Numerical simulation of a bubble rising in shear-thinning fluids

The motion of a single bubble rising freely in quiescent non-Newtonian viscous fluids was investigated experimentally and computationally. The non-Newtonian effects in the flow of viscous inelastic fluids are modeled by the Carreau rheological model. An improved level set approach for computing the incompressible two-phase flow with deformable free interface is used. The control volume formulation with the SIMPLEC algorithm incorporated is used to solve the governing equations on a staggered Eulerian grid. The simulation results demonstrate that the algorithm is robust for shear-thinning liquids with large density (ρ₁/ρ_g up to 10³) and high viscosity (η₁/η_g up to 10⁴). The comparison of the experimental measurements of terminal bubble shape and velocity with the computational results is satisfactory. It is shown that the local change in viscosity around a bubble greatly depends on the bubble shape and the zero-shear viscosity of non-Newtonian shear-thinning liquids. The shear-rate distribution and velocity fields are used to elucidate the formation of a region of large viscosity at the rear of a bubble as a result of the rather stagnant flow behind the bubble. The numerical results provide the basis for further investigations, such as the numerical simulation of viscoelastic fluids.     

Numerical simulation of contraction and expansion flows of Langmuir monolayers

Langmuir monolayers consist of amphiphilic molecules at the air–water interface and can be modeled as two-dimensional fluids. Earlier experiments [D.J. Olson, G.G. Fuller, J. Non-Newtonian Fluid Mech. 89 (2000) 187–207] on 4:1 contraction and 4:1 expansion flows have been simulated using an integral constitutive equation of the K-BKZ type, suitably modified to account for strain-thickening in the planar extensional viscosity. The model has been used to fit linear viscoelastic data (G′ and G″) and the shear viscosity (η_S), while the amount of strain-hardening is assumed, due to lack of experimental data. The simulations are in good agreement with the experiments on Newtonian monolayers, which show no vortices in the contraction but large inertial vortices in the expansion. For the viscoelastic monolayer (a poly-octadecyl methacrylate or PODMA), the opposite is true. The contraction flow shows vortices, while in the expansion flow the vortex activity is substantially reduced compared with the Newtonian one. The viscoelastic behavior is well captured by the model, provided that substantial strain-thickening is exhibited by the monolayer in planar extension. The latter behavior is very much like that for a branched LDPE melt, which also shows big vortices due to strain-hardening in planar as well as in uniaxial extension.     

Viscosity enhancement in the flow of hydrolysed poly(acrylamide) saline solutions around spheres: implications for enhanced oil recovery

We have studied dilute aqueous solutions of hydrolysed poly(acrylamide), in various ionic environments, in flow around single spheres and around two spheres aligned on the axis of flow. The spheres are held on flexible cantilevers, while the polymer solutions, or solvent, are drawn past at controlled flow rates. We estimate the specific viscosities of the various solutions as a function of the strain rate over strain rates encompassing both the shear thinning and extension thickening regimes. For flow of solutions without added salts around a single sphere, we observe shear thinning followed by a significant increase in the non-Newtonian viscosity with increasing strain rate. The shear thinning reduces the maximal extensional viscosities of the solutions, which has important implications regarding the effectiveness of hydrolysed poly(acrylamide) in oil field applications. For flow of polymer solutions around two axially aligned spheres, we observe a significant reduction in the non-Newtonian forces experienced by the downstream sphere in comparison to the upstream sphere. We consider that this is salient to the understanding of non-Newtonian viscosification in porous media flow.     

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium

An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semi-analytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity are shown in several graphs for different values of the porous parameters \({\sigma _1 =(\mu /\tilde {\mu }) (a/\sqrt{K_1 })}\) and \({\sigma _2 =(\mu /\tilde {\mu }) (a/\sqrt{K_2 })}\) , where a is the radius of the sphere, μ is the dynamic viscosity of the fluid, \({\tilde {\mu }}\) is an effective or Brinkman viscosity, while K ₁ and K ₂ are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ ₁ and σ ₂.     

Review of the entry flow problem: Experimental and numerical

This paper is concerned with a review of both experimental and numerical studies of axisymmetric and planar entry flows which have been considered as test problems for the numerical simulation of viscoelastic fluids. The test of the method is usually based upon whether the numerical model predicts vortices in the entry corners. However, it is not clear as to whether one should observe vortices for all viscoelastic fluids. Polyacrylamide solutions and Boger fluids exhibit vortices in axisymmetric flow and the size of the vortex does increase with fluid elasticity. However, the vortex is nearly suppressed in planar entry flow. On the other hand, not all polymer melts are found to exhibit vortices in either axisymmetric or planar entry flow. It is our belief that the origin of vortices is not related to the elasticity based on shear flow propertes but to the behavior of the transient extensional viscosity. Certain polymer melts such as low density polyethylene exhibit vortices in both planar and axisymmetric flow along with unbounded stress growth at the start up of extensional flow. It is believed that the constitutive equations used in the numerical simulation must reflect this extensional behavior if vortices are to be predicted. A review of the numerical simulations concerned with entry flow shows that there is considerable doubt about the accuracy of the predictions for most of the studies. Even for those where the numerical solution is thought to be accurate, the magnitude of the stream function associated with the vortices is usually very low. None of the differential models used to date predicts strain hardening extensional viscosity, but those which are thought to predict vortices do rise more rapidly to the steady-state extensional viscosity values with time. It is recommended that the search of test fluids be widened beyond polymer solutions as there may already exist a number of polymer melts which behave similarly to the predictions of existing constitutive equations.     

Molecular orientation in non-Newtonian flow of dilute polymer solutions around spheres

A novel approach is presented to study the benchmark problem of flow around spheres in model dilute solutions of monodisperse samples of atactic polystyrene in di-octyl phthalate. Spheres are held stationary on flexible cantilevers of known spring-constant, k, while the polymer solutions are pumped past at controlled flow rates, allowing access to a wide range of Deborah number. In this way the non-Newtonian forces experienced by the spheres can be measured as a function of Deborah number, while detailed observations and measurements of birefringence are made, enabling assessment of macromolecular strain and orientation. In addition the flow field around a sphere has been measured in an a-PS solution. Experiments have been performed on a single sphere and on two spheres axially aligned in the direction of flow. The extensional flow around the downstream stagnation point of the single sphere is found to play a pivotal role in the development of molecular strain and stress, resulting in flow modification and subsequent non-Newtonian behaviour. The flow birefringence in the wake is found to modify severely the flow around a second, downstream, sphere, affecting the non-Newtonian forces encountered by the second sphere. This provides an explanation for the time interval dependent terminal velocity often observed when two spheres follow the same path through viscoelastic liquids.     

Experimental study on the capillary thinning of entangled polymer solutions

The transient elongation behavior of entangled polymer and wormlike micelles (WLM) solutions has been investigated using capillary breakup extensional rheometry (CaBER). The transient force ratio X = 0.713 reveals the existence of an intermediate Newtonian thinning region for polystyrene and WLM solutions prior to the viscoelastic thinning. The exponential decay of X(t) in the first period of thinning defines an elongational relaxation time λ _x which is equal to elongational relaxation time λ _e obtained from exponential diameter decay D(t) indicating that the initial stress decay is controlled by the same molecular relaxation process as the strain hardening observed in the terminal regime of filament thinning. Deviations in true and apparent elongational viscosity are discussed in terms of X(t). A minimum Trouton ratio is observed which decreases exponentially with increasing polymer concentration leveling off at Tr_min = 3 for the solutions exhibiting intermediate Newtonian thinning and Tr_min ≈ 10 otherwise. The relaxation time ratio λ _e/ λ _s, where λ _s is the terminal shear relaxation time, decreases exponentially with increasing polymer concentration and the data for all investigated solutions collapse onto a master curve irrespective of polymer molecular weight or solvent viscosity when plotted versus the reduced concentration c[ η], with [ η] being the intrinsic viscosity. This confirms the strong effect of the nonlinear deformation in CaBER experiments on entangled polymer solutions as suggested earlier. On the other hand, λ _e ≈λ _s is found for all WLM solutions clearly indicating that these nonlinear deformations do not affect the capillary thinning process of these living polymer systems.     

Negative wake in the uniform flow past a cylinder

The upstream/downstream streamline shift and the associated negative wake generation (streamwise velocity overshoot in the wake) in a viscoelastic flow past a cylinder are studied in this paper, for the Oldroyd-B, UCM, PTT, and FENE-CR fluids, using the Discrete Elastic Viscous Split Stress Vorticity (DEVSS-ω) scheme (Dou HS, Phan-Thien N (1999). The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity vorticity (DAVSS-ω) formulation. J Non-Newtonian Fluid Mech 87:47–73). The numerical algorithm is a parallelized unstructured Finite Volume Method (FVM), running under a distributed computing environment through the Parallel Virtual Machine (PVM) library. It is demonstrated that both the normal stress and its gradient are responsible for the negative wake generation and streamline shifting. Fluid extensional rheology plays an important role in the generation of the negative wake. The negative wake can occur in flows where the fluid extensional viscosity does not increase rapidly with strain rate. The formation of the negative wake does not depend on whether the streamlines undergo an upstream or a downstream shift. Shear-thinning viscosity weakens the velocity overshoot and while shear-thinning first normal stress coefficient enhances the velocity overshoot. Wall proximity is not necessary for the velocity overshoot; however, it enhances the strength of the negative wake. For the Oldroyd-B fluid, the ratio of the solvent viscosity to the zero-shear viscosity plays an important role in the streamline shift. In addition, mesh dependent behaviour of normal stresses along the centreline at high De in most cylinder/sphere simulations is due to the convection of normal stress from the cylinder to the wake, which results in the maximum of the normal stress being located off the centreline by a short distance at high De.     

Flow structures around an equilateral triangle arrangement of three spheres

This paper represents the results of an experimental study on the flow structure around a single sphere and three spheres in an equilateral-triangular arrangement. Flow field measurements were performed using a Particle Image Velocimetry (PIV) technique and dye visualization in an open water channel for a Reynolds number of Re = 5 × 10³ based on the sphere diameter. The distributions and flow features at the critical locations of the contours of the velocity fluctuations, the patterns of sectional streamlines, the vorticity contours, the turbulent kinetic energy, the Reynolds stress correlations and shedding frequency are discussed. The gap ratios (G/D) of the three spheres were varied in the range of 1.0 ⩽ G/D ⩽ 2.5 where G was the distance between the sphere centers, and D was the sphere diameter which was taken as 30 mm. Due to the interference of the shedding shear layers and the wakes, more complex features of the flow patterns can be found in the wake region of the two downstream spheres behind the leading sphere. For G/D = 1.25, a jet-like flow around the leading sphere through the gap between the two downstream spheres occurred, which significantly enhanced the wake region. It was observed that a continuous flow development involving shearing phenomena and the interactions of shedding vortices caused a high rate of fluctuations over the whole flow field although most of the time-averaged flow patterns were almost symmetric about the two downstream spheres.     

Non-Newtonian viscosity measurements in the intermediate shear rate range with the falling-ball viscometer

An attempt to use the falling-ball experiment to measure the non-Newtonian viscosity in the intermediate shear rate range was successfully accomplished by combining the direct experimental observations with a simple analytical model for the average shear and shear rate at the surface of a sphere. The viscosity data of aqueous solutions of Carbopol-960, carboxymethyl cellulose, polyethylene oxide and polyacrylamide obtained from the falling-ball viscometer gave good agreement with those from other viscometers, confirming the general applicability of the analytical approach.In the experiments with the highly viscoelastic polyacrylamide solutions the terminal velocity was observed to be dependent on the time interval between the dropping of successive balls. This time-dependent phenomenon was used to determine characteristic diffusion times of the concentrated solutions of polyacrylamide. These values were, in turn, compared with characteristics relaxation times determined by the Powell-Eyring model.The experimental program revealed that the falling-ball viscometer has very limited utility for the measurement of the steady shear viscosity of aqueous polymer solutions.     

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.     

The rheology of suspensions of vinylon fibers in polymer liquids. II. Suspensions in polymer solutions

Summary The rheological properties of vinylon fiber suspensions in polymer solutions were studied in steady shear flow. Shear viscosity, first normal-stress difference, yield stress, relative viscosity, and other properties were discussed. Three kinds of flexible vinylon fibers of uniform length and three kinds of polymer solutions as mediums which exhibited remarkable non-Newtonian behaviors were employed. The shear viscosity and relative viscosity ( _r ) increased with the fiber content and the aspect ratio, and depended upon the shear rate. Shear rate dependence of _r was found only in the low shear rate region. This result was different from that of vinylon fiber suspensions in Newtonian fluids. The first normal-stress difference increased at first slightly with increasing fiber content but rather decreased and showed lower values for high content suspensions than that of the medium. A yield stress could be determined by using a modified equation of Casson type. The flow properties of the fiber suspensions depended on the viscosity of the medium in the suspensions under consideration.With 16 figures and 1 table     

Weakly-imposed Dirichlet boundary conditions for non-Newtonian fluid flow

We propose a new formulation for weakly imposing Dirichlet boundary conditions in non-Newtonian fluid flow. It is based on the Gerstenberger–Wall formulation for Newtonian fluids [1], but extended to non-Newtonian fluids. It uses a stabilization term in the weak form that is independent from the actual fluid model used, except for an adjustable parameter κ, having the physical dimension of a viscosity. The new formulation is tested, combined with an extended finite element method, for the flow past a cylinder between two walls using both a generalized Newtonian and a viscoelastic fluid. It is shown that the convergence is optimal for the generalized Newtonian fluid by comparing with a converged boundary-fitted solution using traditional strong boundary conditions. Also the solution of the viscoelastic fluid compares very well with a traditional solution using a boundary-fitted mesh and strong Dirichlet boundary conditions. For both fluid models we also test various values of the κ parameter and it turns out that a value equal to the zero-shear-viscosity gives good results. But, it is also shown that a wide range of κ values can be chosen without sacrificing accuracy.     

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid–liquid crystalline (LC) polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie–Ericksen theory is described by the first Rivlin–Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion–extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extrudate of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a series of new anisotropic non-Newtonian fluid problems can be addressed. The project supported by the National Natural Science Foundation of China (10372100, 19832050) (Key project). The English text was polished by Yunming Chen.     

Scaling relations for elongational flow of polystyrene melts and concentrated solutions of polystyrene in oligomeric styrene

A consistent model of the rheology of polymer melts and concentrated solutions is presented, based on the idea that the pressures exerted by a polymer chain on the walls of an anisotropic confinement are anisotropic (Doi and Edwards. The Theory of Polymer Dynamics, Oxford University Press, 1986). In a tube model with variable tube diameter, chain stretch and tube diameter reduction are related, and at deformation rates larger than the inverse Rouse time τ _R, the chain is stretched and its confining tube becomes increasingly anisotropic. Tube diameter reduction leads to an interchain pressure in the lateral direction of the tube (Marrucci and Ianniruberto. Macromolecules 37:3934-3942, 2004). Chain stretch is balanced by interchain tube pressure in the lateral direction, which is proportional to the third power of stretch, and by a spring force in the longitudinal direction of the tube, which is linear in stretch. Analyzing elongational viscosity data of Huang et al. (Macromolecules 46:5026-5035, 2013a; ACS Macro Letters 2:741-744, 2013b) shows that dilution of polystyrene by oligomeric styrene does not change the relative interchain tube pressure. Based on this extended interchain pressure concept, scaling relations for linear viscoelasticity and elongational viscosity of polystyrene melts and concentrated solutions of polystyrene in oligomeric styrene are presented based exclusively on the relaxation modulus of a reference polymer melt, the volume fraction of polymer in the solution, and the time-molar-mass shift as well as the time-temperature shift caused by the reduction of the glass transition temperature T _g of the polymer in a solution relative to T _g of the melt.     

Extrudate swell in some dilute elastic solutions

An experimental programme has been carried out to obtain data for the isothermal swelling of dilute viscoelastic fluids based on the polyisobutylene-polybutene system. The fluid was thus a constant viscosity one and so the observed swell was predominantly due to elastic effects alone. To establish the experimental rig and procedure, preliminary results for the creeping flow of a Newtonian fluid, based on a polybutene, were acquired. The Newtonian results, showing a creeping flow swell of 13.8%, demonstrated that the rig and procedure were satisfactory for the formation of free, isothermal jets. The viscoelastic results were quite unexpected, showing that dilute elastic solutions can exhibit significant swelling despite the fact that the recoverable shear is small (S _R < 0.1). These results qualitatively confirm a recent numerical study [1].     

Molecular weight polydispersity effects on the melt viscoelasticity of styrene-acrylonitrile random copolymers

A series of polydisperse SAN (styrene-co-acrylonitrile) random copolymers was studied by means of oscillatory rheometry in the rubbery plateau zone and in the terminal zone. The plateau modulus, the Newtonian viscosity, and the critical frequencies for the onset of non-Newtonian behavior were extracted from the experimental data. All these viscoelastic quantities consistently indicate that the tail of molecular weights below approximately M _e (the entanglement spacing) acts as a solvent for the rest of the polymer with M>M _e .