Phan-Thien Tanner Research Articles

Flow Dynamics of PTT and FENE-P Viscoelastic Fluids in Circular and Flat Ducts: An Analytical Study

The main aim of this research work is to establish, under fully developed conditions, an analytical solution for viscoelastic fluids flow obeying the constitutive Phan–Thien–Tanner (PTT) and Finely Extensible Nonlinear Elastic-Peterlin (FENE-P) models. In fact, a unified formulation for laminar, stationary and fully developed flow in the cases of circular and flat ducts geometries is obtained for velocity profile involving PTT and FENE-P models. In this investigation, the effect of pressure gradient on velocity profile, shear stress, shear rate and shear viscosity is explored. Effects of pressure gradient and Weissenberg number on normal stress are also studied and discussed.

PHAN-THIEN-TANNER NANOFLUID FLOW WITH GOLD NANOPARTICLES THROUGH A STENOTIC ELECTROKINETIC AORTA: A STUDY ON THE CANCER TREATMENT

The present study deals with a new novel scheme to cure a deadly disease, i.e., cancer, by adding gold nanoparticles (AuNPs) in the blood. Gold nanoparticles have promising properties that make them supreme compared with other nanoparticles for drug delivery. The hyaluronic acid-coated cisplatin conjugated AuNPs have been used to transport cisplatin into the targeted place and gained a synergistic impact by utilizing laser treatment. The Phan-Thien-Tanner (PTT) fluid model is contemplated to examine the rheology of blood moving through a tapered artery along with stenosis. Two forms of the PTT fluid model have been discussed explicitly to analyze the flow pattern. First, the impact of electrostatic potential and constriction of gold nanoparticles are considered. Second, lubrication theory is applied to formulate mathematical modeling. The resulting nonlinear system in the implicit forms is solved numerically using the Rung-Kutta-Merson scheme (RKMS) with a combination of shooting technique. The impact of velocity, wall shear stress, temperature, and nanoparticle concentration is deliberated against numerous leading parameters. A comparative analysis is also shown through graphs betwixt exponential and linearized models. Moreover, different cases of arteries have also been discussed through graphs. It is noticed that the effects of electroosmotic parameter m and the Brownian motion diffusion parameter Br will oppose the fluid motion, which enhances the shear stress. The thermophoresis and Brownian motion parameters generate a higher temperature, further rising by the golden nanoparticles because of their sizeable atomic number. A significant change in the gold nanoparticle concentration is noticed after increasing the Joule heating parameter because the gold nanoparticles absorb the internal heat and lean to propagate in the converse stream.

An a posteriori error estimator based on least-squares finite element solutions for viscoelastic fluid flows

<p style='text-indent:20px;'>We present a posteriori error estimator strategies for the least-squares finite element method (LS) to approximate the exponential Phan-Thien-Tanner (PTT) viscoelastic fluid flows. The error estimator provides adaptive mass weights and mesh refinement criteria for improving LS solutions using lower-order basis functions and a small number of elements. We analyze an a priori error estimate for the first-order linearized LS system and show that the estimate is supported by numerical results. The LS approach is numerically tested for a convergence study and then applied to the flow past a slot channel. Numerical results verify that the proposed approach improves numerical solutions and resolves computational difficulties related to the presence of corner singularities and limitations arising from the exorbitant number of unknowns.</p>

Investigations into the Complete Spreading Dynamics of a Viscoelastic Drop on a Spherical Substrate.

We study the spreading dynamics of a sphere-shaped elastic non-Newtonian liquid drop on a spherical substrate in the capillary-driven regime. We use the simplified Phan-Thien-Tanner model to represent the rheology of the elastic non-Newtonian drop. We consider the drop to be a crater on a flat substrate to calculate the viscous dissipation near the contact line. Following the approach compatible with the capillary-viscous force balance, we establish the evolution equation for describing the temporal evolution of the contact line during spreading. We show that the contact line velocity obtained from the theoretical calculation matches well with our experimental observations. Also, as confirmed by the present experimental observations, our analysis deems efficient to capture the phenomenon during the late stage of spreading for which the effect of line tension becomes dominant. An increment in the viscoelastic parameter of the fluid increases the viscous dissipation effect at the contact line. It is seen that the higher dissipation effect leads to an enhancement in the wetting time of the drop on the spherical substrate. Also, we have shown that the elastic nature of the fluid leads to an increment in the dynamic contact angle at any temporal instant as compared to its Newtonian counterpart. Finally, we unveil that the phenomenon of the increasing contact angle results in the time required for the complete wetting of drop, which becomes higher with increasing viscoelasticity of the fluid. This article will fill a gap still affecting the existing literature because of the unavailability of experimental investigations of the spreading of the elastic non-Newtonian drop on a spherical substrate.

A microstructure model for viscoelastic–thixotropic fluids

A microstructure model to describe the viscoelasticity and thixotropy properties of complex fluids is proposed. The model is based on the Lodge–Yamamoto network theory and is an extension of the Phan-Thien–Tanner model, with a kinetic process in which specific forms of creation and destruction rates are assumed. The final equation is simple with a small number of empirical parameters required and can be conveniently employed in engineering simulations. The predictions based on the model in a variety of shear and oscillatory shear flows are given. The stress response obtained from the model prediction agrees well with experiments on both shear and oscillatory flow histories.

On the use of the Lambert function in solving non-Newtonian flow problems

We consider unidirectional flows of ideal or regularized Bingham fluids as well as viscoelastic fluids for which analytical solutions can be derived in terms of the Lambert W function. Explicit expressions are derived for the radius of the yielded region in partially yielded circular and axial Couette flows. Analytical solutions are also derived for the velocity and the volumetric flow rate in the plane and axisymmetric Poiseuille flows of a Windhab fluid, which is a combination of the Bingham and Papanastasiou models, and for the shear stress in the plane Couette flow of an exponential Phan–Thien–Tanner fluid. Finally, the Lambert function is used to solve the Poiseuille flow of a power-law fluid and the Newtonian circular Couette flow with wall slip and non-zero slip yield stress by means of a regularized slip equation, which is valid for any value of the wall shear stress.

Influence of Temperature and Pressure on Viscoelastic Fluid Flow in a Plane Channel

The hydrodynamics of a steady-state nonisothermal flow of a viscoelastic polymer medium in a plane channel and heat transfer in it under boundary conditions of the first kind have been investigated. Fluid flow with a low Reynolds number and a high Peclet number was investigated, which made it possible to neglect the gravity and inertial forces, as well as the longitudinal thermal conductivity of the medium. From the rheological viewpoint, the polymer melt represents a viscoelastic fluid; therefore the Phan-Thien–Tanner fluid model was used as a rheological model of the fluid, with viscosity depending on temperature and pressure. A high-viscosity medium was considered; therefore a dissipation term was included into the equation of the energy of its flow. With the use of the indicated rheological model the velocity profile of fluid flow was obtained in an explicit form from the equation of fluid motion. It has been established that the dependence of the fluid viscosity on temperature and pressure exerts a noticeable influence on the distribution of the Nusselt number and of bulk temperature of the fluid along the channel length. It is shown that account for the temperature dependence of fluid viscosity leads to a decrease in the role of energy dissipation of its flow in the process of flow heating and that, conversely, the dependence of the fluid viscosity on pressure considerably enhances the dissipation effect. The problem has been solved numerically by the method of finite differences.

State of the‐Art for Extrudate Swell of Molten Polymers: From Fundamental Understanding at Molecular Scale toward Optimal Die Design at Final Product Scale

AbstractPolymer extrudate swell is a rheological phenomenon taking place after polymer melt flow emerges the die exit of extrusion equipment, due to flow redistributions and molecular stress relaxations. The associated die design is based on the strategy of inhibiting or compensating the swelling behavior downstream the die exit. This contribution comprehensively highlights the current knowledge and plethora of experimental and modeling tools to tune extrudate swell of polymer melts. The novelty lies in the combined consideration of the impact and interplay of i) the processing parameters, ii) the die geometry, iii) the macromolecular characteristics (e.g., molar mass distribution and branching level), and iv) the additive macroscopic properties. Furthermore, the prediction of extrudate swell through various flow geometries by semiempirical expressions and fundamental viscoelastic constitutive models is compared. Specific focus is on the differential Phan–Thien–Tanner (PTT), the molecular‐based Double‐Convected POM–POM model (DCPP), and the integral Kaye–Bernstein–Kearsley–Zapas (K‐BKZ) model and their usefulness for 3D swelling descriptions, indicating the optimum model to understand the fundamental relation of molecular parameters and viscoelasticity. The relevance of such detailed analysis is also illustrated by including case studies addressing design at the final product scale.

Start-up and cessation of steady shear and extensional flows: Exact analytical solutions for the affine linear Phan-Thien–Tanner fluid model

Exact analytical solutions for start-up and cessation flows are obtained for the affine linear Phan-Thien–Tanner fluid model. They include the results for start-up and cessation of steady shear flows, of steady uniaxial and biaxial extensional flows, and of steady planar extensional flows. The solutions obtained show that at start-up of steady shear flows, the stresses go through quasi-periodic exponentially damped oscillations while approaching their steady-flow values (so that stress overshoots are present); at start-up of steady extensional flows, the stresses grow monotonically, while at cessation of steady shear and extensional flows, the stresses decay quickly and non-exponentially. The steady-flow rheology of the fluid is also reviewed, the exact analytical solutions obtained in this work for steady shear and extensional flows being simpler than the alternative formulas found in the literature. The properties of steady and transient solutions, including their asymptotic behavior at low and high Weissenberg numbers, are investigated in detail. Generalization to the multimode version of the Phan-Thien–Tanner model is also discussed. Thus, this work provides a complete analytical description of the rheology of the affine linear Phan-Thien–Tanner fluid in start-up, cessation, and steady regimes of shear and extensional flows.

Effects of elasticity on unsteady forced convective heat transfer of viscoelastic fluid around a cylinder in the presence of viscous dissipation

The effects of elasticity on flow structures and forced convection heat transfer of a viscoelastic fluid around a cylinder were studied numerically using a finite volume method for the first time. In addition, the effects of viscous dissipation on flow and heat transfer parameters were studied over a wide range of Brinkman number (Br). The accurate non-linear Phan-Thien–Tanner model was used to describe the viscoelastic behavior of a polymer solution with a high retardation ratio (β = 0.6) over a wide range of Weissenberg (Wi) and Reynolds numbers (Re). The fluid properties were considered to be temperature-dependent. The results showed that the role of elastic effects in vortex shedding is a depreciator for Wi < 0.5 and an amplifier for Wi > 0.5. The Nusselt number was monotonically increased up to 52.2% with increasing Wi, β, Prandtl number (Pr), and Br over a wide range. The elastic forces affected the physical parameters by contrasting with the viscous forces so that the effects of elastic forces were weaker for high Reynolds numbers. Both hydrodynamic and thermal boundary layers were thinner in a viscoelastic fluid, compared to a Newtonian fluid. In addition, a correlation for variations of the drag coefficient with Wi and two correlations for variations of drag coefficient and Strouhal number with Re for Newtonian and viscoelastic fluid were proposed. Finally, the effects of Wi, Pr, Br, and model parameters on the Nu, St, drag, and lift coefficient and the distribution of stress components, velocity, and temperature were examined.

The impact of upstream contraction flow on three-dimensional polymer extrudate swell from slit dies

The relevance of upstream contraction for extrudate swell behavior of polymer melts is numerically investigated for slit dies with an aspect ratio of 10 and a length-to-height ratio ranging from 2.5 to 40. Three-dimensional (3D) flow simulations are performed with ANSYS Polyflow software at different flow rates and a constant temperature of 200°C, selecting a differential multimode Phan-Thien-Tanner (PTT) constitutive model to describe the viscoelastic behavior of polypropylene. It follows that decreasing the die length results in an opposite trend of the swell ratio in the width and height directions. The width swell ratio decreases up to 28%, whereas the middle height swell ratio increases up to 55% and the edge height contraction increases with 37%, all highlighting the 3D anisotropy of the extrudate swell deformation. However, especially for low flow rates, the area swell ratio remains almost unchanged and cannot be used to capture the anisotropic swelling behavior. This is predominantly attributed to the similar relative redistribution degree of the axial velocity (vx). An analysis of the transversal (vy) and lateral velocity (vz) combined with the stress field variation is performed as well to fully relate the anisotropic extrudate swell behavior to the various die lengths. The current study therefore contributes to completing the understanding of the relation of slit die design parameters and 3D extrudate flow under isothermal conditions, covering both developing and fully developed flow.

Streaming-potential-mediated pressure-driven transport of Phan-Thien-Tanner fluids in a microchannel.

Streaming potential mediated pressure driven electrokinetic transport of Phan-Thien-Tanner fluids in a slit type parallel plate microchannel is studied analytically and semianalytically. Without adopting the traditional considerations of Debye-Hückel linearization approximation for low surface potentials, exact analytical solutions are obtained for the electrostatic potential distribution, velocity, and volumetric flow rates taking into account the full Poisson-Boltzmann equation. The influences of interfacial electrokinetics and viscoelasticity on the streaming potential development, polymeric stress components, shear viscosity, and the hydroelectric energy conversion efficiency are incorporated concurrently. Major findings indicate that the magnitude of the induced streaming potential, volumetric flow rates, and the energy conversion efficiency increases up to a threshold limit of zeta potential of ζ≤6, however, it follows an asymptotic reduction at the other end of higher zeta potentials 6<ζ≤10. The polymeric stress components and shear viscosity follow a similar trend in the regime of 1≤ζ≤10, which is primarily governed by the streaming potential field. In contrast, the transverse averaged shear viscosity in the range 1≤ζ≤10 obeys an opposite trend by yielding an inverted parabolic shape. Amplification in the Stern layer conductivity yields a progressive reduction in the streaming potential magnitude and the hydroelectric energy conversion efficiency. The effect of the fluid viscoelasticity designated by the Weissenberg number exhibits a linear enhancement in streaming potential, flow rates, and the energy conversion efficiency. Moreover, we show that with the optimal combinations of surface charging and fluid viscoelasticity, it is possible to accomplish a giant augmentation in the hydroelectric energy conversion efficiency and flow rates. The analytical and semianalytical results presented in this investigation are believed to be worthy not only to cater deeper understanding in micro- and nanofluidic transport characteristics but also will act as functional design instrument for the future generation of energy efficient narrow fluidic devices.

Unconditional finite amplitude stability of a viscoelastic fluid in a mechanically isolated vessel with spatially non-uniform wall temperature

We investigate finite amplitude stability of spatially inhomogeneous steady state of an incompressible viscoelastic fluid which occupies a mechanically isolated vessel with walls kept at spatially non-uniform temperature. For a wide class of incompressible viscoelastic models including the Oldroyd-B model, the Giesekus model, the FENE-P model, the Johnson–Segalman model, and the Phan–Thien–Tanner model we prove that the steady state is stable subject to any finite perturbation.

Evaluating predictability of various constitutive equations for MAOS behavior of entangled polymer solutions

To predict the nonlinear viscoelastic behavior of polymer melts and solutions under medium amplitude oscillatory shear (MAOS) flow, we summarized known analytical MAOS material functions and derived new analytical solutions from several continuum- or molecular-based constitutive models [Johnson–Segalman, Phan-Thien–Tanner, Leonov, Larson, Pom-Pom, coupled double-convection-reptation with chain stretch (cDCR-CS), and Rouse linear entangled polymers (Rolie-Poly)]. All constitutive models discussed in this study were classified as time-strain separable or inseparable models. Whereas linear viscoelastic behavior of all constitutive models under small amplitude oscillatory shear flow was identical, the frequency-dependent signatures of four MAOS moduli (first-harmonic elastic and viscous MAOS moduli and third-harmonic elastic and viscous MAOS moduli) were model-specific, though all analytical solutions exhibited negative first-harmonic elastic MAOS moduli. In particular, tube-based constitutive models displayed a sign change in first-harmonic viscous MAOS modulus when chain retraction of the models after deformation was not instantaneous. In addition, two models for entangled linear polymers, that is, the Rolie-Poly and cDCR-CS models, displayed a second sign change in third-harmonic elastic MAOS modulus when finite chain extensibility was introduced. These results suggest that chain relaxation dynamics can be reflected in signatures of MAOS moduli. We extended single-mode predictions into multimode versions to predict the behavior of weakly or moderately entangled polystyrene solutions (effective entanglement number < 5). All models described the two elastic MAOS moduli well, but different predictability was obtained for viscous MAOS moduli. Therefore, viscous MAOS moduli can be a measure of evaluating the predictability of constitutive models for MAOS.

Asymmetric flows of complex fluids past confined cylinders: A comprehensive numerical study with experimental validation

Three non-Newtonian constitutive models are employed to investigate how fluid rheological properties influence the development of laterally asymmetric flows past confined cylinders. First, simulations with the shear-thinning but inelastic Carreau–Yasuda model are compared against complementary flow velocimetry experiments on a semidilute xanthan gum solution, showing that shear-thinning alone is insufficient to cause flow asymmetry. Next, simulations with an elastic but non-shear-thinning finitely extensible non-linear elastic dumbbell model are compared with experiments on a constant viscosity solution of poly(ethylene oxide) (PEO) in an aqueous glycerol mixture. The simulations and the experiments reveal the development of an extended downstream wake due to elastic stresses generated at the stagnation point but show no significant lateral asymmetries of the flow around the sides of the cylinder. Finally, the elastic and shear-thinning linear Phan–Thien–Tanner (l-PTT) model is compared with experimental velocimetry on a rheologically similar solution of PEO in water. Here, at low flow rates, lateral symmetry is retained, while the growth of a downstream elastic wake is observed, in qualitative similarity to the non-shear-thinning elastic fluids. However, above a critical flow rate, the flow bifurcates to one of the two stable and steady laterally asymmetric states. Further parameter studies with the l-PTT model are performed by varying the degrees of shear-thinning and elasticity and also modifying the confinement of the cylinder. These tests confirm the importance of the coupling between shear-thinning and elasticity for the onset of asymmetric flows and also establish stability and bifurcation diagrams delineating the stable and unstable flow states.

Evaluating the exit pressure method for measurements of normal stress difference at high shear rates

A challenge for polymer rheology is the reliable determination of shear dependent first normal stress difference (N1 values) at high shear rates (>10 s−1). Here, we evaluate the correctness of the commonly applied exit pressure method focusing on polypropylene and high and low density polyethylene melts at 200 °C. It is demonstrated that the linear extrapolation of pressure values toward the die exit, which is a key step in the application of the exit pressure method, is affordable to determine N1 values despite that these extrapolated exit pressure values are characterized by a relative deviation of 25%–40%. The validity of the exit pressure method is further supported by an excellent match with rheological data from the Laun rule (exponent close to 0.7) and a representative simulation of extrudate swelling data in the width and height direction, considering tuned parameters for the Phan–Thien–Tanner constitutive model. Also, the absence of a significant viscous heating effect near the die exit is highlighted based on numerical analysis.

Newtonian and viscoelastic fluid flows through an abrupt 1:4 expansion with slip boundary conditions

In this work, we present a systematic numerical investigation of the 1:4 planar expansion creeping flow under the influence of slip boundary conditions for Newtonian and viscoelastic fluids, the latter modeled by the simplified Phan–Thien–Tanner constitutive model. The linear and nonlinear Navier slip laws were considered with the dimensionless slip coefficient kl* varying in the range 0, 4500 and the slip exponents m = 0.5, 1, and 2. The simulations were carried out for a low Reynolds number, Re = 0.001, and for Deborah numbers (De) between 0 and 100. Convergence could not be achieved for higher values of the Deborah number and large values of the slip coefficient due to the large stress gradients near the singularity point (reentrant corner). The results obtained allow us to conclude that for all De, the increase in slip velocity leads to vortex suppression. The flow characteristics are described in detail for low values of the Deborah number, De ≤ 5, while for higher De the main features are only shown for specific values of the slip coefficient. These results find application in polymer processing, where the use of lubricants that migrate to the wall is common, which promotes slip.

Computationally efficient viscoelastic flow simulation using a Lagrangian-Eulerian method and GPU-acceleration

A recently proposed Lagrangian-Eulerian method for viscoelastic flow simulation is extended to high performance calculations on the Graphics Processing Unit (GPU). The two most computationally intensive parts of the algorithm are implemented for GPU calculation, namely the integration of the viscoelastic constitutive equation at the Lagrangian nodes and the interpolation of the resulting stresses to the cell centers of the Eulerian grid.In the original CPU method, the constitutive equations are integrated with a second order backward differentiation formula, while with the proposed GPU method the implicit Euler method is used. To allow fair comparison, the latter is also implemented for the CPU. The methods are validated for two flows, a planar Poiseuille flow of an upper-convected Maxwell fluid and flow past a confined cylinder of a four-mode Phan Thien Tanner fluid, with identical results.The calculation times for the methods are compared for a range of grid resolutions and numbers of CPU threads, revealing a significant reduction of the calculation time for the proposed GPU method. As an example, the total simulation time is roughly halved compared to the original CPU method. The integration of the constitutive equation itself is reduced by a factor 50 to 250 and the unstructured stress interpolation by a factor 15 to 60, depending on the number of CPU threads used.

Non-Newtonian effects on draw resonance in film casting

In this paper, the influence of non-Newtonian material properties on the draw resonance instability in film casting is investigated. Viscoelastic models of infinite width film casting are derived systematically following an asymptotic expansion and using two well-known constitutive equations: the Giesekus model and the simplified Phan–Thien/Tanner (PTT) model. Based on a steady state analysis, a numerical boundary condition for the inlet stresses is formulated, which suppresses the unknown deformation history of the die flow. The critical draw ratio in dependence of both the Deborah number and the nonlinear parameters is calculated by means of linear stability analysis. For both models, the most unstable instability mode may switch under variation of the control parameters, leading to a non-continuous change in the oscillation frequency at criticality. The effective elongational viscosity, which depends exclusively on the local Weissenberg number, is analyzed and identified as crucial quantity as long as the Deborah number is not too high. This is demonstrated by using a generalized Newtonian fluid model to approximate the PTT model. Based on such a generalized Newtonian fluid model, the effects of strain hardening and strain thinning are finally explored, revealing two opposing mechanisms underlying the non-Newtonian stability behavior.

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.