Upper-convected Maxwell Constitutive Equation Research Articles

Centrifugal spinning of polymeric solutions: Experiments and modelling

We assess some of the main parameters affecting the performance of Centrifugal Spinning (CS), a method for making nanofibres, using laboratory experiments and modelling. Rheologically characterized polymer solutions of various concentrations are propelled into curved jet trajectories by an in-house CS device and they are visualized via high speed imaging, while final dried fibre morphology diameter distributions are characterized by scanning electron microscopy. Our experiments provide validations of a comprehensive string model, based on the viscoelastic upper convected Maxwell (UCM) constitutive equation for the polymer, allowing us to examine dependence of the CS process on the key dimensionless numbers. We experimentally analyze the effects of the polymer concentration, rotation speed, nozzle diameter, spinneret radius and nozzle angle, and correspondingly use the model to analyze the effects of the Weissenberg number (Wi), Rossby number (Rb), nozzle diameter ratio (ϵa0), spinneret radius ratio (ϵs0), and nozzle angle (α0) on the fibre trajectory and radius. Finally, we use a simple scaling analysis to classify the flow regimes into stable and unstable with respect to jet instabilities and breakup. The results of this analysis agree with the experiments in finding fibres stable at large capillary and Weissenberg numbers.

Viscoelastic fluids with pressure-dependent viscosity; exact analytical solutions and their singularities in Poiseuille flows

We study the pressure-driven unidirectional flow of a viscoelastic fluid with pressure-dependent viscosity in a straight channel and a circular tube. The upper convected Maxwell (UCM) constitutive equation is utilized to model the response of the viscoelastic fluid under deformation. The starting point of the analysis is based on the second-order, symmetric conformation tensor. As such, the final model is slightly different from previous models in terms of the extra-stress tensor as the primary variable. The solution of the governing equations is found analytically for all the dependent variables except for a constant which is introduced by the separation of variables method. A physical singular point of the solution is also revealed explicitly. The singularity is taken into account in order to construct very accurate high-order asymptotic solutions for the unknown constant. Further reprocessing of the asymptotic solutions by means of non-linear techniques which accelerate the convergence of series, result in more accurate analytical expressions even close to the singularity of the solutions.

Research on the Hydrodynamics of the Mechanism of Viscous-Elastic Fluids Displacing Residual Oil Droplets in Micro Pores

In order to analyze the microscopic stress field acting on residual oil droplets in micro pores, calculate its deformation, and explore the hydrodynamic mechanism of viscous-elastic fluids displacing oil droplets, the viscous-elastic fluid flow equations in micro pores are established by choosing the Upper Convected Maxwell constitutive equation; the numerical solutions of the flow field are obtained by volume control and Alternate Direction Implicit methods. From the above, the velocity field and microscopic stress field; the forces acting on residual oil droplets; the deformations of residual oil droplets by various viscous-elastic displacing fluids and at various Wiesenberg numbers are calculated and analyzed. The result demonstrated that both the normal stress and horizontal force acting on the residual oil droplets by viscous-elastic fluids are much larger compared to that of inelastic fluid; the distribution of normal stress changes abruptly; under the condition of the same pressure gradient in the system under investigation, the ratio of the horizontal forces acting on the residual oil droplets by different displacing fluids (which are water, a power-law fluid with a power law index of 0.6 and a viscous-elastic fluid with a Wiesenberger Number of 0.3 respectively) is about 1:8:20, which means that under the above conditions, the driving force on a oil droplet is 20 times higher for a viscous-elastic fluid compared to that of a Newtonian Fluid. This force must be considered when studying the mechanism of isolated oil droplets mobilization. Furthermore, the viscous-elastic fluids will cause the residual oil droplet to deform much more significantly, beneficial to the mobilization of residual oil droplets. The above results are very beneficial to understanding the mechanism of the mobilization of residual oil droplets by viscous-elastic fluids; the conclusions are supportive of the mechanism that viscous-elastic driving fluids can increase the Displacement Efficiency. This should be of help in designing new chemicals and selecting Enhanced Oil Recovery systems.

New insights into the use of multi-mode phenomenological constitutive equations to model extrusion film casting process

This article is concerned with the effect of the individual viscoelastic relaxation modes of a polymer melt on its behavior in polymer melt extrusion film casting process. We compare the predicted versus experimentally obtained film necking or neck-in profile as a function of draw ratio. The predicted necking profile was obtained using well-established one-dimensional isothermal flow kinematics and consisted of using two different phenomenological constitutive equations, upper convected Maxwell and Phan-Thien–Tanner, with a discrete spectrum of relaxation times. The numerical simulations, containing the two different phenomenological constitutive equations, provided an insight into the effect of the slow and the fast relaxing modes on the stresses, strains, and strain/extensional rates that develop in the molten polymer film as it is stretched from the die exit to the chill-roll. The slow relaxing modes follow trends that are directly proportional to strain (similar to Hookean solids), whereas the fast relaxing modes follow trends that are directly proportional to the stretch rate (in accordance with Newton’s law of viscosity). Comparing the numerical predictions with the experiments showed that predictions using the upper convected Maxwell constitutive equation best described the long-chain branched polymers (like low-density polyethylene, which shows extensional strain hardening) in the extrusion film casting process. On the other hand, predictions using the Phan-Thien–Tanner constitutive equation best described the linear polymers (like linear low-density polyethylene, which does not show noticeable extensional strain hardening) in the extrusion film casting process.

On the modelling of the shear thickening behavior in micellar solutions

The steady and transient nonlinear rheological behaviors of dilute rod-like micellar solutions are predicted here with a particular case of the generalized Bautista–Manero–Puig (BMP) model that consists of the upper-convected Maxwell constitutive equation and a dissipative power-dependent kinetic equation, which takes into account the formation and disruption of shear-induced structures (SISs). This model has been derived using the extended irreversible thermodynamic (EIT) formalism. In steady shear, the model predicts a Newtonian region at low shear rates and a characteristic shear rate (\( {\dot{\upgamma}}_{\mathrm{c}} \)) at which shear thickening develops. In the shear thickening region, the model predicts either a reentrant zone, which is caused by multi-valued shear stresses when the data are collected with a shear stress-controlled mode or a continuous increase in the shear stress-shear rate flow curve, when the data are collected with a shear rate-controlled mode; in this region, two coexisting phases are predicted. The coexisting phases at the same shear rate in the two-phase envelope and the spinodal-like region were evaluated from the extended Maxwell equal-area criterion, which was calculated from the equal values of the two minima of the plot of the extended Gibbs free energy versus shear rate. At higher shear rates, the model predicts a transition to shear thinning, and under transient flows, an induction time and a saturation time are obtained from the predicted and the experimental data as detailed in the text. The magnitudes of both, the induction and the saturation times, diminish as the shear rate departs from \( {\dot{\upgamma}}_{\mathrm{c}} \), but only the induction time decreases according to a power law with shear rate. The conditions under which these rheological responses arise are derived and justified in detail with the BMP model. The model predictions are compared with experimental data of two dilute micellar solutions. The model parameters were determined from a set of independent experiments without fitting.

A unified multicomponent stress-diffusion model of drug release from non-biodegradable polymeric matrix tablets

We propose a new transport model of drug release from hydrophilic polymeric matrices, based on Stefan-Maxwell flux laws for multicomponent transport. Polymer stress is incorporated in the total mixing free energy, which contributes directly to the diffusion driving force while leading to time-dependent boundary conditions at the tablet interface. Given that hydrated matrix tablets are dense multicomponent systems, extended Stefan-Maxwell (ESM) flux laws are adopted to ensure consistency with the Onsager reciprocity principle and the Gibbs-Duhem thermodynamic constraint. The ESM flux law for any given component takes into account the friction exerted by all other species and is invariant with respect to reference velocity, thus satisfying Galilean translational invariance. Our model demonstrates that penetrant-induced plasticization of polymer chains partially or even entirely offsets the steady decline of chemical potential gradients at the tablet-medium interface that drive drug release. Utilizing a Flory-Huggins thermodynamic model, a modified form of the upper convected Maxwell constitutive equation for polymer stress and a Fujita-type dependence of mutual diffusivities on composition, depending on parameters, Fickian, anomalous or case II drug transport arises naturally from the model, which are characterized by quasi-power-law release profiles with exponents ranging from 0.5 to 1, respectively. A necessary requirement for non-Fickian release in our model is that the matrix stress relaxation time is comparable to the time scale for water diffusion. Mutual diffusivities and their composition dependence are the most decisive factors in controlling drug release characteristics in our model. Regression of the experimental polymer dissolution and drug release profiles in a system of Theophylline/cellulose (K15M) demonstrate that API-water mutual diffusivity in the presence of excipient cannot generally be taken as a constant.

Numerical Calculation of the Horizontal Stress Difference of Viscous-Elastic Fluid Acting on the Residual Oil in Micro Pore

In order to analyze the microscopic theory of viscous-elastic fluid flooding residual oil, the flow equation of polymer solution in the micro pore can be derived by selecting upper-convected Maxwell constitutive equation, continuity equation and motion equation. Then, the flow velocity field and stress field can be calculated under the boundary condition, and with the theory of stress tensor, the horizontal stress difference of polymer solution acting on the residual oil can be calculated. The results show that the greater the elasticity of viscous-elastic fluid is, the wider the flow channel is, the greater the horizontal stress difference is. The force acting on residual oil by viscous-elastic fluid can be increased by increasing the concentration of the polymer solution.

Numerical Calculation of Viscous-Elastic Fluid Flooding Residual Oil Film in the Complex Pore

In order to analyze the stress and deformation of different permeability of residual oil film in the complex pore, which are affected by the viscous-elasticity of the fluid, the hydrodynamic displacement mechanism is explored from the stand-point of hydrodynamics, that is, the residual oil film displaced by alternating injection of different concentrations of the polymer solution, viscous-elastic fluid flow equation is established in the complex pore by choosing continuity equation, motion equation and the upper convected Maxwell constitutive equation. The flow field is computed by using the method of numerical analysis. Not only the stress and deformation of residual oil film on the different permeability of micro pores, but also the analysis of the flooding mechanism of alternating injection of different concentrations of the polymer solution is got. The results show that the larger the viscous-elasticity of polymer solution is, the bigger the normal deviatoric stress acting on the residual oil film is; the distribution of normal deviatoric stress has the abrupt change. The stronger the viscous-elasticity of the polymer solution is, the bigger the horizontal stress difference acting on the residual oil film is and the more obvious the deformation is; the high-concentration polymer solution is suitable for high-permeability micro pores. Low-concentration polymer solution is suitable for medium and low-permeability micro pores. Alternating injection of polymer solution can improve Volumetric Sweep Efficiency and increase the deformation of residual oil film, which is conducive to enhancing oil recovery.

Mechanical Analysis of Viscous-Elastic Fluid Acting on Residual Oil in the Micro Pore

In order to analyze the normal deviatoric stress that viscous-elastic fluid acting on the residual oil under the situation of different flooding conditions and different permeabilities, Viscous-elastic fluid flow equation is established in the micro pore by choosing the continuity equation, motion equation and the upper-convected Maxwell constitutive equation, the flow field is computed by using numerical analysis, the forces that driving fluid acting on the residual oil in micro pore are got, and the influence of flooding conditions, pore width and viscous-elasticity of driving fluid on force is compared and analyzed. The results show that: the more viscous-elasticity of driving fluid increases, the greater the normal deviatoric stress acting on the residual oil increases; using constant pressure gradient flooding, the lager the pore width is, the greater normal deviatoric stress acting on the residual oil will be.

Causes Analysis of Eccentric Wear of Sucker Rod in Produced Fluid with Polymer

After polymer flooding was put into effect in DaQing oilfield, eccentric wear of sucker rod and tubing has been so serious that it made the rods break. This is because production liquid produced from normal wells is pure viscous fluid while liquid produced from polymer flooding oil wells is viscoelastic fluid. The flow equation of viscoelastic fluid in eccentric annulus was established in cylinder coordinate system, and the velocity distribution was solved. Based on upper-convected Maxwell constitutive equation, the normal stress calculating model of viscoelastic fluid acting on sucker rods was established. The normal stress of viscoelastic fluid acting on inner rod in eccentric annulus was measured under different experimental conditions indoors. The result proves that normal stress is the main reason of eccentric wear.

Application of the log-conformation tensor to three-dimensional time-dependent free surface flows

The numerical simulation of flows of highly elastic fluids has been the subject of intense research over the past decades with important industrial applications. Therefore, many efforts have been made to improve the convergence capabilities of the numerical methods employed to simulate viscoelastic fluid flows. An important contribution for the solution of the High-Weissenberg Number Problem has been presented by Fattal and Kupferman [J. Non-Newton. Fluid. Mech. 123 (2004) 281–285] who developed the matrix-logarithm of the conformation tensor technique, henceforth called log-conformation tensor. Its advantage is a better approximation of the large growth of the stress tensor that occur in some regions of the flow and it is doubly beneficial in that it ensures physically correct stress fields, allowing converged computations at high Weissenberg number flows. In this work we investigate the application of the log-conformation tensor to three-dimensional unsteady free surface flows. The log-conformation tensor formulation was applied to solve the Upper-Convected Maxwell (UCM) constitutive equation while the momentum equation was solved using a finite difference Marker-and-Cell type method. The resulting developed code is validated by comparing the log-conformation results with the analytic solution for fully developed pipe flows. To illustrate the stability of the log-conformation tensor approach in solving three-dimensional free surface flows, results from the simulation of the extrudate swell and jet buckling phenomena of UCM fluids at high Weissenberg numbers are presented.

Numerical prediction of three-dimensional time-dependent viscoelastic extrudate swell using differential and algebraic models

This study investigates the numerical simulation of three-dimensional time-dependent viscoelastic free surface flows using the Upper-Convected Maxwell (UCM) constitutive equation and an algebraic explicit model. This investigation was carried out to develop a simplified approach that can be applied to the extrudate swell problem. The relevant physics of this flow phenomenon is discussed in the paper and an algebraic model to predict the extrudate swell problem is presented. It is based on an explicit algebraic representation of the non-Newtonian extra-stress through a kinematic tensor formed with the scaled dyadic product of the velocity field. The elasticity of the fluid is governed by a single transport equation for a scalar quantity which has dimension of strain rate. Mass and momentum conservations, and the constitutive equation (UCM and algebraic model) were solved by a three-dimensional time-dependent finite difference method. The free surface of the fluid was modeled using a marker-and-cell approach. The algebraic model was validated by comparing the numerical predictions with analytic solutions for pipe flow. In comparison with the classical UCM model, one advantage of this approach is that computational workload is substantially reduced: the UCM model employs six differential equations while the algebraic model uses only one. The results showed stable flows with very large extrudate growths beyond those usually obtained with standard differential viscoelastic models.

Numerical Solution of the Upper-Convected Maxwell Model for Three-Dimensional Free Surface Flows

This work presents a finite difference technique for simulating three-dimensional free surface flows governed by the Upper-Convected Maxwell (UCM) constitutive equation. A Marker-and-Cell approach is employed to represent the fluid free surface and formulations for calculating the non-Newtonian stress tensor on solid boundaries are developed. The complete free surface stress conditions are employed. The momentum equation is solved by an implicit technique while the UCM constitutive equation is integrated by the explicit Euler method. The resulting equations are solved by the finite difference method on a 3D-staggered grid. By using an exact solution for fully developed flow inside a pipe, validation and convergence results are provided. Numerical results include the simulation of the transient extrudate swell and the comparison between jet buckling of UCM and Newtonian fluids.

Stability analysis of shear banding flow with the BMP model

The Bautista–Manero–Puig (BMP) model, consisting of the upper-convected Maxwell constitutive equation coupled to a kinetic equation that takes into account structural changes induced by flow, predicts the basic features of shear banding flow in polymer-like micellar solutions. In this work, the Lyapunov stability analysis applied to this model is used to determine the regions of stability and instability under conditions of shear banding flow. Results indicate that the steady state is reached very slowly within the meta-stable regions and quite rapidly in the homogeneous flow regions as well as in the unstable region where the slope of the constitutive flow curve is negative. Moreover, the Lyapunov stability criterion suggests the locus of the spinodal curve and the existence of a critical point for specific values of temperature and surfactant concentration. In addition, a criterion to set the stress plateau is derived from Extended Irreversible Thermodynamic (EIT) that consists in the equality of the minima in extended Gibbs free energy of the stable flow branches. This approach relates the EIT criterion for the stress plateau to the stability analysis for shear banding flow.

On the rheological modeling of associative polymers

The viscoelastic behavior of a commercial hydrophobic alkali-soluble emulsion (HASE) associative polymer in small-amplitude oscillatory, steady, and unsteady simple-shear flows is analyzed with a model that couples the upper-convected Maxwell constitutive equation with a kinetic equation that accounts for structural changes induced by the flows. A spectrum of relaxation times is considered in the prediction of rheological properties to account for the association dynamics between hydrophobic groups along the HASE backbone and physical entanglements. Viscoelastic response is similar to that of a transient network assembled through hydrophobic associations, where the kinetics of chain breakage and reformation is consistent with classical descriptions of transient network formulations. The model accounts for deviations from the Cox–Merz rule and predicts observed limiting behaviors at high strain rates in stress relaxation and inception of shear flow.

Non-Newtonian effects on lubricant thin film flows

An analysis of non-Newtonian effects on lubrication flows is presented based on the upper-convected Maxwell constitutive equation, which is the simplest viscoelastic model having a constant viscosity and relaxation time. By employing characteristic lubricant relaxation times in all order of magnitude analysis, a perturbation method is developed to analyze the flow of a non-Newtonian lubricant between two surfaces. The effect of viscoelasticity on the lubricant velocity and pressure fields is examined, and the influence of minimum film thickness on lubrication characteristics is investigated. Numerical simulations show a significant enhancement in the pressure field when the minimum film thickness is sufficiently small. This mechanism suggests that viscoelasticity does indeed produce a beneficial effect on lubrication performance, which is consistent with experimental observations.

Shear-Banded Flow and Transient Rheology of Cationic Wormlike Micellar Solutions

Linear oscillatory as well as transient and steady shear measurements for micellar solutions of dodecyltrimethylammonium bromide (DTAB) and sodium salicylate (NaSal) as a function of the salt-to-surfactant concentration ratio (CSALT/CDTAB) are presented. Our results indicate that, for molar ratios of salt to surfactant (CSALT/CDTAB) smaller than 1, the micellar solutions follow closely Maxwell behavior with a single relaxation time, that is, they are in the fast-breaking regime, and exhibit shear banding above a critical shear rate. When this ratio is greater than 1, micellar solutions behave as semidilute polymer solutions with a spectrum of relaxation times and no stress plateau is observed in steady shear. The results are analyzed with the Granek−Cates model (for linear response) and with a simple model that consists of the upper convected Maxwell constitutive equation coupled to a kinetic equation to account for the breaking and reformation of the micelles (nonlinear behavior). The stress plateau and the critical shear rates are determined from an extended irreversible thermodynamic analysis.

Nonaffine network structural model for molten Low-Density polyethylene and High-Density Polyethylene in oscillatory shear

We propose molten polymer’s entanglement network deformation to be nonaffine and use transient network structural theory with the revised Liu’s kinetics rate equation and the revised upper convected Maxwell constitutive equation to establish a nonaffine network structural constitutive model for studying the rheological behavior of molten Low-Density Polyethylene (LDPE) and High-Density Polyethylene (HDPE) in oscillatory shear. As a result, when the strain amplitude or frequency increases, the shear stress amplitude increases. At the same time, the accuracy of the nonaffine network model is higher than that of affine network model. It is clear that there is a small amount of nonaffine network deformation for LDPE melts which have long-chain branches, and there is a larger amount of nonaffine network deformation in oscillatory shear for HDPE melts which has no long-chain branches. So we had better consider the network deformation nonaffine when we establish the constitutive equations of polymer melts in oscillatory shear.

Non-isothermal viscoelastic flow computations in an axisymmetric contraction at high Weissenberg numbers by a finite volume method

This paper presents a numerical study of non-isothermal viscoelastic flows in a 4:1 axisymmetric abrupt contraction. The model retained for the simulations is an upper convected Maxwell (UCM) constitutive equation whose temperature dependence is described by a WLF equation. General thermodynamical framework leads to introduce the complex energy conversion mechanism from elastic to thermal energy occurring in viscoelastic fluid flow. The mass, momentum, constitutive and energy equations are discretized using a finite volume method on a staggered grid with upwind scheme for the convective-type terms. A decoupled algorithm stabilized by a pseudo-transient stress term and an elastic viscous stress splitting (EVSS) formulation are employed. Convergence is reached by a fixed-point iteration. The Stokes problem is solved by a fast augmented Lagrangian method and convective-type equations by a bi-conjugate gradient stabilized iterative procedure. Elastic effects are investigated in both isothermal and non-isothermal flow situations. Thermodynamical behaviour related to pure energy elasticity and pure entropy elasticity is considered. Various temperature boundary conditions corresponding to an external cooling of the flow are prescribed at the wall in order to investigate the temperature dependence in flows with large temperature gradients. Without encountering any upper limit for convergence, the present method provides solutions up to Weissenberg number We=10.00 and is able to take into account great temperature changes. General difficulties involved in thermal control processing are underlined.

Assessing a flow-based finite element model for the sintering of viscoelastic particles

The finite element method is applied to model viscoelastic sintering as a flow-based process. Momentum and mass conservation in the creeping flow limit are solved along with the upper convected maxwell constitutive equation implemented via the DEVSS-G method to determine the coalescence rate and shape of two touching spheres of equal size. While extra stresses in the neck region of viscolelastic particles showed marked differences from those in viscous (Newtonian) particles, material flows and particle shapes were nearly identical for viscoelastic and viscous systems, indicating that deviations from viscous flow predictions observed in physical systems is due predominantly to the early contact behavior described by purely elastic particle deformations. The mechanistic picture for viscoelastic sintering suggested by this work invokes elastic-dominated behavior for early stage neck growth with a transition to viscous-dominated effects at later stages. This work also demonstrates that sintering of materials possessing any degree of history dependence cannot be adequately described from a continuum-based flow model but must also include accounting of the earliest stages of contact and neck growth.