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Thin liquid sheets are ubiquitous in many industrial processes, such as atomization and curtain coating. In the latter, the thickness of the deposited layer is limited by the breakup of the liquid curtain below a critical flow rate. The stability of a liquid sheet depends on the disturbance characteristics, sheet thickness and fluid properties. Experimental analyses have shown that thinner stable liquid curtains can be obtained with viscoelastic liquids; however, the underlining physical mechanisms associated with the increased stability are not fully understood. This work presents a numerical analysis of the effect of viscoelasticity on the stability of a thin liquid sheet. We first derive linear stability criteria for both planar and axisymmetric perturbations of Newtonian and Oldroyd-B liquids. The time evolution of planar and axisymmetric perturbations in an Oldroyd-B liquid sheet is evaluated using asymptotic expansion of the flow variables and a fully-implicit time integration scheme. The breakup time is calculated as a function of Deborah number and polymer to solvent viscosity ratio. The results show that the liquid rheological behavior does not change the linear stability criterion, however it has a strong effect on the growth rate of the disturbance and consequently on the breakup time.

The smaller the scales on which complex fluids are studied, the more fluctuations become relevant, e.g. in microrheology and nanofluidics. In this paper, a general approach is presented for including fluctuations in conformation-tensor based models for viscoelasticity, in accordance with the fluctuation-dissipation theorem. It is advocated to do this not for the conformation tensor itself, but rather for its so-called contravariant decomposition, in order to circumvent two major numerical complications. These are potential violation of the positive semi-definiteness of the conformation tensor, and numerical instabilities that occur even in the absence of fluctuations. Using the general procedure, fluctuating versions are derived for the upper-convected Maxwell model, the FENE-P model, and the Giesekus model. Finally, it is shown that the fluctuating viscoelasticity proposed here naturally reduces to the fluctuating Newtonian fluid dynamics of Landau and Lifshitz [L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6 of Course of Theoretical Physics, Pergamon Press, Oxford, 1959], in the limit of vanishingly small relaxation time.

The effect of viscoelastic forces on the hole growth speed in a liquid curtain and on the critical flow rate below which a liquid curtain breaks was studied by high-speed visualization. Different Boger-like aqueous solutions were used as the viscoelastic liquids. The expansion of a hole initiated within the liquid curtain leads to the curtain breakup. The results show a strong stabilizing effect of the viscoelastic forces; the minimum flow rate for stable curtain falls drastically by adding small amounts of high molecular weight polymer. The results reveal that the mechanisms for the increased stability are that the initiation of the hole is delayed and the speed at which it expands is also reduced, as the viscoelastic forces become stronger.

This study aims to investigate the elastic properties of blood for pulsatile arterial hemodynamics in terms of their effects on velocity, vorticity and stress fields, as well as the aneurysm rupture risk indicators. The blood flow in a developed fusiform aneurysm is numerically simulated using the modified Oldroyd–B viscoelastic fluid model which includes shear thinning effects, and the results are compared with Newtonian and purely shear thinning Carreau model predictions. Fluid-solid interaction method with arbitrary Lagrangian–Eulerian formulation is adopted in the simulations, where the arterial wall is modeled as a linearly elastic material. The viscoelastic fluid displays significant differences in the vortical field, where the rotation direction of the vortices is reversed, and additional secondary and tertiary vortices appear. The effects of increasing fluid elasticity on hemodynamics are further investigated comparing large and medium vessel geometries. As inertia is decreased viscoelastic effects become more pronounced for the medium artery model, and the difference in the velocity and vortical fields between the Newtonian and Carreau models are reduced, due to the decrease in shear thinning effects. For the modified Oldroyd–B model, an increase in the fluid's elasticity level, observed through the medium artery model, affects the vortical fields as the primary vortex grows and shifts towards the distal end, and the risk indicators increase compared to the large artery geometry and inelastic rheological models. Along the interface at the vessel wall, von Mises stresses are also observed to increase with fluid's elasticity. It is also shown that the hyperelastic vessel wall assumption leads to small differences in the flow structure and the risk indicators with respect to the linearly elastic vessel wall case for the modified Oldroyd–B model.

The flow dynamics of an upper-convected-Maxwell (UCM) falling film down a flexible vertical wall is studied in the limit of weak viscoelasticity. A set of Benney-like weakly nonlinear equations for the film thickness and wall deflection, which is valid for small flow rate, is derived based on the long-wave theory. It shows that the unstable role of liquid viscoelasticity is equivalent to that of the flow inertia. A set of asymptotic evolution equations valid for moderate flow rate is obtained based on the integral theory. The linear instability property of the system is examined by using a normal-mode analysis. It shows that the liquid viscoelasticity acts to destabilize the falling film even for the flow with inertia being negligible. The nonlinear evolution equations for the moderate flow rate are solved numerically. The spatio-temporal evolutions of the liquid–air interface and flexible wall are examined. It is concluded that the liquid viscoelasticity plays a role to strengthen the dispersion of the initial imposed perturbation. It can promote the traveling speed of the solitary-like humps and suppress the front-running ripples at the same time. Both the wall damping and wall tension acts to suppress the fluctuations of the flexible wall. However, they play different roles in the evolution of the liquid–air interface.

Roll-to-roll patterning of small-scale features on a rapidly moving web is an industrially important process with a wide array of commercial applications both old and new. Examples include magazine printing and more recently the pattering of flexible electronics. Among the many existing web coating techniques for large-scale fabrication, slit die and gravure coating are the most commonly used. In gravure coating, an engraved roller with a regular array of shallow cavities/cells is used to pick up fluid from a bath. It is then passed through a flexible doctoring blade in order to meter off excess fluid before printing the fluid onto a flexible substrate. Here we present an experimental investigation into the effect that viscoelasticity has on the dynamics of liquid transfer from an idealized gravure cell to a flat rigid substrate. Although the dynamics of the actual gravure coating process is quite complex, we chose to study a simplified process by imposing an extensional flow using a modified filament stretching rheometer in which one of the endplates is replaced by a cell containing a single truncated conical gravure cell. The deformation and stretching of the resulting liquid bridges, the motion of the contact line within the gravure cell and the total amount of fluid removed from the gravure cell are studied as a function of the imposed stretch rate, the fluid rheology, and the geometry of the gravure cell. Two different viscoelastic solutions of high molecular weight polyethylene oxide in water were studied and compared to a series of Newtonian fluids. The results show that the primary impact of viscoelasticity is the addition of an elastic stress which increases the tension along the liquid bridge and significantly increases the bridge lifetime. For stretches where the gravure cell was placed on the bottom and the top plate moved vertically, viscoelasticity was found to significantly reduce the amount of fluid transferred to the top plate. However, by placing the gravure cell on top and reversing the relative direction of the inertial and gravitational stresses, viscoelasticity was found to significantly increase the amount of fluid transferred. Increasing the stretch rate was found to amplify these observations. Finally, increasing the contact angle between the fluid and the gravure cell and decreasing the aspect ratio of the gravure cell were both found to increase the amount of fluid transferred.

This paper investigates the role of viscoelasticity on the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. There is much debate in the literature over the cause of the jump discontinuity, which is observed once the bubble exceeds a certain critical volume. In this paper, the employment of some choice modelling assumptions allows insights into the mechanisms of the jump discontinuity which cannot be ascertained experimentally. The ambient fluid is assumed incompressible and the flow irrotational, with viscoelastic effects included through the stress balance on the bubble surface. The governing equations are solved using the boundary element method. Some Newtonian predictions are discussed before investigating the role of viscoelasticity. The model predicts the trademark cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies.

Effects of drop and matrix viscoelasticity on the retraction of a sheared drop are numerically investigated. Retraction of an Oldroyd-B drop in a Newtonian matrix is initially faster and later slower with increasing drop Deborah number. The observed behavior is explained using an ordinary differential equation model representing the dominant balance between various forces during retraction. The initial faster relaxation of viscoelastic drops is due to viscoelastic stresses pulling the drop interface at the tips inward. The later slower retraction is due to the slowly-relaxing viscoelastic forces at the equator, where they act against the capillary force. The drop inclination decreases substantially during retraction unlike in a Newtonian case. Matrix viscoelasticity slows the relaxation of a Newtonian drop because of the increasingly slow relaxation of highly stretched polymers near the drop tip with increasing Deborah number. Increasing the ratio of polymeric to total viscosity further accentuates the viscoelastic effects in both cases. For an Oldroyd-B drop in an Oldroyd-B matrix, a competition between the dispersed and the continuous phase elasticities, represented by their ratio, determines the dynamics; larger values of the ratio leads again to initial faster and later slower retraction.

The influence of matrix and droplet viscoelasticity on the steady deformation and orientation of a single droplet subjected to simple shear is investigated microscopically. Experimental data are obtained in the velocity–vorticity and velocity–velocity gradient plane. A constant viscosity Boger fluid is used, as well as a shear-thinning viscoelastic fluid. These materials are described by means of an Oldroyd-B, Giesekus, Ellis, or multi-mode Giesekus constitutive equation. The drop-to-matrix viscosity ratio is 1.5. The numerical simulations in 3D are performed with a volume-of-fluid algorithm and focus on capillary numbers 0.15 and 0.35. In the case of a viscoelastic matrix, viscoelastic stress fields, computed at varying Deborah numbers, show maxima slightly above the drop tip at the back and below the tip at the front. At both capillary numbers, the simulations with the Oldroyd-B constitutive equation predict the experimentally observed phenomena that matrix viscoelasticity significantly suppresses droplet deformation and promotes droplet orientation. These two effects saturate experimentally at high Deborah numbers. Experimentally, the high Deborah numbers are achieved by decreasing the droplet radius with other parameters unchanged. At the higher capillary and Deborah numbers, the use of the Giesekus model with a small amount of shear-thinning dampens the stationary state deformation slightly and increases the angle of orientation. Droplet viscoelasticity on the other hand hardly affects the steady droplet deformation and orientation, both experimentally and numerically, even at moderate to high capillary and Deborah numbers.

The effect of viscoelasticity on the interfacial dynamics of air displacing a viscoelastic fluid under the presence of gravity, i.e., the dip coating flows is examined. A stabilized finite element method coupled with a pseudo-solid domain mapping technique is used to carry out the computations. The fluid is modeled by the Finitely Extensible Non-linear Elastic Chilton–Ralison (FENE-CR) constitutive equation. Simulations at various Ca and Bo are performed in order to determine the limiting condition for dip coating where the flow characteristics become independent of Bo. For all values of Ca and Bo studied, the flow is characterized by recirculation near the interface. To this end the film thickness scaled with the capillary length, as a function of Wi, at low Ca and high Bo collapses onto a single curve, and agrees with the analytical expression for the film thickness in the low Wi limit. As the value of Ca is increased, the corresponding value of Bo that is required to collapse the results onto a single curve, i.e., the dip coating flow limit, is correspondingly higher. For a fixed Ca and Wi, increasing Bo results in a decrease in the film thickness, an increase in the size of the recirculation region and an increase in the strain rates subsequently leading to an increase in the normal stresses. We show that the interfacial dynamics in the dip coating flow are qualitatively similar to those observed in the Hele-Shaw flow. Specifically, at low Wi, film thinning occurs and as the value of Wi is increased, the formation of normal elastic stress boundary layers in the capillary transition region is observed. This is accompanied by a sharp increase in the film thickness and a compression of the air–liquid interface in the capillary transition region.

We study the effect of viscoelasticity on the fluid dynamics of slot coating flow of dilute polymer solutions. The fluid is modeled by the Oldroyd-B and FENE-P equations in a conformation tensor formulation. The fully coupled equations of the flow are solved by the DEVSS-TG finite element method together with the elliptic domain mapping method to capture the unknown free surface. We observe that dilute solutions, where the presence of polymer molecules affects the flow field, behave qualitatively differently from ultra-dilute solutions, where the presence of polymer molecules does not alter the flow field. In dilute solutions: (1) the stagnation point on the free surface moves towards the static contact line and the recirculation zone shrinks with increasing Weissenberg number; (2) once the stagnation point reaches the static contact line, the hoop stress on the free surface changes sign from negative to positive, which destabilizes the flow; (3) the region of most severe polymer stretch and distortion also moves to the static contact line. The field variables, such as velocity, velocity gradient and conformation tensor, become singular due to a geometric singularity at the static contact line which leads to the failure of the computational method. In contrast, in ultra-dilute solutions, the computations fail when the mesh can not capture steep stress boundary layers at the free surface. The low-flow limit of inertialess slot coating flow is examined in terms of the Elastocapillary number; the coating window for uniform coating narrows as the liquid grows more elastic.

In linear viscoelasticity, an important consideration is the Boltzmann causal integral equation σ ( t ) = ∫ − ∞ t G ( t − τ ) γ ˙ ( τ ) d τ , which defines how the stress σ ( t ) at time t depends on the earlier history of the strain rate γ ˙ ( t ) via the relaxation modulus (kernel) G ( t ) . The Boltzmann model of linear viscoelasticity is an appropriate model for materials that simultaneously exhibit viscous and elastic behaviour. In the formulation of a Boltzmann model, the key consideration is the choice of the relaxation modulus. In a wide variety of applications, including the modelling of the glassy state of dense matter, polymer dynamics, and bone and muscle rheology, the Kohlrausch function has proved to be more appropriate in modelling the associated relaxation and decay processes than the standard exponential function. It is therefore an appropriate practical choice for the relaxation modulus. In this paper, the relaxation modulus is modelled as the sum of Kohlrausch functions. It is established theoretically how the unknown parameters in such models can be recovered using moments of the measured stress, and of the corresponding known applied strain rates. For a single Kohlrausch relaxation modulus, the proposed methodology is quite simple to implement. Three moments of both the measured stress and the applied strain-rate are evaluated numerically. A simple algorithm then determines the Kohlrausch parameters as defined by these three moments. Simulations are performed to assess and confirm the utility of the numerical performance of these two steps.

The primitive chain network model of entangled polymers, where primitive chains and entanglements are assigned in real 3-D space, is a possible step in the multi-scale path from atomistic simulations to macroscopic constitutive equations. In any coarse-grained model of a polymeric system, however, determination of the basic parameters is a crucial step. By fitting the linear viscoelastic response of several melts of monodisperse linear polymers taken from the literature, we here determine the corresponding basic parameters of the primitive chain network model. We concentrate in particular on the value of the mean molar mass between entanglements, and on how such a mass is related to plateau modulus from our simulations. We consistently find values that fall around 60% of the entanglement molecular weight as obtained from Ferry's definition, and we envisage that such a result is due to Brownian fluctuations of the entanglement nodes.

A pseudo-solid domain mapping technique, based on a finite strain model, coupled with a DEVSS finite element formulation is applied to study the effects of viscoelasticity on free surface displacement flows. The flow type analyzed is the displacement of a dilute polymeric solution, modeled by a FENE-CR constitutive equation, in a Hele Shaw Cell. Our study indicates the presence of two distinct flow regimes. In the absence of gravity, a recirculation flow at low Ca (Ca 1.0), are observed. In the recirculation flow we observe the formation of elastic stress boundary layers in the capillary transition region (the thickness of which decrease with increasing elasticity), an increase in the hydrodynamic film thickness with increasing elasticity and a meniscus invasion when the stresses in the elastic boundary layer become very large. We have also qualitatively studied the effect of channel wall divergence on the flow dynamics and ascertained that the elastic stress boundary layer formation is a purely local phenomenon and largely independent of geometric considerations. For the bypass flow, in addition to the elastic stress boundary layer in the capillary transition region, an additional elastic stress boundary layer near the bubble tip is observed. As the Wi is increased, film thickening is observed but no meniscus invasion occurs. We have also studied the effect of gravity on both flows. Addition of gravity in recirculation flow results in a film thinning effect and an increase in the normal stresses in the capillary transition regime. For the bypass flow, even a small amount of gravity introduces recirculation. Finally, we discuss how both flows are elastically unstable and plausible mechanisms for the onset of instability are suggested.

A Hamiltonian framework of non-equilibrium thermodynamics is adopted to construct a set of dynamical continuum equations for a polymer blend with matrix viscoelasticity and a narrow droplet size distribution that is assumed to obey a Weibull distribution function. The microstructure of the matrix is described in terms of a conformation tensor. The variable droplet distribution is described in terms of two thermodynamic variables: the droplet shape tensor and the number density of representative droplets. A Hamiltonian functional in terms of the thermodynamic variables is introduced and a set of time evolution equations for the system variables is derived. Sample calculations for homogenous flows and constant droplet distribution are compared with data of a PIB/PDMS blend and a HPC/PDMS blend with high viscoelastic contrast. For the PIB/PDMS blend, satisfactory predictions of the flow curves are obtained. Sample calculations for a blend with variable droplet distribution are performed and the effect of flow on the rheology, droplet morphology, and on the droplet distribution are discussed. It is found that deformation can increase or decrease the dispersity of the droplet morphology for the flows investigated herein.

The effect of fluid viscoelasticity on the draw resonance dynamics of melt spinning has been examined using White-Metzner and Phan Thien-Tanner fluid models into the governing equations of the process, in a continued effort to study the effects of process conditions and material properties on draw resonance, following up the earlier study [J. Non-Newtonian Fluid Mech. 87 (1999) 165] dealing with the effect of spinline cooling on the same draw resonance. Whether or not the fluid viscoelasticity stabilizes melt spinning has turned out to coincide with whether or not the spinline tension sensitivity decreases with the increasing fluid viscoelasticity. This is because the spinning stability is always enhanced by a decrease in tension sensitivity to process disturbances and this tension sensitivity was then found in the said earlier study to be moving opposite to the level of the spinline tension: the higher spinline tension, the smaller tension sensitivity. It has been found in the present study that the effect of fluid viscoelasticity on spinning stability can be classified into two diametrically different kinds: for extension-thickening fluids an increase in viscoelasticity increases tension, decreases tension sensitivity and, thus, stabilizes the spinning, whereas it decreases tension, increases tension sensitivity and, thus, destabilizes the spinning of extension-thinning fluids.

We study the behaviour of a single integral constitutive equation, capable of providing analytic expressions for the viscoelastic stress in extensional flows of a variety of deformation histories and geometries, ranging from uniaxial to equibiaxial. It is based on the use of a stress damping function, with a power-law dependence on the elongation, λ: h(λ) = 1/λ n . The parameter n (0 ≤ n ≤ 2) signifies the nonlinear viscoelastic character of the material and, therefore, is an inverse measure of network connectivity strength of the underlying microstructure. This renders the constitutive approach applicable to incompressible polymers of a variable degree of branching, strain hardening and stress thinning behavior. Methods of connecting n with the macromolecular architecture and the alignment strength of the flow are also explored. © 2006 Elsevier B.V. All rights reserved.

Capillary instability of a liquid cylinder can arise when either the interior or exterior fluid is a gas of negligible density and viscosity. The shear stress must vanish at the gas-liquid interface but it does not vanish in irrotational flows. Joseph and Wang [D.D. Joseph, J. Wang, The dissipation approximation and viscous potential flow, J. Fluid Mech. 505 (2004) 365] derived an additional viscous correction to the irrotational pressure. They argued that this pressure arises in a boundary layer induced by the unphysical discontinuity of the shear stress. Wang et al. [J. Wang, D.D. Joseph, T. Funada, Pressure correction for potential flow analysis of capillary instability of viscous fluids, J. Fluid Mech. 522 (2005) 383] showed that the dispersion relation for capillary instability in the Newtonian case is almost indistinguishable from the exact solution when the additional pressure contribution is included in the irrotational theory. Here we extend the formulation for the additional pressure to potential flows of viscoelastic fluids in flows governed by linearized equations, and apply this additional pressure to capillary instability of viscoelastic liquid filaments of Jeffreys type. The shear stress at the gas–liquid interface cannot be made to vanish in an irrotational theory, but the explicit effect of this uncompensated shear stress can be removed from the global equation for the evolution of the energy of disturbances. This line of thought allows us to present the additional pressure theory without appeal to boundary layers. The validity of this purely irrotational theory can be judged by comparison with the exact solutions of Navier–Stokes equations. Here we show that our purely irrotational theory is in remarkably good agreement with the exact solution in linear analysis of the capillary instability of a viscoelastic liquid cylinder. © 2005 Elsevier B.V. All rights reserved.

Particles suspended in viscoelastic media are known to develop string-like structures when being sheared at sufficiently high shear rates. The occurrence of these strings has been attributed to normal stress effects, and it has been suggested that a critical Weissenberg number of 10 governs their appearance. This assumption is tested here, using suspensions of monodisperse polystyrene spheres. The level of viscosity and elasticity, as well as the degree of shear thinning of the suspending fluid, have been varied over a wide range of values. The critical Weissenberg number for the onset of string formation, as detected by microscopy, was not constant but ranged from 0.5 to 16.5 in contrast with literature results. Suspensions in highly elastic Boger fluids did not display any alignment at all, even at Weissenberg numbers as high as 260. Small angle light scattering (SALS) was used to calculate an alignment factor. It provides an objective and quantitative measure for the degree of alignment of the particles. It also makes it possible, in some cases, to determine precisely the interparticle spacing in the strings. The kinetics of string formation can be studied as well, they seem to be strain-controlled. SALS experiments with different gap spacings have been used to demonstrate that the string formation occurs in the bulk of the liquid rather than at the walls of the cell.

One-to-one molar ratio sodium salicylate (5 mM)/Arquad S-50 (5 mM) solution was investigated in comparison to the sodium salicylate (12.5 mM)/Arquad S-50 (5 mM) or 2.5:1 solution, which was previously reported to be drag reducing yet non-viscoelastic [9] . Results showed a dramatic influence of the counterion to surfactant concentration ratio on the viscoelasticity of the surfactant drag reducing systems. The 1:1 solution has normal surfactant drag reducer characteristics such as viscoelastic properties and thread-like micellar networks and branches. This is the first cationic surfactant/counterion drag reducing system found, for which excess counterion makes the solution non-viscoelastic, while the microstructure of thread-like micellar networks is apparently unchanged.

In a companion paper [Macromolecules 35 (2002) 2689], we have addressed the direct problem of predicting the linear viscoelastic response of entangled linear polymers from their molecular weight distribution, using a suitable model inspired by reptation theory. By comparing the theoretical results to experimental data for a variety of samples of different nature (polystyrene, polycarbonate, high-density polyethylene) and distribution (monomodal and multimodal), we found that the time-dependent diffusion reptation model proposed by des Cloizeaux, suitably modified to treat short chains and to include Rouse processes, is capable of quantitative predictions. In the present paper, we use the modified des Cloizeaux model to address the inverse problem of predicting molecular weight distribution from dynamic moduli. A parametric approach is implemented to deal with the ill-posedness of the problem. Results are given for most of the samples studied in [Macromolecules 35 (2002) 2689]. They are in quantitative agreement with size-exclusion chromatography data. In particular, the proposed methodology is able to resolve small amounts of short chains in bimodal blends containing large amounts of long chains. (C) 2002 Elsevier Science B.V. All rights reserved.

The Doi–Edwards (D–E) theory for entangled polymer dynamics is extended to account for polymer chain stretching in fast flows. Our analysis is based on a new estimate for the non-equilibrium size of the tube under a macroscopic deformation and yields an optimum primitive chain contour length E · u 1/2 times greater than the equilibrium (no-flow) value. This finding implies that polymer chains subjected to nonlinear step strain will only retract partially and should therefore maintain significant chain stretch up to the time the primitive chain escapes the deformed tube. Primitive chain retraction is also shown to reduce the number of entanglements between a deformed polymer chain and its neighbors, yielding significant enhancement of molecular relaxation at high deformations. Various steady-state and transient stress predictions based on this partial strand extension concept are compared with existing experimental data.

Large amplitude oscillatory shear (LAOS) and exponential shear can be used to study the nonlinear viscoelastic properties of molten polymers. Using a kinetic network theory, the sensitivity of the nonlinear viscoelasticity of a polymer melt to small changes in the linear spectrum was calculated and compared with the sensitivity of the linear response. In LAOS, the strain amplitude and phase angle of the first harmonic were no more sensitive to changes in the relaxation time than the linear counterparts. The sensitivity of the amplitudes and phase angles of all the harmonics to changes in the linear spectrum decreases with strain amplitude. In exponential shear, the sensitivity of the maximum of the exponential shear viscosity to small changes in the relaxation time was found to decrease with increasing strain rate. However, increasing strain amplitude (in LAOS) and strain rate (in exponential shear) were both found to amplify subtle differences in the kinetic parameters of the model. Thus large shearing deformations do not significantly amplify subtle changes in the relaxation spectrum, but they do amplify subtle differences in the kinetic rate constants.

It is shown that motions with zero acceleration in incompressible materials consist of velocity fields which are steady or unsteady in the Eulerian sense. Moreover, in these motions all Rivlin-Erickson tensors of order three and higher are zero. Consequently, it is shown that there are non-viscometric flows in which Rivlin-Ericksen tensors of order three and higher are all zero. An example of a homogeneous motion with an unsteady velocity field is given to highlight this fact and a general method to generate such flows, which may be non-homogeneous, is also indicated.

A macroscopic continuum mechanical model for incompressible side chain nematic polymers, under isothermal conditions is given. The model is a synthesis of a transient network model and the standard nematorheological model. Simplifications in the model yield constitutive equations that are identical to well-known rheological models for polymer melts and for low-molar-mass nematics. The model is used to derive the anisotropic relaxation modulus, and the orientation-dependent storage and loss moduli for small-amplitude oscillatory shear. The model is partially validated with existing small-amplitude oscillatory shear data. The present model is able to predict and explain the observed differences in the storage and loss moduli between the isotropic and shear flow-aligned nematic phase of a side chain nematic polymer.

A modified Stokesian dynamics technique is used in an oscillatory shear fleld to obtain the elastic and storage modulus of the suspension of spherical particles in a Newtonian liquid. Electrostatic or steric repulsive forces between suspended particles result in a viscoelastic behavior of a concentrated suspension. The mechanism for energy storage is a particle microstructure phenomenon: the compression of the electrostatic potential by hydrodynamic forces stores energy.

A strain-coupling integral constitutive equation for non-linear viscoelasticity is evaluated using double-step shear strain experiments. Evaluation of the functions of the theory is discussed, and it is shown that the theory provides satisfactory descriptions of strain and time variations for doublestep stress relaxation experiments.

This paper investigates the geophysically relevant problem of thermal convection in a Maxwell medium. The case of convection in a 2-dimensional cavity heated from below is solved numerically for infinite Prandtl number and constant material parameters. Within a finite difference context, different approximation schemes for the advective terms in the material equations are explored. Using an upwind approximation, solutions for an upper convected Maxwell model up to Deborah numbers De ≈ 1.0 are possible, where the vigour of the convective flow is reduced to approximately 50% of its Newtonian equivalent. At this De the normal stress pattern seems to become singular at the boundaries. For De < 0.5 it is shown that the results are mainly controlled by the choice of the free parameters, a, of a Johnson-Segalman model.

An experimental study investigated the effects of glass fiber suspensions on the onset of instability in the non-Newtonian fluid flow in the concentric spherical gaps between an inner rotating hemisphere and an outer stationary whole sphere. Glass fibers with different aspect ratios were mixed with a macromolecule polymeric solution to obtain different suspension fluids. For comparison, the pure macromolecule polymeric fluid was also investigated. The torques on the inner sphere were measured for various spherical gaps and various rotational Reynolds numbers. The onset of instability of the polymeric fluid flows was delayed by adding glass fibers to the polymeric solution for all tested gap ratios.

The interaction of fluid yield stresses and viscoelasticity upon the Weisenberg effect has been investigated in several ways. Firstly, a simple theoretical model for the Weissenberg effect has been developed (assuming the Weissenberg hypothesis) and used to predict surface profiles for solutions without yield stresses and for solutions of similar rheologies but with yield stresses superimposed. Secondly, the surface profiles for pairs of polymer solutions were observed and compared. Each comparison consisted of two solutions of similar experimentally determined rheologies except that one solution exhibited a yield stress whereas the other did not. Thirdly, the model for the Weissenberg effect was used to predict the surface profiles for these pairs of solutions using their experimentally determined rheologies. The results of each of these analyses were in agreement, i.e., the Weissenberg effect is suppressed when yield stresses and stagnant fluid regions are present. In many processes, for example fluid mixing, it is important to know whether a fluid is viscoelastic or not and the Weissenberg effect is an indication of this viscoelasticity. The effects of stirred tank scale-up criteria upon the presence of a noticeable Weissenberg effect around a rotating impeller shaft are discussed.

It is shown that the Bueche-Ferry hypothesis that polymers obey Rouse dynamics on a short time-scale holds quite generally for polymeric liquids both in the linear and the non-linear regimes of viscoelasticity. The hypothesis is shown to be connected with the principle of virtual work in simple fluid theory, and with the stress optical law. Based on these arguments, a general form for the stress optical law is presented.

The rheological equation of state derived in Part I on the basis of relaxation equations of chain dynamics is analyzed for the steady and transient shear and uniaxial elongational flows of monodisperse polymers. The effect of superslow relaxation processes associated with basic macro-molecular motions that occur on a characteristic scale essentially greater than the so-called distance between entanglements was investigated in these flows. It is shown that the relaxation times in the region of linear and non-linear viscoelasticity are self-consistent. Theoretical predictions are in good agreement with the experimental data for melts of nearly monodisperse flexible polymers.

The viscoelastic properties of a polymer solution vary with both the frequency and the amplitude of the oscillatory shear rate. In steady flow the viscosity is non-Newtonian, changing with the shear rate. A rheological model is developed which describes the interrelationship between these properties. This model is based upon multiple relaxation processes ordered as in a generalized Maxwell model. However, these processes are truncated at higher shear rates in a manner dependent upon the product of the shear rate and the relaxation time. Both the steady flow and the oscillatory flow properties are controlled by the same relaxation processes. It is shown that the critical condition marking the onset of shear rate dependence of the viscoelasticity is that the shear rate is equal to either the reciprocal of the longest relaxation time or the radian frequency, whichever is larger.

Based on the Doi-Edwards theory and the primitive chain length fluctuation effect similar to those proposed by Doi, a general linear viscoelastic theory has been developed for monodisperse flexible linear polymers. It contains four dynamic modes: Rouse motion between two adjacent entanglement points, (μ A ( t )), chain slippage through entanglement links (μ X ( t )), primitive chain length fluctuation (μ B ( t )) and reptational motion correction for the chain length fluctuation effect (μ C ( t )). An additional dynamic process, chain tension relaxation, μ * B ( t, E ), as described by Doi exists in the non-linear regime. The measured stress relaxation moduli ( G ( t )) in both linear and non-linear regimes are compared as analysed by the theories. The comparison sheds light on how the polymer molecular dynamics change from the linear regime to the non-linear regime in affecting the polymer's viscoelastic properties. As a logical consequence of the very good quantitative G ( t ) line shape analysis over an extensive range of MW, the proposed general linear viscoelastic theory explains the MW dependence of the zero-shear viscosity (η 0 ∝ M 3.4 for M > M C and η 0 ∝ M for M M C ) and the steady-state compliance J e . A constitutive equation containing both μ * B ( t, E ) and μ C ( t ) processes has been obtained. When MW is greater than ≈ 10 M e , a maximum followed by a dip starts to appear on the flow curve calculated from the constitutive equation. It is shown by comparing theory with experiment that the maximum point corresponds to the onset of the slip—stick melt flow instability.

The intrinsic spinnability of a fluid is herein defined in terms of the asymptotic behavior of the velocity profile of a spin-threadline at high elongation rates under such conditions that the profile is determined solely by the material rheology. Spinnability can be categorized as draw-ratio-limited, threadline length-limited, residence-time-limited and unlimited; this indexing is linked to the dependence of the spinning viscosity upon strain rate, thinning materials being the less spinable. A method for deriving bounds on spinnability directly from the constitutive equation for single-relaxation-time integral models is presented, and, as an example, the lower-convected Maxwell model is shown to have length-limited spinnability.

We classify and derive a canonical form for a linearization of the upper convected Maxwell model for steady viscoelastic fluid flow. We show that there are always two families of characteristic curves which follow the streamlines and across which the solution may be discontinuous. If the product of the Weissenberg number and the Reynolds number is greater than one, then two additional families of characteristic curves exist.

Rheological equations of state for the concentrated solutions and melts of high polymers are derived by applying a structural approach. The dynamics of a macromolecule are considered on the basis of the fundamental model of the polymer chain, e.g., the bead-spring model. The drag forces describing correlations of motion macromolecules are determined by means of the relaxation equations. The oscillatory shearing flow of the melts is studied on the basis of the equations derived. Expressions for the dynamic modulus and relaxation times are determined. As can be judged from the form of the dependence of the dynamic modulus on frequency, the relaxation time distribution is the same as in real materials. The relaxation spectrum of high polymers has a terminal zone with abnormally long relaxation times.

A numerical study on the impingement and spreading of an isothermal viscoelastic droplet on a solid surface is presented in this work. The time-dependent incompressible Navier–Stokes equations are used to describe the fluid flow in the liquid droplet, whereas the viscoelasticity in the moving droplet is described by the Giesekus constitutive equation. A finite element scheme with the arbitrary Lagrangian–Eulerian (ALE) approach is proposed to solve the coupled time-dependent incompressible Navier–Stokes equation and the Giesekus constitutive equation in a time-dependent domain. In addition, a three-field formulation based on the Local Projection Stabilization (LPS) is used in the numerical scheme. The stabilized scheme allows us to use equal order interpolation spaces for the velocity and the viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and the pressure. The coupled system is solved by a monolithic approach in a 3D-axisymmetric configuration. In addition to the mesh convergence study, parametric studies of the Weissenberg number, Newtonian solvent ratio, polymeric viscosity, Reynolds number and the equilibrium contact angle are performed to demonstrate the effects of viscoelasticity on the flow dynamics of the droplet on wetting surfaces. The numerical study shows that the wetting diameter of the droplet increases with an increase in the viscoelasticity of the fluid. Further, the viscoelastic effects during the spreading process increases with an increase in the Reynolds number. Moreover, the viscoelastic effects on the flow dynamics are not influenced by the equilibrium contact angle.

Direct numerical simulations (DNS) of Newtonian and drag-reducing viscoelastic turbulent channel flows generate crucial data to analyze and understand the nature of turbulence and its modification due to viscoelasticity. Such an understanding is necessary to enable better drag reduction technologies. Previously (Samanta et al., Phys. Fluids, 21, 115106, 2009) we have shown that viscoelasticity can significantly alter the non-Gaussian character of the probability density functions (PDFs) of many of the turbulent statistics in the flow. In this work, we investigate further the non-Gaussian characteristics of the PDFs generated in viscoelastic turbulence by focusing on their tail behavior. In particular, we show that many of the tails of the PDFs of the velocity and its derivatives obey a power law, as opposed to an exponential one exhibited by Gaussian PDFs, with power law indices that sometimes, for viscoelastic flows, are low enough to imply unbounded fourth- (kurtosis), or even third-(skewness) moments of the distribution, what we can call “strong fat tails” behavior. We use Hill's estimator to evaluate the power law index of the tails, as well as statistical inference methods to perform hypothesis testing for the appropriateness of the power law fit. We have found that fat tails corresponding to power law tail distribution behavior are observed in the PDFs for both Newtonian and viscoelastic cases. However, viscoelasticity leads to PDFs that exhibit consistently fatter tails than the Newtonian ones and it is only in the presence of viscoelasticity that turbulence leads to “strong fat tails” as defined above. We infer that drag-reducing viscoelastic turbulence involves more infrequent and more correlated and more extreme events which makes DNS of viscoelastic turbulent flows much more computationally demanding than Newtonian DNS, requiring finer spatial resolutions, longer integration times, and larger computational domain sizes.

We study the stability of pressure-driven channel flow of a thixotropic-viscoelasto-plastic fluid. Several recent experiments have shown that channel flows of shear-thinning polymer solutions can be linearly unstable even at very low Reynolds numbers. We use the Bautista–Manero–Puig model (BMP) to attempt to capture the physics of these instabilities. We obtain an analytic solution for the steady-state velocity profile, dependent on our fluid parameters, that is able to predict a large variety of base states. We derive dimensionless groups to compare the effects of viscoelasticity, thixotropy and plasticity on the flow stability. We find that sinuous perturbations are slightly more unstable than varicose modes, and we identify the values of our model parameters for which the instability has its strongest effects. We conclude that dominant thixotropy can stabilise the flow, but instability occurs when the characteristic timescale of viscoelasticity is much longer than both those of plasticity and thixotropy. The most dangerous situation is a plug flow with an apparent yield surface just within the channel, and we find that the growth rate of the instability scales with the rate of thixotropic structure recovery.

A stabilized finite element scheme is developed for computations of buoyancy driven 3D-axisymmetric viscoelastic two-phase flows with insoluble surfactants. The numerical scheme solves the Navier-Stokes equations for the fluid flow, Giesekus constitutive equation for the effects of viscoelasticity and simultaneously an evolution equation for the surfactant concentration on the interface. The interface is tracked by the coupled arbitrary Lagrangian-Eulerian (ALE) and Lagrangian approach. The interface-resolved moving meshes allow accurate incorporation of the interfacial tension force, Marangoni forces and the jumps in the material properties. Further, the tangential gradient operator technique is used to handle the curvature approximation in a semi-implicit manner. An one-level Local Projection Stabilization (LPS), which is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh is used to stabilize the model equations. The stabilized numerical scheme allows us to use isoparametric second order conforming finite elements enriched with cubic bubble functions for velocity and viscoelastic stress, second order finite elements for surfactant concentration and discontinuous first order finite element for pressure. A number of computations are performed for a Newtonian drop rising in a viscoelastic fluid column and a viscoelastic drop rising in a Newtonian fluid column with insoluble surfactants on the interface. The influence of the Marangoni number, initial surfactant concentration and Peclet number on the dynamics of the rising drop are analyzed. The numerical study shows that a viscoelastic drop rising in a Newtonian fluid column develops an indentation around the rear stagnation point with a dimpled shape without insoluble surfactants. The presence of insoluble surfactants forces the drop to rise slowly but the drop at the tail end is pulled up more. However, a Newtonian drop rising in a viscoelastic fluid column experiences an extended trailing edge with a cusp-like shape without insoluble surfactants. The presence of surfactants pulls the tail end of the drop up slightly and makes the tail flatter with/without small undulations depending on the magnitude of the surfactant concentrations.

A viscoelastic model for cellulose nanofiber (CNF) suspensions using a population balance equation for fiber floc aggregation-breakage was proposed. A Krieger–Dougherty model was used to describe the dependence of viscosity on the volume fraction of flocs, and a White–Metzner type model was used to represent the effect of viscoelasticity. The rheological properties of the present model were investigated, and this model was confirmed to describe shear thinning in viscosity and viscoelastic responses of stress growth, depending on the relaxation time. Furthermore, flows of CNF suspensions through a circular tube were analyzed using the present model. The velocity profile changes with time according to the temporal change in the distribution of the effective volume fraction of fiber flocs. The numerical prediction of the velocity profile captures the typical properties of shear-thinning fluids. The development rate of the floc size distribution depends on the velocity gradient and affects the temporal changes of both velocity and stress fields. The development of the stress field is delayed longer for a longer relaxation time. These phenomena affect the temporal change in the floc size distribution and hence also affect the growth of the velocity field through the effective volume fraction of flocs. These results indicated that the present model can describe the viscoelastic behavior of CNF suspensions in flows.

In this paper the small-amplitude oscillation and linear instability of an electrically charged liquid droplet suspended in a vacuum is studied. The liquid is assumed to be viscoelastic and leaky dielectric. The problem is linearized and a characteristic equation is derived. The complex frequency is solved numerically as a function of the relevant dimensionless parameters. The effects of surface charge and the electrical properties of the liquid on the oscillation and instability characteristics of the droplet are investigated for several cases of viscoelasticity and the quadrupole mode n = 2 . In the case of large or small elasticities, increasing the charge level leads to a decrease in the critical Ohnesorge number at which a supercritical bifurcation occurs and narrows down the interval of the Ohnesorge number for oscillation. For large viscosities, oscillation can be induced by elasticity when the relative stress relaxation time is between 0.004 and 10 approximately, and the electric field may diminish this upper limit to a certain extent. The existence of the electric field also gives rise to new aperiodic branches first appearing at large viscosities or elasticities with smaller damping rates. The effect of finite conductivity on the oscillation behavior of the droplet is rather complicated. Most significantly, when the electrical relaxation time is of the order of the capillary time or a little smaller, new supercritical and subcritical bifurcations with new intervals that favor the appearance of oscillation occur at moderate/relatively large viscosities or elasticities. In addition, it is found that the Rayleigh limit expressed in the form χ = n + 2 where χ the electrical Bond number and n is the order of the mode is identical with that for a perfectly conducting, inviscid droplet. Above the Rayleigh limit, the viscoelastic droplet becomes unstable, and high-order modes may be dominant over the quadrupole one when the charge level is high enough.

A phenomenological constitutive model that accounts for thixotropy, viscoelasticity and yielding behavior is presented. It uses the fluidity to specify the microscopic state. The model is composed of two differential equations, one tensorial equation that relates the stress to the rate of strain and one scalar evolution equation for the fluidity. The equation for stress is a modified version of the Oldroyd-B model in which the relaxation and retardation times are functions of the fluidity. In contrast to the existing phenomenological models for thixotropic materials, the evolution equation that describes the microscopic state only involves material functions that are directly measurable by means of standard rheological tests.

The dynamics of streaky structures in a viscoelastic pipe flow of FENE-P fluids is studied numerically. The values of Weissenberg number ( W i , polymer relaxation time over flow turn-over time) and viscosity ratios are varied to evaluate the effects of viscoelasticity in the pipe. For Reynolds numbers 3000 ∼ 3500 (based on pipe radius and maximum laminar velocity) near the transition boundary, we select finite-amplitude two-dimensional rolls as initial conditions and add infinitesimal three-dimensional perturbations to trigger the transition. The 2D streamwise rolls are highly unstable to small 3D disturbances, which helps to better understand the evolution of large scale structures. The stronger temporal intermittency in viscoelastic flows makes it more difficult to identify a successful transition; we found that quantities directly related to viscoelasticity can serve as better indicators of transition to turbulence. The role of polymer stress is studied through the kinetic energy budget and it is found that polymer torque opposes the vorticity and become more dominant for higher W i , consistent with previous studies. The results show that the secondary instability of the streaky structures can be enhanced by viscoelasticity at low W i , while at high W i viscoelasticity delays the transition to turbulence, sometimes leading to relaminarization.

Lubricants can exhibit significant viscoelastic effects due to the addition of high molecular weight polymers. The overall behavior of the mixture is vastly different from a simpler Newtonian fluid. Therefore, understating the influence of viscoelasticity on the load carrying capacity of the film is essential for lubricated contacts. A new modeling technique based on lubrication theory is proposed to take into account viscoelastic effects. As a result, we obtain a modified equation for the pressure, i.e. the viscoelastic Reynolds (VR) equation. We have first examined a parabolic slider to mimic a roller bearing configuration. An increase of the load carrying capacity is observed when polymers are added to the lubricant. Furthermore, our results are compared with existing models based on the lubrication approximation and direct numerical simulations (DNS). For small Weissenberg number ( W i ), i.e. the ratio between the polymer relaxation time and the residence time scale, VR predicts the same pressure of the linearized model, in which ϵ W i is the perturbation parameter ( ϵ is the ratio between the vertical length scale and the horizontal length scale). However, the difference grows rapidly as viscoelastic effects become stronger. Excellent quantitative and qualitative agreement is observed between DNS and our model over small to moderate Weissenberg number. While DNS is numerically unstable at high values of the Weissenberg number, VR does not have the same issue allowing to capture the evolution of the stress and pressure also when the viscoelastic effects are strong. It is shown that even in high shear flows, normal stresses have the largest impact on load carrying capacity and thus cannot be neglected. Furthermore, the additional pressure due to viscoelasticity comprises two components, the first one due to the normal stress and the second one due to the shear stress. Afterwards, the methodology used for the parabolic slider is extended to a plane slider where, instead, the load decreases by adding polymers to the fluid. In particular, under the effect of the polymers surface slopes enhance the rate at which pressure gradients increase, whereas curvature opposes this along the contact. Therefore, the increase of the load carrying capacity observed for viscoelastic lubricants is due to its shape close to the inlet, which is steeper than the plane slider.