Kinematics and normal stress differences of a viscoelastic fluid undergoing wiggle flow

An experimental program was carried out to determine the laminar regime kinematics and normal stress differences of a viscoelastic fluid in wiggle flow employing non-contact measurement techniques. The viscoelastic fluid was a 5% by weight solution of polyisobutylene dissolved in Primol 355, a high purity mineral oil.The kinematics were determined by Laser-Doppler Anemometry and compared with the data obtained for a Newtonian fluid, Primol 355, under identical flow conditions. It was found that the normalized axial velocity versus axial position curves along the centerline for both fluids superimposed at very low flow rates, an experimental verification that a viscoelastic fluid behaves like a Newtonian fluid under very low shear rates. However, at higher flow rates the behaviour of the viscoelastic fluid curves changed appreciably whereas the Newtonian fluid curves did not change at all. Thus, the effect of flow rate on viscoelastic fluid behaviour was also experimentally established.The normal stress differences were determined using a stress-birefringence apparatus. Data obtained along the centerline clearly exhibited a delayed growth of stress which should be attributed to the expected memory effects in viscoelastic fluid flow.     

Squeezing a viscoelastic liquid from a wedge: an exact solution

The paper reports an exact solution for the squeezing flow from a wedge of a general viscoelastic liquid. To obtain numerical values for the field variables, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. It is found that both these features are important in this transient flow; stress overshoot is responsible for a stiffer response of the fluid (compared to the inelastic case) at moderate time —at large time, shear-thinning dominates and the fluid behaves like an inelastic fluid. On the other hand, the Oldroyd-B fluid always predicts a softer response than the Newtonian one. Furthermore, there is a limiting Weissenberg number above which one component of the stresses of the Oldroyd-B fluid increases unboundedly with time. This limiting Weissenberg number is approximately $\text{sol2}\text{3}$.     

Transient deformation of an inviscid inclusion in a viscoelastic extensional flow

The transient deformation of a bubble in a viscoelastic extentional flow is analyzed by means of a finite element algorithm for viscoelastic moving boundary problems. Using the Oldroyd-B constitutive model, we find that bubbles in a viscoelastic fluid deform to the same steady-state configurations as bubbles in a Newtonian fluid at equal values of the far-field extensional stresses (corresponding to different stretch rates). Vapor bubbles in a developed extensional flow collapse more readily in the viscoelastic liquid than bubbles in Newtonian fluids because of the large compressive stresses associated with the viscoelastic liquid.     

Surfactant spreading on a thin weakly viscoelastic film

A mathematical model is presented for surfactant-driven thin weakly viscoelastic film flows on a flat, impermeable plane. The Oldroyd-B constitutive relation is used to model the viscoelastic fluid. Lubrication theory and a perturbation expansion in powers of the Weissenberg number (We) are employed, which give rise to non-linear coupled evolution equations governing the transport of insoluble surfactant and thin liquid film thickness. Spreading on a Newtonian film is recovered to leading order and corrections to viscoelasticity are obtained at order We. These equations are solved numerically over a wide range of viscosity ratio (ratio of solvent viscosity to the sum of solvent and polymeric viscosities), pre-existing surfactant level and Peclet number (Pe). The effect of viscoelasticity on surfactant transport and fluid flow is investigated and the mechanisms underlying this effect are explored. Shear stress, streamwise normal stress and the temporal rate of change of extra shear stress generated from gradients in surfactant concentration dominate thin viscoelastic film flows whereas only shear stresses play a role in Newtonian thin film flows. Our results also reveal that, for weak viscoelasticity, the influence of viscosity ratio on the evolution of surfactant concentration and film thickness can be significant and varies considerably, depending on the concentration of pre-existing surfactant and surfactant surface diffusivity.     

Break-up of a viscoelastic liquid jet

Applying Green's continuum theory of a slender body, the process of liquid jet break-up is analysed for a viscoelastic upper-convected Jeffreys fluid. In contrast to a Newtonian liquid an enforced growth of the perturbation is received from a linear analysis. A non-linear numerical analysis shows the viscosity-dependent filament formation between growing droplets of the viscoelastic liquid. The radius of these filaments decreases in an uniaxial extensional flow.     

Viscoelastic effects in double‐pipe single‐pass counterflow heat exchangers

The conducto‐convective heat loss from a viscoelastic liquid, in the core of a double‐pipe heat exchanger arrangement, to a cooler Newtonian fluid flowing in the outer annulus is investigated with direct numerical simulations. A numerical algorithm based on the finite difference method is implemented in time and space with the Giesekus constitutive model for the viscoelastic liquids. The flow of both the annulus and core‐fluids is considered to be Poiseuille flow, driven by respective pressure gradients. In general, the results show that a viscoelastic core‐fluid leads to slightly lower (albeit comparable) attainable temperatures in the core‐fluid stream as compared with a corresponding Newtonian fluid. Copyright © 2008 John Wiley & Sons, Ltd.     

Operability windows in viscoelastic slot coating flows using a simplified viscoelastic-capillary model

The viscoelastic-capillary model to predict approximately coating windows for the stable operations of viscoelastic coating liquids is derived using a lubrication approximation in slot coating processes. Pressure distributions and velocity profiles for viscoelastic liquids based on the Oldroyd-B and Phan-Thien and Tanner (PTT) models are solved in the coating bead region considering the Couette-Poiseuille flow feature and the pressure jumps at upstream and downstream menisci. Practical operating limits for the uniform coating of rheologically different liquids that are free from leaking and bead break-up defects are constructed under various conditions, incorporating the position of the upstream meniscus as an important indicator while determining limits. The shift of the uniform operating range shows different patterns for the Oldroyd-B liquid with a constant shear viscosity and the PTT liquid with a shear-thinning nature in comparison with the Newtonian case. The windows predicted by the simplified model are corroborated with experimental observations for one Newtonian and two viscoelastic liquids.     

Flow of a shear-thinning liquid in a cylindrical container with a rotating end wall

The flow patterns produced by rotating one end wall of a circular cylinder completely filled with a strongly shear-thinning viscoelastic liquid have been investigated using the laser-induced fluorescence flow visualization technique. An intense toroidal vortex is produced in the vicinity of the rotating end wall with outward spiraling flow over the end wall itself. This vortex drives a second countercirculating vortex of low intensity in the region of the stationary end wall. Under some circumstances an axial jet of fluid is observed moving away from the rotating end wall. This jet showed evidence of instability, whereas all flows were otherwise completely steady. The double-vortex structure is different from those recently observed in either a Newtonian or slightly shear-thinning liquid or in the low Reynolds number flow of an elastic liquid. There are, however, similarities with older work for a viscoelastic liquid at relatively high Reynolds numbers. The observations highlight the suitability of the cylinder/rotating end wall configuration as a sensitive test case for computational work.     

The propagation of vorticity in a viscoelastic fluid

A layer of constant vorticity exists in an infinite space of incompressible isotropic viscoelastic fluid for which the shear stress for rectilinear shearing flows depends linearly on the history of the velocity gradient. At some instant of time the forces maintaining this flow are removed. The subsequent time-dependent vorticity field is calculated explicitly in the particular cases when the fluid is Newtonian and when it is Maxwellian. The limiting case in which the vorticity layer becomes a vortex sheet is also calculated.     

Transient free‐surface flow of a viscoelastic fluid in a narrow channel

The interplay between inertia and elasticity is examined for transient free‐surface flow inside a narrow channel. The lubrication theory is extended for the flow of viscoelastic fluids of the Oldroyd‐B type (consisting of a Newtonian solvent and a polymeric solute). While the general formulation accounts for non‐linearities stemming from inertia effects in the momentum conservation equation, and the upper‐convected terms in the constitutive equation, only the front movement contributes to non‐linear coupling for a flow inside a straight channel. In this case, it is possible to implement a spectral representation in the depthwise direction for the velocity and stress. The evolution of the flow field is obtained locally, but the front movement is captured only in the mean sense. The influence of inertia, elasticity and viscosity ratio is examined for pressure‐induced flow. The front appears to progress monotonically with time. However, the velocity and stress exhibit typically a strong overshoot upon inception, accompanied by a plug‐flow behaviour in the channel core. The flow intensity eventually diminishes with time, tending asymptotically to Poiseuille conditions. For highly elastic liquids the front movement becomes oscillatory, experiencing strong deceleration periodically. A multiple‐scale solution is obtained for fluids with no inertia and small elasticity. Comparison with the exact (numerical) solution indicates a wide range of validity for the analytical result. Copyright © 2004 John Wiley & Sons, Ltd.     

Analytic results for viscoelastic flow in a porous tube

We consider the problem of an upper-convected Maxwell fluid that is injected into or sucked from a cylindrical tube with porous walls. Many salient features of the flow are revealed by solving for the stresses assuming Newtonian kinematics. It is thereby found that the stress components have eigensolutions that in suction are eliminated by boundary conditions imposed on the centerline, which is the upstream boundary for suction. For injection, the centerline is the downstream boundary; boundary conditions applied at the centerline then do not eliminate the eigensolutions. As a result, numerical integration of the differential equations starting at the centerline is stable for suction, unstable for injection. This conclusion applies whether or not Newtonian kinematics are assumed. It is shown here that the eigensolution in injection is eliminated, and the integration stabilized, if integration is started at a position of zero radial flow outside the tube wall, in the region of “pre-history”, and carried out in the direction of flow, towards the centerline of the tube. The solution obtained in this way shows steep stress gradients that are driven by a biaxial extensional near-singularity in the region of pre-history.     

Finite element computations of viscoelastic two-phase flows using local projection stabilization

A three-field local projection stabilized (LPS) finite element method is developed for computations of a three-dimensional axisymmetric buoyancy driven liquid drop rising in a liquid column where one of the liquid is viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier-Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian-Eulerian (ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. Interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. A one-level LPS based on an enriched approximation space and a discontinuous projection space is used to stabilize the numerical scheme. A comprehensive numerical investigation is performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor, and the Eötvös number on the drop dynamics are analyzed. The numerical study shows that beyond a critical Capillary number, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape and also exhibits a negative wake phenomena. However, a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape.     

On the fully developed tube flow of a class of non-linear viscoelastic fluids

The fully developed pipe flow of a class of non-linear viscoelastic fluids is investigated. Analytical expressions are derived for the stress components, the friction factor and the velocity field. The friction factor which depends on the Deborah and Reynolds numbers is substantially smaller than the corresponding value for the Newtonian flow field with implications concerning the volume flow rate. We show that non-affine models in the class of constitutive equations considered such as Johnson-Segalman and some versions of the Phan-Thien-Tanner models are not representative of physically realistic flow fields for all Deborah numbers. For a fixed value of the slippage factor they predict physically admissible flow fields only for a limited range of Deborah numbers smaller than a critical Deborah number. The latter is a function of the slippage.     

An experimental investigation of the flow about a sphere rotating in a rivlin-ericksen fluid

Tangential and radial velocity profiles were measured for the flow about a sphere rotating slowly in a Newtonian fluid, contained in a rectangular tank. Velocities were determined from enlarged streak photographs of aluminium particles moving in a collimated “sheet” of light, at several planes through the flow field. Similar velocity profiles were measured for the flow of a 1.50% Natrosol 250 H solution about two spheres of different diameters rotating in two different sized rectangular tanks. A set of velocity distributions were also measured for a sphere rotating in a 0.9% Natrosol 250 H solution. A dye tracer study of the flow about a sphere rotating in this liquid is presented as well. Both Natrosol solutions exhibited viscoelastic behaviour. The Newtonian fluid study was carried out at a Reynolds number of 1.2 and the viscoelastic fluid studies were within the Reynolds number range of 0.05–1.24.The zero shear viscosities of the Natrosol solutions were measured using the falling-sphere method. The non-Newtonian material parameters were obtained by fitting the theoretical curves to the measured velocity data. The values of the elastic and shear thinning parameters for the two liquids obtained in the different geometrical and dynamical situations are compared.     

A 2D boundary element method for simulating the deformation of axisymmetric compound non‐Newtonian drops

The boundary integral formulation of the solution to the Stokes equations is used to describe the deformation of small compound non‐Newtonian axisymmetric drops suspended in a Newtonian fluid that is subjected to an axisymmetric flow field. The non‐Newtonian stress is treated as a source term in the Stokes equations, which yields an extra integral over the domains containing non‐Newtonian material. By transforming the integral representation for the velocity to cylindrical co‐ordinates and performing the integration over the azimuthal direction analytically, the dimension of the problem can be reduced from three to two. A boundary element method for the remaining two‐dimensional problem aimed at the simulation of the deformation of such axisymmetric compound non‐Newtonian drops is developed. Apart from a numerical validation of the method, simulation results for a drop consisting of an Oldroyd‐B fluid and a viscoelastic material are presented. Moreover, the method is extended to compound drops that are composed of a viscous inner core encapsulated by a viscoelastic material. The simulation results for these drops are verified against theoretical results from literature. Moreover, it is shown that the method can be used to identify the dominant break‐up mechanism of compound drops in relation to the specific non‐Newtonian character of the membrane. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical simulation of non-linear elastic flows with a general collocated finite-volume method

This paper reports the development and application of a finite-volume based methodology for the calculation of the flow of fluids which follow differential viscoelastic constitutive models. The novelty of the method lies on the use of the non-staggered grid arrangement, in which all dependent variables are located at the center of the control volumes, thus greatly simplifying the adoption of general curvilinear coordinates. The pressure–velocity–stress decoupling was removed by the development of a new interpolation technique inspired on that of Rhie and Chow, AIAA 82 (1982) 998. The differencing schemes are second order accurate and the resulting algebraic equations for each variable are solved in a segregated way (decoupled scheme). The numerical formulation especially designed for the interpolation of the stress field was found to work well and is shown to be indispensable for accurate results. Calculations have been carried out for two problems: the entry flow problem of Eggleton et al., J. Non-Newtonian Fluid Mech. 64 (1996) 269, with orthogonal and non-orthogonal meshes; and the bounded and unbounded flows around a circular cylinder. The results of the simulations compare favourably with those in the literature and iterative convergence has been attained for Deborah and Reynolds numbers similar to, or higher than, those reported for identical flow problems using other numerical methods. The application of the method with non-orthogonal coordinates is demonstrated. The entry flow problem is studied in more detail and for this case differences between Newtonian and viscoelastic fluids are identified and discussed. Viscoelasticity is shown to be responsible for the development of very intense normal stresses, which are tensile in the wall region. As a consequence, the viscoelastic fluid is more intensely decelerated in the wall region than the Newtonian fluid, thus reducing locally the shear rates and the role of viscosity in redeveloping the flow. A layer of high stress-gradients is formed at the wall leading edge and is convected below and away from the wall; its effect is to intensify the aforementioned deviation of elastic fluid from the wall.     

Experimental and linear viscoelastic stress distribution measurements of high density polyethylene flowing into and within a slit

We report experimental measurements fo the centreline stress build up and relaxation as molten polyethelene flows into a slit die. The time-dependent extensional stress distribution is obtained using flow birefringence techniques and these observations complement the corresponding velocity field measurements already reported. Experimental measurements of the linear viscoelastic storage and loss modulus are obtained and, from these results, the polymers are characterized in terms of a modulus spectrum. Using this modulus spectrum together with a Maxwell-type constitutive equation and the experimental centreline kinematics, we find that it is possible to simulate successfully the experimentally observed stress distributions. Our results indicate that it is essential to include the polymers' broad spectrum of relaxation times when considering time dependent flow problems.     

Squeezing a viscoelastic fluid from a cone

The paper reports an exact kinematics for the squeezing flow from a cone of a general viscoelastic fluid. To obtain numerical values for the stresses, a network model that allows stress overshoot and shear-thinning in the start-up of a shear flow is adopted. Both these features are important in this flow. For the special case of an Oldroyd-B fluid it is shown that there is a limiting Weissenberg number above which at least one component of the stresses increases unboundedly with time.     

Numerical study on the dynamics of droplet passing through a cylinder obstruction in confined microchannel flow

The droplet dynamics passing through a cylinder obstruction was investigated with direct numerical simulations with FE-FTM (Finite Element-Front Tracking Method). The effect of droplet size and capillary number (Ca) was studied for both Newtonian and viscoelastic fluids. In the case of Newtonian droplet immersed in Newtonian medium, the droplet breakup induced by the geometric hindrance depends on the droplet size. As Ca increases, the short droplets (1.3 times longer than the channel width) break up while passing through the obstruction. However, the breakup does not occur for longer droplets (1.8 times longer than the channel width). When the viscoelastic fluid characterized by the Oldroyd-B model is considered, the Newtonian droplet immersed in viscoelastic medium breaks up into two smaller droplets while passing through the cylinder obstruction with increasing De_m (Deborah number of the medium). We also show that the normal stress difference plays a key role on the droplet breakup and the droplet extension. The normal stress difference is enhanced in the negative wake region due to the droplet flow, which also promotes droplet extension in that region. This numerical study provides information not only on underlying physics of the droplet flows passing through a cylinder obstruction but also on the useful guidelines for microfluidic applications.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.