Oldroyd-B Constitutive Model Research Articles

Lagrangian Split-Step Method for Viscoelastic Flows.

This research addresses and resolves current challenges in meshless Lagrangian methods for simulating viscoelastic materials. A split-step scheme, or pressure Poisson reformulation of the Navier-Stokes equations, is introduced for incompressible viscoelastic flows in a Lagrangian context. The Lagrangian differencing dynamics (LDD) method, which is a thoroughly validated Lagrangian method for Newtonian and non-Newtonian incompressible flows, is extended to solve the introduced split-step scheme to simulate viscoelastic flows based on the Oldroyd-B constitutive model. To validate and evaluate the new method's capabilities, the following benchmarks were used: lid-driven cavity flow, droplet impact response, 4:1 planar sudden contraction, and die swelling. These findings highlight the LDD method's effectiveness in accurately simulating viscoelastic flows and capturing large deformations and memory effects. Even though the extra stress was directly modeled without any regularization approach, the method produced stable simulations for high Weissenberg numbers. The stability and performance of the the Lagrangian numerics for complex temporal evolution of material properties and stress responses encourage its use for industrial problems dealing with polymers.

Simulation of viscoelastic free-surface flows with the Particle Finite Element Method

Viscoelastic fluids are central in numerous applications from polymer manufacturing to the pharmaceutical industry and biological research. However, since analytical solutions are generally not available or too complex, it is common practice to study free-surface viscoelastic flows through numerical simulation techniques. This work proposes the use of the so-called particle finite element method (PFEM), a Lagrangian approach combining standard FEM techniques with a remeshing strategy. The PFEM is able to efficiently handle mesh distortion and to accurately track the free-surface evolution. Therefore, it is exploited in this work to deal with large displacements problems in the context of nonlinear viscoelasticity. An implementation of the Oldroyd-B constitutive model in the PFEM framework is here presented including details regarding how to deal with the transfer of the internal variables during remeshing events. Additionally, an innovative approach to impose unilateral Dirichlet boundary conditions ensuring optimal mass conservation is presented. The implementation is verified with two free-surface highly viscous benchmark flows: the impacting drop and the jet buckling problems. The results show perfect agreement with those obtained with other numerical techniques. The proposed framework opens the way for using PFEM in various applications, ranging from polymer extrusion to more sophisticated scenarios involving viscoelastic and viscoelasto-plastic constitutive laws.

Benchmarking the immersed boundary method for viscoelastic flows

We present and analyze a series of benchmark tests regarding the application of the immersed boundary (IB) method to viscoelastic flows through and around non-trivial, stationary geometries. The IB method is widely used to simulate biological fluid dynamics and other modeling scenarios in which a structure is immersed in a fluid. Although the IB method has been most commonly used to model systems involving viscous incompressible fluids, it also can be applied to visoelastic fluids and has enabled the study of a wide variety of dynamical problems including the settling of vesicles and the swimming of elastic filaments in fluids modeled by the Oldroyd-B constitutive equation. In the viscoelastic context, however, relatively little work has explored the accuracy or convergence properties of this numerical scheme. Herein, we present benchmarking results for an IB solver applied to viscoelastic flows in and around non-trivial geometries using either the idealized Oldroyd-B constitutive model or the more physically realistic, polymer-entanglement-based Rolie-Poly constitutive equations. We use two-dimensional numerical test cases along with results from rheology experiments to benchmark the IB method and compare it to more complex finite element and finite volume viscoelastic flow solvers. Additionally, we analyze different choices of regularized delta function and relative Lagrangian grid spacings which allow us to identify and recommend the key choices of these numerical parameters depending on the present flow regime.

Effect of elasticity on the induced charge electro-osmotic mixing of viscoelastic fluids in a micromixer with a conductive cylinder

Mixing of reagents in microfluidics is critical, and currently, the focus is on Newtonian fluids, but these reagents are often viscoelastic fluids. In this study, a micromixer containing a conductive cylinder is proposed based on the principle of induced charge electro-osmosis (ICEO). The Oldroyd-B constitutive model was chosen to characterize the flow properties of viscoelastic fluids, and the Poisson–Boltzmann model was used to describe the ion distribution in the electrolyte. The impact of the elasticity number (El) of viscoelastic fluids on the mixing efficiency, velocity, and vortex in the micromixer was studied. The results show that the mixing efficiency is only 55.41% when El = 0 (Newtonian fluid), and the mixing efficiency reaches 99.08% when El = 50. As El increases from 0 to 50, the cross-sectional average velocity at 50 μm from the micromixer exit decreases from 160 to 26.1 μm/s. Furthermore, the vortices around the conductive plate generated by the ICEO phenomenon begin to fluctuate at El = 7.

Mixing performance of an expansive mixer on viscoelastic solutions under alternating current electric field

Development of effective technologies for mixing viscoelastic solutions with reagents is still a challenge, which plays an important role in the fields of disease diagnosis, water desalination, and immunoassay. The purpose of this paper is to numerically study the mixing performance of an expansive mixer on the viscoelastic solution under the impact of an alternating current (AC) electric field through the analyzation of mixing efficiency, the concentration distribution, and average outlet velocity. A poly acrylic acid (PAA) water solution with different concentrations characterized by the Oldroyd-B constitutive model is used as a mixed medium, and an open-source toolbox called rheoTool based on finite volume algorithm is applied to solve the governing equations. We quantitatively determine the mixing efficiency for different PAA concentrations, AC electric field strengths, and frequencies and compare the mixing efficiency created by AC and direct current (DC) electric fields. Our investigation indicates that the mixing efficiency of the mixer diminishes with the PAA concentrations but enhances with the AC electric field strengths for a relatively high polymer concentration, and AC electric fields with different frequencies lead to a larger mixing efficiency than DC electric fields for a low polymer concentration. Additionally, the influence of the model parameters on the average outlet velocity and concentration distribution is discussed at length. Our numerical simulation results show that the mixer has an excellent mixing performance for both Newtonian and viscoelastic solutions, with a mixing efficiency of more than 99% at a low polymer concentration and a stable outflow velocity.

Induced charge electro-osmotic mixing performance of viscoelastic fluids in microchannels with an electrically conductive plate

Micromixers have important applications in lab-on-a-chip, biomanufacturing, and chemical engineering. In this study, a micromixer with a conductive barrier plate based on the induced charge electro-osmosis is proposed. The Oldroyd-B constitutive model was chosen to characterize the flow characteristics of viscoelastic fluids, and the Poisson–Boltzmann model was used to characterize the electrokinetic properties. The effects of the installation of the conductive plate, the concentration of the polymer, and the shape of the conductive plate on the mixing were studied based on the finite volume method. The mixing efficiency of the viscoelastic fluids is 78.3% when a non-conductive plate is placed in the micromixer. However, placing a conductive plate increases the mixing efficiency to 89.8%. As the polymer concentration increases, the mixing efficiency increases, which is attributed to the elastic instability. As the curvature of the conductive plate increases from 0° to 360°, the mixing efficiency of the Newtonian fluid increases by 2.82%, while that of the polyacrylamide solutions at concentrations of 100 and 250 ppm increases by 5.31% and 1.97%, respectively.

Vortex of viscoelastic fluid electroosmotic flow at the micro-nanochannel interface

The research on ion transport and vortex formation of viscoelastic fluids at the interface of micro-nanochannels may provide a new solution to improve the efficiency of analyte preconcentration in the ion depletion zone. The electroosmotic flow of a viscoelastic fluid in a nanochannel connecting two reservoirs was numerically investigated using the Navier-Stokes model, the Poisson-Nernst-Planck model, and the Oldroyd-B constitutive model. The results show that viscoelasticity suppresses the vortices in the ion depletion zone and breaks the symmetry of the vortices. Viscoelasticity also imparts a time dependence of vortex structure and velocity. When cKCl is 1 mM, the ratio of the average velocity at the upstream to the average velocity at the downstream for the viscoelastic fluid with the PAA concentration of 1000 ppm reaches 1.42, while it is only 0.70 for the Newtonian fluid. This study has positive significance for the development of nanochannel-based biosensing technology.

Mixing performance of the induced charge electro-osmosis micromixer with conductive chamber edges for viscoelastic fluid

Enhancing the micro-channel flow mixing is always a difficult problem. In this study, a micromixer based on induced charge electro-osmosis is proposed. A T-shaped micromixer, which has a chamber with conductive surfaces in the channel, is chosen. Due to the electro-osmotic effect of the induced charge, the induced potential is generated on the conductive surface. The Oldroyd-B constitutive model is chosen to characterize the flow characteristics of polyacrylamide solution, and an open-source solver named rheoTool based on the finite-volume method is used. The effect of the chamber shape, the chamber size, the conductive edge numbers in the chamber, and the applied electric intensity on the mixing efficiency are investigated. The results show that the micromixer with conductive edges in the chamber has better mixing effect because of the vortices. At the same time, compared with other shapes, the micromixer with diamond chamber has the best mixing effect, and the mixing efficiency reaches 79.51%. In addition, the mixing efficiency of one conductive edge in the diamond chamber is 4.39% higher than that of the two conductive edges chamber. It is found that increasing the chamber size will improve the mixing efficiency, and the mixing efficiency increased by 12.76% with the increase in chamber size. On the other hand, when increasing the electric field intensity from 100 to 200 V/cm, the mixing efficiency will decrease.

Heat transfer enhancement by pulsating flow of a viscoelastic fluid in a microchannel with a rib plate

ABSTRACT In order to improve the heat transfer performance of the microchannel heat exchanger, a composite heat transfer enhancement method was proposed. Viscoelastic fluid was used as working fluid in pulsating flow condition, and rib plates were added to the microchannel to bring extra disturbance to the flow. The Oldroyd-B constitutive model of the viscoelastic fluid was used in the numerical simulation, and the flow field, temperature field, Nusselt number (Nu), and pressure drop were analyzed when the average Reynolds number (Re) is 10. Both Strouhal number (St) and amplitude are important factors affecting heat transfer, but they have an insignificant influence on pressure drop at low Reynolds number. The St = 0.125 and amplitude A = 0.8 are better parameters. The increase of Weissenberg number (Wi) will cause the vortex to split into several subsidiary vortexes during its development, which will also develop to various positions in the channel, thus further enhancing the heat transfer. When the Wi is in the range of 1 ~ 5, the performance evaluation criteria rises at a relatively fast rate, from 1 to 1.404.

Data-driven modeling of heterogeneous viscoelastic biofilms.

Biofilms are typically heterogeneous in morphology, structure, and composition, resulting in nonuniform mechanical properties. The distribution of mechanical properties, in turn, determines the biofilm behavior, such as deformation and detachment. Most biofilm models neglect biofilm heterogeneity, especially at the microscale. In this study, an image-based modeling approach was developed to transform two-dimensional optical coherence tomography (OCT) biofilm images to a pixel-scale non-Newtonian viscosity map of the biofilm. The map was calibrated using the bulk viscosity data from rheometer tests, based on assumed maximum and minimum viscosities and a relationship between OCT image intensity signals and non-Newtonian viscosity. While not quantitatively measuring biofilm viscosity for each pixel, it allows a rational spatial allocation of viscosities within the biofilm: areas with lower cell density, for example, voids, are assigned lower viscosities, and areas with high cell densities are assigned higher viscosities. The spatial distribution of non-Newtonian viscosity was applied in an established Oldroyd-B constitutive model and implemented using the phase-field continuum approach for the deformation and stress analysis. The heterogeneous model was able to predict deformations more accurately than a homogenous one. Stress distribution in the heterogeneous biofilm displayed better characteristics than that in the homogeneous one, because it is highly dependent on the viscosity distribution. This study, using a pixel-scale, image-based approach to map the mechanical heterogeneity of biofilms for computational deformation and stress analysis, provides a novel modeling approach that allows the consideration of biofilm structural and mechanical heterogeneity. Future research should better characterize the relationship between OCT signal and viscosity, and consider other constitutive models for biofilm mechanical behavior.

Purely elastic linear instabilities in parallel shear flows with free-slip boundary conditions

We perform a linear stability analysis of viscoelastic plane Couette and plane Poiseuille flows with free-slip boundary conditions. The fluid is described by the Oldroyd-B constitutive model, and the flows are driven by a suitable body force. We find that both types of flow become linearly unstable, and we characterise the spatial structure of the unstable modes. By performing a boundary condition homotopy from the free-slip to no-slip boundaries, we demonstrate that the unstable modes are directly related to the least stable modes of the no-slip problem, destabilised under the free-slip situation. We discuss how our observations can be used to study recently discovered purely elastic turbulence in parallel shear flows.

Unsteady Magnetohydrodynamics Stagnation-Point Flow of Fractional Oldroyd-B Fluid Over a Stretching Sheet With Modified Fractional Fourier's Law

AbstractThe aim of the article is to research the unsteady magnetohydrodynamic stagnation-point flow of fractional Oldroyd-B fluid over a stretched sheet. According to the distribution characteristics of pressure and magnetic field near the stagnation point, the momentum equation based on a fractional Oldroyd-B constitutive model is derived. Moreover, the modified fractional Fourier's law considering thermal relaxation-retardation time is proposed, which applies in both the energy equation and the boundary condition of convective heat transfer. A new finite difference scheme combined with the L1 algorithm is established to solve the governing equations, whose convergence is confirmed by constructing an exact solution. The results indicate that the larger relaxation parameters of velocity block the flow, yet the retardation parameters of velocity show the opposite trend. It is particularly worth mentioning that all the temperature profiles first go up slightly to a maximal value and then descend markedly, which presents the thermal retardation characteristic of Oldroyd-B fluid. Additionally, under the effects of temperature's retardation and relaxation parameters, the intersection of the profiles far away from stretching sheet demonstrates the thermal memory characteristic.

Induced charge electroosmosis characteristics of viscoelastic fluid around a metal cylinder

The mixing of chemical analysis reagents at the microscale is critical. Most of the biological fluids and polymer solutions used in the field of biomedicine belong to the category of non-Newtonian fluids which have received limited attention in the research of induced charge electroosmosis (ICEO). In this study, the flow behavior of viscoelastic fluid described by the Oldroyd-B constitutive model in a microfluidic sample preparation mixer under different conditions is studied by numerical simulation. The nonlinear Poisson-Nernst-Planck equation which controls the distribution of potential and ion concentration is used and solved by the finite volume method. This paper analyses the influence of polar voltage, electrical double layer (EDL) thickness, and Weissenberg (Wi) number on the ICEO characteristics of viscoelastic fluid around a metal cylinder. The most obvious finding to emerge from this study is that the Wi number of polymer solutions has a significant impact on fluid velocity. Due to the shear thickening characteristics, the fluid flow rate will decrease as the Wi number increases, respectively, under various voltage conditions used. Similarly, the decrease of EDL thickness and the increase of applied voltage can result in the rise of the charge density close to the wall and the corresponding increase of the coulomb force, therefore resulting in the rise of fluid velocity. A linear relationship between ICEO velocity and electric field strength squared is observed in all the examined PAA solutions. The fluid velocity affects the relative movement between different streamlines which will cause the fluid micro-elements to deform, stretch, and fold, and then increase the interface area between the fluids to be mixed, thereby increasing the mixing effect. This research might explore services for the detection of viscoelastic fluids such as biological fluids and develop high-performance microfluidic preparation mixers for viscoelastic fluids.

Comparative Response of Newtonian and Non-Newtonian Fluids Subjected to Exothermic Reactions in Shear Flow

A comparative computational study of the thermal response of Newtonian and non-Newtonian fluids subjected to exothermic reactions is conducted in simple shear flow. The investigations conducted in this study are of fundamental importance to industrial and biological applications in which heat generation minimization is important, such as in heat exchangers, in lubrication, and in internal medicine. Specifically, the comparative investigations central to this study are conducted on four types of fluids, namely; Newtonian fluids, generalized Newtonian fluids, viscoelastic fluids, and generalized viscoelastic fluids. The Oldroyd-B constitutive model is used for the viscoelastic fluids and a Carreau viscosity constitutive model is used to describe the viscosity shear-rate dependence of the generalized fluids. Efficient semi-implicit finite difference methods are employed to obtain the numerical solutions to the governing systems of equations. The computational methodologies are implemented in the MATLAB software. The sensitivity of the fluid temperature and the polymer stresses to increases in shear-thinning characteristics as well as to increases in polymeric properties is investigated. In general, it is observed that, at comparative parameter values, the viscoelastic fluids give the best resistance to temperature increases, followed by the generalized viscoelastic fluids, followed in turn by the Newtonian fluids, and with the generalized Newtonian fluids recording the highest temperature increases.

Stabilization of a smoothed finite element semi-implicit coupling scheme for viscoelastic fluid–structure interaction

We propose in this work a stabilization approach for semi-implicit coupling of viscoelastic fluid–structure interaction (VFSI) using the cell-based smoothed finite element method (CS-FEM). The viscoelastic fluid and nonlinear solid equations are spatially discretized by the CS-FEM and then are semi-implicitly coupled via a partitioned solution strategy. The current semi-implicit coupling framework depends on a second-order characteristic-based split (CBS(B)) scheme that solves the Navier–Stokes equations together with the Oldroyd-B constitutive model in the fractional-step manner. To enhance the stability of the semi-implicit coupling algorithm, the discrete elastic-viscous split stress-gradient (DEVSS-G) procedure is introduced into the explicit stage while the stabilized pressure gradient projection (SPGP) is earmarked for the implicit stage. Moreover, the iterated end-of-step velocity begins with the intermediate velocity during the subiterations. The DEVSS-G/CBS(B)-SPGP technique is readily applied to the CBS-based partitioned semi-implicit coupling algorithm for VFSI. Visible improvements in stabilization and efficiency are revealed in a benchmark test. • A semi-implicit coupling scheme is proposed for viscoelastic fluid–structure interaction. • DEVSS-G/CBS(B)-SPGP stabilization is presented for the semi-implicit scheme. • VFSI system is entirely formulated by cell-based smoothed finite element method. • The developed technique boosts the stability and efficiency considerably.

Simulation of Oldroyd-B Viscoelastic Fluid in Axisymmetric Straight Channel by Using a Hybrid Finite Element/Volume Method

In this study, incompressible viscoelastic fluid through the axisymmetric circular channel is simulated with Oldroyd-B model. The simulation is performed based on a hybrid finite volume/element method, which consists of Taylor-Galerkin finite element discretisation, and a cell vertex fluctuation-distribution finite volume method. In this context, the momentum and continuity equations are treated with a finite element method, while a finite volume approach is applied to solve the Oldroyd-B constitutive model. Analytical expressions are presented for the velocity and stress components in fully developed channel flow of Oldroyd-B fluid. For this complex fluid, we see an excellent agreement between the analytic and the numerical solutions. The study of axisymmetric circular channel problem based on a hybrid numerical method represents a great challenge. The novelty here is to study the temporal convergence-rate of the system solution that is taken to be steady state, incompressible, axisymmetric, and laminar, which did not address by researchers previously. Here, the rate of convergence for all solution components is presented, where a large level of convergence is appeared for stress compared to the other solution components. Moreover, the pressure drops and stress response across the flow are provided with respect to difference in solvent-fraction and Weissenberg number . A significant effect from the viscoelastic parameters upon the level of the stress has been detected, while for the pressure response the change is semi-modest. For the stress response the findings reveal that, with decreasing solvent-fraction , the maxima level of stress components are strongly amplifies.

Comparative study on numerical performances of log-conformation representation and standard conformation representation in the simulation of viscoelastic fluid turbulent drag-reducing channel flow

In this paper, a new derivation process of the log-conformation governing equation for viscoelastic fluid flows is presented by using the Taylor series definition of the matrix logarithm. Based on the log-conformation representation (LCR) and standard conformation representation (CR) methods, the turbulent drag-reducing channel flow of viscoelastic fluid described by the Oldroyd-B constitutive model is simulated by the finite difference method. The comparison illustrates that the turbulent drag reduction (DR) effect under the condition of a low Weissenberg number (Wi = 1) or moderate Weissenberg number (Wi = 5) can be successfully reproduced by the CR method but is very difficult to be obtained by the LCR method at the same grid resolution if the commonly used interpolation approaches in the computing domain (i.e., log domain) are employed. Further research reveals that the interpolation method of log-conformation tensor involved is one of the dominant reasons responsible for the disability to obtain a turbulent DR effect by using the LCR method. If the interpolation is performed in a physical domain, the turbulent DR effect can be reproduced by using the LCR method. If the interpolation involved in the CR method is carried out in a log domain, the turbulent DR phenomenon can still be simulated but with a weakened DR effect. In sum, this study demonstrates that the commonly used interpolation approaches in the log domain should be responsible for the poor performance of the LCR method.

A lattice Boltzmann modeling of viscoelastic drops’ deformation and breakup in simple shear flows

The deformation and breakup of viscoelastic drops in simple shear flows of Newtonian liquids are studied numerically. Our three-dimensional numerical scheme, extended from our previous two-dimensional algorithm, employs a diffusive-interface lattice Boltzmann method together with a lattice advection–diffusion scheme, the former to model the macroscopic hydrodynamic equations for multiphase fluids and the latter to describe the polymer dynamics modeled by the Oldroyd-B constitutive model. A block-structured adaptive mesh refinement technique is implemented to reduce the computational cost. The multiphase model is validated by a simulation of Newtonian drop deformation and breakup under an unconfined steady shear, while the coupled algorithm is validated by simulating viscoelastic drop deformation in the shear flow of a Newtonian matrix. The results agree with the available numerical and experimental results from the literature. We quantify the drop response by changing the polymer relaxation time λ and the concentration of the polymer c. The viscoelasticity in the drop phase suppresses the drop deformation, and the steady-state drop deformation parameter D exhibits a non-monotonic behavior with the increase in Deborah number De (increase in λ) at a fixed capillary number Ca. This is explained by the two distribution modes of the polymeric elastic stresses that depend on the polymer relaxation time. As the concentration of the polymer c increases, the degree of suppression of deformation becomes stronger and the transient result of D displays an overshoot. The critical capillary number for unconfined drop breakup increases due to the inhibitive effects of viscoelasticity. Different distribution modes of elastic stresses are reported for different De.

The cell-based smoothed finite element method for viscoelastic fluid flows using fractional-step schemes

This paper generalizes the cell-based smoothed finite element method (CS-FEM) to viscoelastic fluid flows under the fractional-step umbrella. The characteristic-based split (CBS) scheme is employed to decouple the Navier–Stokes equations and the Oldroyd-B constitutive model. Moreover, the discrete elastic-viscous-split-stress (DEVSS) procedure and stabilized pressure gradient projection (SPGP) are introduced into a second-order CBS framework. The projections of velocity and pressure gradients therein combine to form the DEVSS-G/CBS(B)-SPGP algorithm that further stabilizes the discretized model. Both proposed schemes are suitable for use in CS-FEM because they accept equal low-order interpolations for primitive variables. The spatial discretization is based upon bilinear four-node quadrilateral elements where all gradient-related items are smoothed without any methodological difficulties. In particular, the cell-based smoothing concept enables accurate evaluation of viscoelastic fluid force. The developed methods are validated against previously published data for several benchmark problems. A satisfactory agreement exists between the present and earlier results.

A lattice Boltzmann method for simulating viscoelastic drops

We report some numerical simulations of multiphase viscoelastic fluids based on an algorithm that employs a diffusive-interface lattice Boltzmann method together with a lattice advection-diffusion scheme, the former used to model the macroscopic hydrodynamic equations for multiphase fluids and the latter to describe the polymer dynamics modeled by the Oldroyd-B constitutive model. The multiphase model is validated by a simulation of Newtonian drop deformation under steady shear. The viscoelastic model is validated by simulating a simple shear flow of an Oldroyd-B fluid. The coupled algorithm is used to simulate the viscoelastic drop deformation in shear flow. The numerical results are compared with the results from conventional methods, showing a good agreement. We study the viscosity (density) ratio effect on the bubble rising in viscoelastic liquids and demonstrate a nonmonotonic relation between the length of the bubble tail and the polymer relaxation time.