dominate the vortex enhancement mechanism in the 4:1 planar contraction flow of the Oldroyd-B fluid.

On the effect of contraction ratio in viscoelastic flow through abrupt contractions

A numerical study of the creeping flow of a PTT fluid through planar sudden contractions was carried out to quantify the effect of contraction ratio upon the flow characteristics (streamlines and size and intensity of recirculation vortices). The relevant governing equations were solved with a finite volume method embodying a new high-resolution scheme (Alves et al. [Int. J. Numer. Meth. Fluids 41 (2003) 47]) for the discretisation of convection terms, which is here explained and shown to yield improved accuracy and robustness. The results of the simulations, in terms of streamline patterns, give further evidence for a lip-vortex enhancement mechanism and are in remarkable agreement with flow visualization photographs from the literature. In addition, the results show that the variation of flow features in the vicinity of the re-entrant corner, such as lip vortex size and streamlines, are dominated by downstream quantities and scale with the common definition for the Deborah number in this flow, while flow characteristics in the salient corner region scale with that Deborah number divided by the contraction ratio.

The flow of an Oldroyd fluid past a reentrant corner: the downstream boundary layer

We reconsider the steady planar flow of an Oldroyd-B fluid within a small distance r of a sharp reentrant corner of angle π/ p, 1/2≤ p<1. Previous theoretical studies of this problem have been unable to resolve a boundary layer near the downstream wall. For a range of angles π/ p, including the benchmark problem p=2/3, and for several non-zero solvent viscosities, we have solved the equations for the downstream boundary layer numerically and additionally found asymptotic results for a frozen stress limit and for high solvent viscosities. For each angle and viscosity we find a one-parameter family of attached ‘potential’ flows, for each of which the stress is proportional to r 2 p−2 and the velocity proportional to r p(3− p)−1 as proposed by Hinch [J. Non-Newtonian Fluid Mech. 50 (1993) 161]. The limiting member of the family has zero shear stress on the upstream wall. The stress is that of a purely elastic neo-Hookean solid in the interior of the flow, with boundary layers on both the upstream and downstream walls having the similarity structure proposed by Renardy [J. Non-Newtonian Fluid Mech. 58 (1995) 83]. The behaviour of these flows is discussed. We propose a form of elastic Kutta condition to determine the separation angle for flows having an upstream lip vortex.

Development of an optimal hybrid finite volume/element method for viscoelastic flows

AbstractA cell‐vertex hybrid finite volume/element method is investigated that is implemented on triangles and applied to the numerical solution of Oldroyd model fluids in contraction flows. Particular attention is paid to establishing high‐order accuracy, whilst retaining favourable stability properties. Elevated levels of elasticity are sought. The main impact of this study reveals that switching from quadratic to linear finite volume stress representation with discontinuous stress gradients, and incorporating local reduced quadrature at the re‐entrant corner, provide enhance stability properties. Solution smoothness is achieved by adopting the non‐conservative flux form with area integration, by appealing to quadratic recovered velocity‐gradients, and through consistency considerations in the treatment of the time term in the constitutive equation. In this manner, high‐order accuracy is maintained, stability is ensured, and the finer features of the flow are confirmed via mesh refinement. Lip vortices are observed for We&gt;1, and a trailing‐edge vortex is also apparent. Loss of evolution and solution asymptotic behaviour towards the re‐entrant corner are also discussed. Copyright © 2003 John Wiley &amp; Sons, Ltd.

The axisymmetric contraction–expansion: the role of extensional rheology on vortex growth dynamics and the enhanced pressure drop

The flow of a polystyrene Boger fluid through axisymmetric contraction–expansions having various contraction ratios (2≤ β≤8) and varying degrees of re-entrant corner curvatures are studied experimentally over a large range of Deborah numbers. The ideal elastic fluid is dilute, monodisperse and well characterized in both shear and transient uniaxial extension. A large enhanced pressure drop above that of a Newtonian fluid is observed independent of contraction ratio and re-entrant corner curvature. Streak images, laser Doppler velocimetry (LDV) and digital particle image velocimetry (DPIV) are used to investigate the flow kinematics upstream of the contraction plane. LDV is used to measure velocity fluctuation in the mean flow field and to characterize a global elastic flow instability which occurs at large Deborah numbers. For a contraction ratio of β=2, a steady elastic lip vortex is observed while for contraction ratios of 4≤ β≤8, no lip vortex is observed and a corner vortex is seen. Rounding the re-entrant corner leads to shifts in the onset of the flow transitions at larger Deborah numbers, but does not qualitatively change the overall structure of the flow field. We describe a simple rescaling of the deformation rate which incorporates the effects of lip curvature and allows measurements of vortex size, enhanced pressure drop and critical Deborah number for the onset of elastic instability to be collapsed onto master curves. Transient extensional rheology measurements are utilized to explain the significant differences in vortex growth pathways (i.e. elastic corner vortex versus lip vortex growth) observed between the polystyrene Boger fluids used in this research and polyisobutylene and polyacrylamide Boger fluids used in previous contraction flow experiments. We show that the role of contraction ratio on vortex growth dynamics can be rationalized by considering the dimensionless ratio of the elastic normal stress difference in steady shear flow to those in transient uniaxial extension. It appears that the differences in this normal stress ratio for different fluids at a given Deborah number arise from variations in solvent quality or excluded volume effects.

A cell-vertex finite volume/element method on triangles for abrupt contraction viscoelastic flows

A stable and accurate implementation is presented of a cell-vertex hybrid finite volume/element scheme based upon triangular meshes. This scheme is novel to this domain and is applied to the numerical solution of Oldroyd model fluids in abrupt planar contraction flows. All important has been the use of non-conservative flux representation, consistency in treatment of transients, fluxes and sources, and a recovery technique for velocity gradients. Linear stress representation, with non-recovered stress gradients, has proved crucial to widening stability thresholds. Solution smoothness at the higher levels of elasticity results, so that converged solutions are attainable. We have also highlighted the importance of incorporating a reduced-integration local discontinuity capturing technique for the re-entrant corner solution. A diminishing lip vortex with mesh refinement is reported. With increasing elasticity, lip vortex growth and a diminishing salient corner vortex is noted. In addition, a trailing-edge vortex is found to accompany the onset of a lip vortex. Agreement with analytical theory is observed through the asymptotic behaviour of velocities and stresses near the re-entrant corner, a region where the loss of evolution of the discrete system is investigated.

Plane contraction flows of upper convected Maxwell and Phan-Thien–Tanner fluids as predicted by a finite-volume method

A finite-volume (FV) procedure is applied to the prediction of two-dimensional (2-D) laminar flow through a 4 : 1 planar contraction of upper convected Maxwell (UCM) and simplified Phan-Thien–Tanner (SPTT) fluids. The method incorporates general coordinates, indirect addressing for easy mapping of complex domains, and is based on the collocated mesh arrangement. Calculations with the UCM model at a Reynolds number of 0.01 were carried out with three consecutively refined meshes which enabled the estimation of the accuracy of the predictions of the main vortex characteristics through Richardson's extrapolation. Converged solutions with the first-order upwind differencing scheme for the convective terms were obtained up to at least De = 8 in the finest mesh, but were limited to De ≤ 1, De ≤ 3 and De ≤ 5 for the fine, medium and coarse meshes, respectively, when using the second-order linear upwind scheme. The predicted flow patterns for increasing Deborah numbers with the UCM model resemble the well known lip vortex enhancement mechanism reported in the literature for constant-viscosity fluids in axisymmetric contractions and shear-thinning fluids in planar contraction, but very fine meshes were required in order to capture the described vortex activity. Predictions with the SPTT model also compared well with the behaviour reported in the literature.

Entry Flow of Polymer Solutions through Multi-Hole Abrupt Contractions.

Flow visualizations and velocity measurements for entry flows of shear-thinning aqueous solutions of polyacrylamide through three kinds of multi-hole contractions have been carried out using the tracer method and LDV technique. In these experiments effects of the number of contractions, distance between contractions and rheology of solutions on the flow structures have been studied. For the multi-hole contractions square arrays of 9 holes and 25 holes were used. In the steady flow regime dimensionless lip vortex length is almost the same for all multi-hole contractions. Swirling-like vortices are also observed for both three-by-three and five-by-five contractions with dimensionless pitch 4, while two-dimensional lip vortices are observed for the three-by-three contraction with dimensionless pitch 8. It is found from LDV measurements that periodic oscillations of the main flow occur at the onset of unsteady flow for all multi-hole contractions and the frequency of the oscillation increases as the Weissenberg number increases. For multi-hole contractions, flow interaction through each contraction is very important factor which decides both steady and unsteady flow structures.

Three dimensional numerical simulations of viscoelastic flows through planar contractions

We present in this paper a fully three dimensional (3D) convergent numerical study of planar viscoelastic contraction flows. A 3D finite volume method (FVM) with the primary variable elastic viscous split stress (EVSS) formulation is employed, and a very efficient 3D block solver coupled with block correction is developed to speed up the convergence rate. Full 3D simulations of viscoelastic flows in 4:1 planar abrupt contractions are carried out using experimental conditions. Upstream vortex patterns comparable with the existing flow visualisation observations are captured using the upper convected Maxwell model for a Boger fluid and the Phan-Thien–Tanner model for a shear thinning fluid. Comprehensive comparisons between numerical simulation results and data measured in the dynamic fields in a 4:1 planar abrupt contraction are made, and the results indicate that the experimental measurements can be quantitatively reproduced if the fluid is well characterised by an appropriate viscoelastic model. It is confirmed numerically that the shear thinning of the fluid reduces the intensity of the singularity of viscoelastic flow near the re-entrant corner. With the Oldroyd-B model, by extensive computations on successively refined meshes with the minimum dimensionless size being 0.16–0.014 on the contraction plane in 2D configuration, it is revealed that, although the asymptotic flow behavior near the re-entrant corner and the build-up of the overall pressure and extensional stresses as well as the kinematic behavior along the centreline are insensitive to mesh refinement, completely different vortex activities may be predicted if the mesh is not sufficiently fine. It is verified numerically that, depending on the flow inertia and rheological properties of fluids, both the lip vortex mechanism and the corner vortex mechanism may be responsible for the vortex activities of viscoelastic fluids in 4:1 planar contraction flow, and the elasticity number E and Mach number M of the flow can be used to determine the vortex mechanism approximately. It is clear that the development process of the vortex activities could be underestimated with 2D simplification, and overpredicted with the creeping flow assumption, particularly when R e>0.5. Therefore, in planar contraction flow analyses, numerical artifacts may be produced with a coarse mesh, and 2D flow simulation is only a good approximation to the fully 3D flow if the upstream aspect ratio W/ H in the experiment is at least 10.