Generalized Newtonian Fluid Research Articles

Inelastic fluid models with an objective stretch rate parameter

This paper presents an extension to the Generalized Newtonian Fluid (GNF) model, where the effects of different flow modes can be discerned. While existing GNF models have proven valuable in simulating processes like molding and extrusion, they often struggle to differentiate between distinct flow modes such as planar extension and simple shear. To address this challenge, we propose a modified GNF model that integrates an objective flow-type parameter, aiming to refine flow characterization. Emphasis is placed on defining the flow-type parameter to be able to transcend viscometric flows, remain frame-indifferent, quantify deformation magnitude, and differentiate between diverse flow modes. Inspired by the new advances in vortex identification in turbulent flow, we introduce a new stretch rate tensor and a new stretch rate parameter that are derived from the real Schur form of the objective velocity gradient tensor. These elements are embedded into the constitutive modeling of non-Newtonian fluid flow. The resulting model is employed to fit polymer melt data from the literature, demonstrating excellent fitting to combined shear and extension data. The basic model uses 5 to 6 parameters for data fitting, and further enhancement may be achieved by incorporating other extracted information of the stretch rate tensor.

Improving the prediction of turbulent kinetic energy for drag reduction in turbulent viscoelastic pipe flow

Reducing turbulence in pipe flows using polymer additives is crucial for industrial applications like crude oil, water, and sewage transportation. While previous research has accurately predicted friction factor and velocity profiles, none has fully understood turbulent kinetic energy (TKE) behavior in such fluids. Authors are now focusing on exploring turbulence models to better understand the TKE behavior. In this research, we have introduced a model to improve the behavior of TKE in a modified generalized Newtonian fluid (GNF). The developed model aims to simulate the viscoelastic effects of fluids that result in drag reduction in turbulent pipe flow. The work is noteworthy as it integrates turbulence and viscoelastic components, offering a comprehensive understanding of the phenomenon. By incorporating the rheological properties of viscoelastic fluids and replacing the damping function with a non-Newtonian alternative proposed by Cruz and Pinho, the Launder–Sharma k–ε turbulence model is now suitable for simulating dilute non-Newtonian viscoelastic fluids. The viscoelastic aspect of the model employs the modified GNF model. The developed model has been subjected to simulations using the computational fluid dynamics software. The results obtained for fluid TKE demonstrate a significant improvement in comparison to our previous research and the findings of other researchers. Furthermore, the model's prediction for the Darcy friction factor has been enhanced, resulting in an average error of only 3.71% in this section. It is noteworthy that the model consistently maintains a high level of accuracy in predicting other essential flow parameters such as mean axial velocity and Reynolds stresses. The provided model advances our understanding of viscoelastic fluid behavior in turbulent pipe flow by applying the modified GNF model.

Implementing vorticity–velocity formulation in a finite difference lattice Boltzmann method for two-dimensional incompressible generalized Newtonian fluids

A finite difference lattice Boltzmann approach is introduced to address the two-dimensional macroscopic equations of velocity–vorticity for generalized Newtonian fluids (GNFs). The study involves equations governing macroscopic momentum, energy, and concentration, along with constitutive models applicable to Newtonian, power-law, and viscoplastic fluids. Subsequently, the lattice Boltzmann method, which recovers these macroscopic equations, is detailed, along with proof of its capability to reproduce the aforementioned equations. In order to evaluate the effectiveness and time efficiency of the method, it is validated against various benchmarks. The results demonstrate the efficacy of the proposed method in successfully solving isothermal, thermal, and solutal problems of GNFs, while significantly reducing computational time compared to our previously suggested approach in this domain.

Modeling flow of Carreau fluids in porous media.

Carreau fluids occur routinely in porous medium systems for a range of applications, and the dependence of the viscosity for such fluids on the rate of strain tensor poses challenges to modeling at an averaged macroscale. Traditional approaches for macroscale modeling such flows have relied upon experimental observations of flows for generalized Newtonian fluids (GNFs) and a phenomenological approach referred to herein as the shift factor. A recently developed approach based upon averaging conservation and thermodynamic equationsfrom the microscale for Cross model GNFs is extended to the case of Carreau fluids and shown to predict the flow through both isotropic and anisotropic media accurately without the need for GNF-flow experiments. The model is formulated in terms of rheological properties, a standard Newtonian resistance tensor, and a length-scale tensor, which does require estimation. An approach based upon measures of the morphology and topology of the pore space is developed to approximate this length-scale tensor. Thus, this work provides the missing components needed to predict Carreau GNF macroscale flow with only rheological information for the fluid and analysis of the pore morphology and topology independent of any fluid flow experiments. Accuracy of predictions based upon this approach is quantified, and extension to other GNFs is straightforward.

Modeling and simulation of bulk viscoelasticity for amorphous polymers in injection molding

Bulk viscoelasticity is not well studied and understood in the field of polymer processing. Its behavior in solid mechanics applications, such as time-dependent bulk modulus and time-dependent thermal expansion, was rarely considered but started to receive attention recently. Bulk viscosity (bulk viscoelasticity in fluid mechanics formulation) has been ignored in polymer processing for decades. Bulk viscosity could play an essential role in compressible polymer melts that undergo substantial volume changes caused by variations in temperature and mechanical pressure during fluid motion and solidification. This study investigates the bulk viscosity of an amorphous polymer, polystyrene (PS), through measurements, modeling, and implementation in an injection molding simulation. Simulation results of cavity pressures and shrinkages are validated with experimental data in a three-plate mold case (part size 300 × 100 × 3 mm3). Results demonstrate that the effects of bulk viscosity reduced mechanical pressure variations during the packing stage in injection molding. However, the cavity pressure predicted by GNF (generalized Newtonian fluid) models with bulk viscosity drops too fast during the holding stage. The current GNF model can neither accurately describe isothermal pressurization (bulk creep) experiment data. A three-element-based constitutive model is proposed to describe bulk viscoelasticity in isobaric cooling and isothermal pressurization PVT (pressure-specific volume–temperature) measurements. This proposed model's predictions of cavity pressure, part weight, and shrinkage agree with the experiments and show significant improvement over the GNF model.

Three-dimensional simulation of capillary rheometry for an estimation of extensional viscosity

Abstract It is difficult to directly measure the steady extensional viscosity of thermoplastic composite materials, especially at high extension rates. The famous Cogswell method was derived analytically from the pressure drop of entrance flow in commercial capillary rheometers for estimating the extensional viscosity. However, using Cogswell´s extensional viscosity has always resulted in over-predictions of pressure drop. Recently, the GNF-X (eXtended Generalized Newtonian Fluid) model with a weighted shear/extensional viscosity was proposed to show the typical extension-induced vortex growth in entrance flow simulations. Under given various values of Trouton’s ratio for extensional viscosity, the GNF-X model is introduced to perform three-dimensional flow simulations of capillary rheometry over a range of apparent shear rates. The difference between the predicted pressure drops and the relevant experimental data is minimized such that the estimation of extensional viscosity is optimized herein.

Three-dimensional simulation of vortex growth within entry flow of a polymer melt

Abstract The corner vortex phenomenon occurring in entry flow is relevant to both polymer rheology and polymer processing. The famous viscoelastic constitutive equations in numerical computations of fluid mechanics have always had limited relevance at low apparent shear rates. However, three-dimensional (3D) simulations of viscoelastic fluids have been rarely found in advanced rheology. Recently, the GNF-X (eXtended Generalized Newtonian Fluid) constitutive equation of the weighted shear/extensional viscosity developed in advanced rheology of complex fluids has been incorporated into state-of-the-art predictive engineering tools. Thereby, 3D numerical simulations of entry flow were performed for a LDPE (low-density polyethylene) melt. As a validation, the predicted vortex streamlines are in good agreement with related experimental observations. More importantly, the simulation results show the vortex growth with respect to apparent shear rates, contraction ratios, and inlet angles. In particular for extensional viscosity, the stronger extension hardening characteristic yields a large vortex size.

Entropy Minimization for Generalized Newtonian Fluid Flow between Converging and Diverging Channels.

The foremost focus of this article was to investigate the entropy generation in hydromagnetic flow of generalized Newtonian Carreau nanofluid through a converging and diverging channel. In addition, a heat transport analysis was performed for Carreau nanofluid using the Buongiorno model in the presence of viscous dissipation and Joule heating. The second law of thermodynamics was employed to model the governing flow transport along with entropy generation arising within the system. Entropy optimization analysis is accentuated as its minimization is the best measure to enhance the efficiency of thermal systems. This irreversibility computation and optimization were carried out in the dimensional form to obtain a better picture of the system’s entropy generation. With the help of proper dimensionless transformations, the modeled flow equations were converted into a system of non-linear ordinary differential equations. The numerical solutions were derived using an efficient numerical method, the Runge–Kutta Fehlberg method in conjunction with the shooting technique. The computed results were presented graphically through different profiles of velocity, temperature, concentration, entropy production, and Bejan number. From the acquired results, we perceive that entropy generation is augmented with higher Brinkman and Reynolds numbers. It is significant to mention that the system’s entropy production grew near its two walls, where the irreversibility of heat transfer predominates, in contrast to the channel’s center, where the irreversibility of frictional force predominates. These results serve as a valuable guide for designing and optimizing channels with diverging–converging profiles required in several heat-transfer applications.

Macroscopic Model for Unsteady Generalized Newtonian Fluid Flow in Homogeneous Porous Media

Macroscopic Model for Unsteady Generalized Newtonian Fluid Flow in Homogeneous Porous Media

Bubble Growth in Generalized-Newtonian Fluid at Low-Mach Number Under Influence of Magnetic Field

We propose the results of a theoretical and mathematical model of the growing bubbles at low Mach number in a generalized Newtonian fluid. The system occurs under the effect of magnetic field and shear stress. The mathematical model is formulated and solved analytically on the basis of mass equation, Navier–Stokes equation, and diffusion equation. Analysis of the results demonstrates the role of magnetic field, shear stress, and electrical conductivity on the bubble growth process in a generalized Newtonian fluid at low Mach number; gives us the bubble radius in non-Newtonian fluid, Newtonian fluid, and viscoelastic fluid in the special cases of the flow index n; and also we get that the effect of magnetic field weakness from the behavior of gas bubble growth.

Velocity–vorticity correlations and the four-layer regime in turbulent channel flow of generalized Newtonian fluids

The data of Arosemena et al. (2021), consisting of turbulent channel flow simulations of generalized Newtonian (GN) fluids, are considered to study the effects of shear-dependent rheology on the nonzero velocity–vorticity correlations and the mean dynamics. In the near-wall region and compared to Newtonian channel flow, the velocity–vorticity products contributing to the turbulent inertia term decrease/increase with shear-thinning/thickening fluid behaviour suggesting that with e.g. shear-thinning rheology, the sublayer streaks are more stable, the near-wall vortical motions are dampened and there is a narrower range of turbulent length scales. The mean momentum balance analysis, on the other hand, revealed that the four-layer structure first recognized by Wei et al. (2005a) remains for all GN fluids and that the shear-dependent rheology only seems to influence the location of the layers. For instance, with shear-thinning behaviour, layers II and III are thicker and there is an increase in the importance of the viscous forces in these intermediate layers. The influence of shear-thinning/thickening fluid behaviour on the extent of the layers II and III is found remarkably similar to an increase/decrease of the Reynolds number for Newtonian channel flow. These findings suggest that the shear-dependent rheology should also be taken into account for proper scaling of the intermediate layers. A potential length scale factor is proposed and its suitability is tested.

Generalized Newtonian fluid constitutive equation for polymer liquids considering chain stretch and monomeric friction reduction for very fast flows modeling

In this work, the recently proposed frame-invariant Generalized Newtonian Fluid (GNF) constitutive equation [M. Zatloukal, “Frame-invariant formulation of novel generalized Newtonian fluid constitutive equation for polymer melts,” Phys. Fluids 32(9), 091705 (2020)] has been modified to provide uniaxial extensional viscosity at a high strain rate limit corresponding to molecular expression for a fully extended Fraenkel chain reported in Ianniruberto et al. [“Melts of linear polymers in fast flows,” Macromolecules 53(13), 5023–5033 (2020)]. It uses basic rheological and molecular parameters together with the ratio of monomeric friction coefficients for equilibrium and fully aligned chains. The modified GNF model was successfully tested by using steady-state uniaxial extensional viscosity data for well-characterized entangled polymer melts and solutions [namely, linear isotactic polypropylenes, poly(n-butyl acrylate), polyisoprenes, and polystyrenes] covering a wide range of strain rates, including those, at which the chain stretch occurs. Only two fitting parameters were sufficient to describe all uniaxial extensional viscosity data, one related to the Rouse stretch time and the other controlling the extensional thinning and thickening behavior at medium and high strain rates. The model was compared to five different advanced viscoelastic constitutive equations, which are based on Doi–Edwards theory and include chain stretch along with a number of important additions. The ability of the proposed GNF model to represent steady uniaxial extensional viscosities under fast flow conditions for entangled polymer fluids has been shown to be superior to the predictions of selected advanced viscoelastic constitutive equations. It is believed that the modified GNF model can be used in the stable modeling of non-Newtonian polymer liquids, especially in very fast steady-state flows where chain stretch begins to occur.

On the Influence of Viscoelastic Modeling in Fluid Flow Simulations of Gum Acrylonitrile Butadiene Rubber.

Computational fluid dynamics (CFD) simulation is an important tool as it enables engineers to study different design options without a time-consuming experimental workload. However, the prediction accuracy of any CFD simulation depends upon the set boundary conditions and upon the applied rheological constitutive equation. In the present study the viscoelastic nature of an unfilled gum acrylonitrile butadiene rubber (NBR) is considered by applying the integral and time-dependent Kaye–Bernstein–Kearsley–Zapas (K-BKZ) rheological model. First, exhaustive testing is carried out in the linear viscoelastic (LVE) and non-LVE deformation range including small amplitude oscillatory shear (SAOS) as well as high pressure capillary rheometer (HPCR) tests. Next, three abrupt capillary dies and one tapered orifice die are modeled in Ansys POLYFLOW. The pressure prediction accuracy of the K-BKZ/Wagner model was found to be excellent and insensitive to the applied normal force in SAOS testing as well as to the relation of first and second normal stress differences, provided that damping parameters are fitted to steady-state rheological data. Moreover, the crucial importance of viscoelastic modeling is proven for rubber materials, as two generalized Newtonian fluid (GNF) flow models severely underestimate measured pressure data, especially in contraction flow-dominated geometries.

Turbulent channel flow of generalized Newtonian fluids at a low Reynolds number

Abstract

Frame-invariant formulation of novel generalized Newtonian fluid constitutive equation for polymer melts

In this work, the frame-invariant formulation of a new generalized Newtonian fluid (GNF) constitutive equation is proposed. Viscosity is considered as a specific function of the second and third invariant of the strain rate tensor and of the second invariant of the objective velocity gradient. The GNF model was successfully tested using experimental data taken from the open literature for different high and low density polyethylene melts with varying amounts of long-chain branches utilizing steady-state shear, uniaxial, and planar extensional viscosities plotted as a function of the strain rate.

A revisitation of generalized Newtonian fluids

Creating an entry vortex growth simulation to show viscous and elastic fluid behaviors has always been a major challenge for the computational rheology of well-characterized polymer melts with extension thickening. For one thing, it is difficult to ensure stable convergence and reliable solutions via the viscoelastic constitutive equations at a relatively high Deborah number. From a revisitation of the generalized Newtonian fluids (GNFs), deriving the weighted viscosity is natural with the coexistence of shear and extensional flow; it is called the eXtended GNF (GNF-X) model. Herein, 3D contraction flow simulations incorporating the GNF-X weighted viscosity were significantly carried out over a wide range of Deborah numbers. Numerical results regarding changes in vortex sizes exhibit the same trend of experimental observations, while the predictive accuracy is enhanced for a quantitative comparison. More importantly, the vortex formation, in terms of the weighted viscosity and the extensional flow, is demonstrated with supporting evidence.

Two-phase flow in microfluidic-chip design of hydrodynamic filtration for cell particle sorting.

As one of the flow-based passive sorting, the hydrodynamic filtration using a microfluidic-chip has shown to effectively separate into different sizes of subpopulations from cell or particle suspensions. Its model framework involving two-phase Newtonian or generalized Newtonian fluid (GNF) was developed, by performing the complete analysis of laminar flow and complicated networks of main and multiple branch channels. To predict rigorously what occurs in flow fields, we estimated pressure drop, velocity profile, and the ratio of the flow fraction at each branch point, in which the analytical model was validated with numerical flow simulations. As a model fluid of the GNF, polysaccharide solution based on Carreau type was examined. The objective parameters aiming practical channel design include the number of the branches and the length of narrow section of each branch for arbitrary conditions. The flow fraction and the number of branches are distinctly affected by the viscosity ratio between feed and side flows. As the side flow becomes more viscous, the flow fraction increases but the number of branches decreases, which enables a compact chip designed with fewer branches being operated under the same throughput. Hence, our rational design analysis indicates the significance of constitutive properties of each stream.

Computational fluid dynamics simulation of the melting process in the fused filament fabrication additive manufacturing technique

Numerical simulation is used to understand the melting and pressurization mechanism in fused filament fabrication (FFF). The results show the incoming fiber melts axisymmetrically, forming a cone of unmelted material in the center surrounded by melted polymer. Details of the simulation reveal that a recirculating vortex of melted polymer is formed at the fiber entrance to the hot end. The large viscosity within this vortex acts to effectively seal the system against back-pressures of order 1000 psi (10 MPa), which are typical under standard printing conditions. The Generalized Newtonian Fluid (GNF) model was appropriate for simulation within the region that melts the fiber, however, a viscoelastic model, the Phan-Thien-Tanner (PTT) model, was required to capture flow within the nozzle. This is due to the presence of an elongational flow as molten material transitions from the melting region (diameter of 3 mm) to the nozzle at the exit (diameter of 0.5 mm). Remarkably, almost half the pressure drop occurs over the short capillary (0.5 mm in length) attached to the end of the converging flow region. Increased manufacturing rates are limited by high pressures, necessitating more consideration in the nozzle design of future FFF printers.

Thixotropic pumping in a cylindrical pipe

We consider the flow of a fluid in a uniform cylindrical pipe, driven by an oscillating pressure gradient or a body force. For a variety of rheological models, solutions can be obtained by integrating ordinary rather than partial differential equations: we illustrate this approach for the thixo-viscoplastic Houska model and the thixo-viscous simplified Moore-Mewis-Wagner model. We present asymptotic results in the limits of small and large Deborah numbers, and numerical results for intermediate Deborah numbers. Under asymmetrical sawtooth forcing, thixotropy leads to the net transport of fluid along the pipe, even when there would be no net transport of the corresponding generalised-Newtonian fluid. We propose the name thixotropic pumping for this novel transport mechanism.

Cuttings Transport Modeling—Part 2: Dimensional Analysis and Scaling

SummaryAs a follow-up to part one of this series (Busch et al. 2018a), we present a holistic dimensional analysis (DA) for the problem of cuttings transport in an annular wellbore element. We account for the non-Newtonian or, more precisely, shear-thinning and yield behavior of the drilling fluid by using non-Newtonian scaling concepts for generalized Newtonian fluids (GNFs) available in the literature.We first perform a nondimensional analysis for the Herschel-Bulkley (HB) material function (also known as the yield power-law [YPL] model) and establish a respective space of nondimensional numbers (Π-space) as well as a generic corresponding nondimensional cuttings-transport-process relationship. In a second step and to ease the derivations, we focus on the power-law (PL) material function and introduce a convenient reference shear rate that allows the evaluation of the reference viscosity in the established nondimensional quantities. Finally, on the basis of the established Π-space for the PL fluid, we generalize the specific PL case to the currently recommended HB material function by means of a local PL approximation, following the concept of Metzner and Reed (1955). Furthermore, we provide the results for both pipe and annuli because the former is often used in solid-transport studies.Compared to the Newtonian case, the obtained Π-spaces are increased by one or two nondimensional numbers accounting for the shear-thinning behavior or yield and shear-thinning behavior, respectively. It follows that when scaling cuttings transport processes, it is not entirely accurate to, for example, represent a Bingham fluid with a PL fluid because these exhibit different rheological behaviors. In addition, we show that the established process relationship must be nonmonomial, in contrast to the monomial forms used in various cuttings transport studies. Furthermore, we demonstrate that typical nondimensional numbers such as Reynolds, Shields, Archimedes, and Froude numbers arise from the analyses and we discuss some relevant differences.The presented Π-spaces might be used for scaling of process parameters as well as for quantitative comparison of results of one cuttings transport study with those of another. In addition, they could be used to scale wellbore parameters to a laboratory setup or vice versa. Finally, the established process relationship provides a framework for cuttings transport correlations by fitting comprehensive experimental data sets and could potentially be used to improve real-time models.