Phan-Thien Tanner Model Research Articles

Hydrodynamic interaction and coalescence of two inline bubbles rising in a viscoelastic liquid

In this paper, direct numerical simulations (DNS) are performed to investigate the inline rise of a pair of three-dimensional (3D) air bubbles in a viscoelastic liquid using the volume-of-fluid approach with an adaptive mesh refinement technique. The exponential Phan-Thien–Tanner model is used as the non-linear viscoelastic constitutive equation for the liquid. The numerical model has been validated by comparison with previously published results, including the terminal velocity jump discontinuity of an isolated bubble rising in a viscoelastic fluid, when its volume exceeds a certain critical value. Focusing on the inline rising bubble pair in such a viscoelastic medium with different configurations, we found that the wake of the small leading bubble attracts a larger trailing bubble, whereas for a supercritical bubble in front of a subcritical bubble, they tend to further separate. Before reaching a critical volume, the two subcritical bubbles remain close to each other after approaching each other, forming a stable chain. For pairs containing a supercritical trailing bubble, however, a drafting–kissing scenario occurs before the bubble–bubble coalescence. The long-range repulsion and the short-range attraction due to fluid elasticity are critical to the aforementioned bubble pair interactions. Interestingly, the terminal rise velocities of the stable bubble chain and the coalesced bubble both increase with the initial spacing. The squeezing flow near the growing bubble neck seems to delay the coalescence process. The capillary wave propagating down to the coalesced bubble tip together with the extensional flow behind the stretched bubble determines the generation of satellite microdroplets along the tail of the coalesced bubble. To the best of our knowledge, this is the first 3D DNS on a bubble pair ascending in viscoelastic fluids.

On peculiar behaviours at critical volumes of a three-dimensional bubble rising in viscoelastic fluids

In this paper, we carried out an investigation of the dynamics of an air bubble rising in viscoelastic liquids with increasing volumes via volume-of-fluid based direct numerical simulations (DNS) with local adaptive mesh refinement techniques. The exponential Phan-Thien Tanner (EPTT) model is adopted for describing the rheological behaviours of the shear-thinning viscoelastic fluid. The well-known jump discontinuity in the terminal rise velocity at a critical bubble volume is captured, which agrees qualitatively with the experimental observation in the literature. On this basis, the numerical simulations have been performed for different rheological properties in wider flow regimes, providing new insights into the local flow field and the stress distribution around the deformable rising bubbles. A characteristic dimensionless regime map has also been proposed to distinguish the peculiar behaviours of a bubble rising in viscoelastic fluids at different Galilei (Ga) and Eötvös (Eo) numbers. We found that at low Ga and low Eo numbers, the large elastic stresses concentrate in the small region near the bubble’s trailing end, leading to the formation of bubble cusp. The appearance of the negative wake is the main reason for the velocity jump to occur. However, for higher Ga of 10, the viscous forces become less dominating. The fluctuations of the polymeric stresses are substantially intensified with increasing bubble volume, and large bubbles experience a pulsating rising velocity with oscillation shapes. On the other hand, at intermediate Ga and high Eo of 10, the modest surface tension induces a cap with a thin skirt trailing the bubble main body at early times. Due to the polymer stretching, the strong extensional flow behind the small bubble eventually forces the bubble tail to break up. The above analyses on the bubble dynamics at constant Morton (Mo) numbers facilitate the extrapolation of the volume effects at different flow regimes and, therefore, will guide the applications in the future.

Global Well-posedness for the Three-Dimensional Generalized Phan–Thien–Tanner Model in Critical Besov Spaces

A new Generalized Phan–Thien–Tanner (GPTT) model is derived from a Lodge–Yamamoto type of network theory for the polymeric fluids. The GPTT model is developed to describe the rheological behavior of the viscoelastic fluids. In this paper, we investigate the initial-value problem for the GPTT model. Under the norm of the initial data is a small perturbation around some particular solution, we prove that the strong solution exists globally in critical Besov spaces.

Spatiotemporal linear stability of viscoelastic free shear flows: Nonaffine response regime

We provide a detailed comparison of the two-dimensional, temporal, and spatiotemporal linearized analyses of the viscoelastic free shear flows (inhomogeneous flows with mean velocity gradients that develop in the absence of boundaries) in the limit of low to moderate Reynolds number and elasticity number obeying four different types of stress–strain constitutive equations: Oldroyd-B, upper convected Maxwell, Johnson–Segalman (JS), and linear Phan-Thien–Tanner (PTT). The resulting fourth-order Orr–Sommerfeld equation is transformed into a set of six auxiliary equations that are numerically integrated via the compound matrix method. The temporal stability analysis suggests (a) elastic stabilization at higher values of elasticity number {shown previously in the dilute regime [Sircar and Bansal, “Spatiotemporal linear stability of viscoelastic free shear flows: Dilute regime,” Phys. Fluids 31, 084104 (2019)]} and (b) a nonmonotonic instability pattern at low to intermediate values of elasticity number for the JS as well as the PTT model. To comprehend the effect of elasticity, Reynolds number, and viscosity on the temporal stability curves of the PTT model, we consider a fourth parameter, the centerline shear rate, ζc. The “JS behavior” is recovered below a critical value of ζc, and above this critical value, the PTT base stresses (relative to the JS model) are attenuated thereby explaining the stabilizing influence of elasticity. The Briggs idea of analytic continuation is deployed to classify regions of temporal stability and absolute and convective instabilities, as well as evanescent modes, and the results are compared with previously conducted experiments for Newtonian as well as viscoelastic flows past a cylinder. The phase diagrams reveal the two familiar regions of inertial turbulence modified by elasticity and elastic turbulence as well as (a recently substantiated) region of elastoinertial turbulence and the unfamiliar temporally stable region for intermediate values of Reynolds and elasticity number.

Approximate analytical solution for the flow of a Phan-Thien–Tanner fluid through an axisymmetric hyperbolic contraction with slip boundary condition

An analytic approximation for the flow of a linear Phan-Thien–Tanner model fluid through an axisymmetric semi-hyperbolic contraction is presented. Such an approximation allows us to compute velocity and pressure response for the flow through axisymmetric contraction geometries; in particular, we have considered here the semi-hyperbolic contraction, which is a geometry where an almost constant extension-rate is reached at different radial positions. In addition, we present a semi-analytic solution capable of representing the exponential version of the selected viscoelastic model; this solution was compared to the results of commercial software, demonstrating the excellent approximation level of the semi-analytic model proposed. Alternatively, for both approaches (linear and exponential Phan-Thien–Tanner), the flow model equations are solved by considering the Navier boundary condition, which allows these models to represent flows with some degree of slip at the geometry wall.

Research on the rheological and flow characteristics of a supramolecular gel in fractures

AbstractLoss circulation is one of the major concerns in drilling and well construction. An effective way to control drilling fluid loss and strengthen the wellbore is to plug fractures and holes with loss circulation materials. In this work, a hydrophobic association supramolecular hydrogel GP‐A developed by n‐dodecylacrylamide, methacrylamide and 2‐acrylamide‐2‐methylpropylsulfonic acid was proposed as a potential lost circulation material for malignant drilling fluid loss. The results show that the initial decomposition temperature of GP‐A was 172°C, and the flow characteristics conform to the Herschel‐Bulkley model with yield stress. Microstructural analysis shows that GP‐A gel with a 3D spatial network, and hydrogen bonds between branched chains form a dynamically recoverable structure. Based on POLYFLOW, the Phan Thien–Tanner model with viscoelastic parameters was used to simulate the viscoelastic flow characteristics of the fluid in fractures. The higher the concentration is, the greaterK, and the higher the inlet driving pressure, while the nonlinear relationship between the driving pressure and the gel slug length is evident. The introduction of supramolecular gel polymer as a loss circulation material is an innovative research topic, which provides a new method to simulate the flow of polymer fluid in fractures.

A study on mixed electro-osmotic/pressure-driven microchannel flows of a generalised Phan-Thien–Tanner fluid

This work presents new semi-analytical solutions for the combined fully developed electro-osmotic pressure-driven flow in microchannels of viscoelastic fluids, described by the generalised Phan-Thien–Tanner model (gPTT) recently proposed by Ferras et al. (Journal of Non-Newtonian Fluid Mechanics, 269:88–99, 2019). This generalised version of the PTT model presents a new function for the trace of the stress tensor—the Mittag–Leffler function—where one or two new fitting constants are considered in order to obtain additional fitting flexibility. The semi-analytical solution is obtained under sufficiently weak electric potential that allows the Debye–Huckel approximation for the electrokinetic fields and for thin electric double layers. Based on the solution, the effects of the various relevant dimensionless numbers are assessed and discussed, such as the influence of $$\varepsilon Wi^2$$ , of the parameters $$\alpha $$ and $$\beta $$ of the gPTT model, and also of $${\bar{\kappa }}$$ , the dimensionless Debye–Huckel parameter. We conclude that the new model characteristics enhance the effects of both $$\varepsilon Wi^2$$ and $${\bar{\kappa }}$$ on the velocity distribution across the microchannels. The effects of a high zeta potential and of the finite size of ions are also studied numerically.

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.

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.

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.

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.

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.

Modeling of Shear Rheological Behavior of Uncured Rubber Melt

AbstractTo describe uncured rubber melt flow, a modified Phan–Thien–Tanner (PTT) model was proposed to characterize the rheological behavior and a viscoelastic one-dimensional flow theory was established in terms of incompressible fluid. The corresponding numerical method was constructed to determine the solution. Rotational rheological experiments were conducted to validate the proposed model. The influence of the parameters in the constitutive model was investigated by comparing the calculated and experimental viscosity to determine the most suitable parameters. The uncured rubber viscosity was 3–4 orders larger than that of plastic and did not have a visible Newtonian region. Compared with the Cross-Williams-Landel-Ferry (Cross-WLF) and original PTT models, the modified PTT model can describe the rheological characteristics in the entire shear-rate region if the parameters are set correctly.

Experimental and numerical study on transient elongational viscosity for PP/LDPE blends

The transient uniaxial elongational viscosity for binary blends composed of polypropylene (PP) and low-density polyethylene (LDPE) was evaluated. A strain hardening behavior is detected for the blends, although LDPE is a dispersed phase. This behavior is attributed to LDPE dispersion deformation; the LDPE forms rigid fibers because of strain hardening. Rheological properties are calculated numerically by the Phan–Thien Tanner model by assuming a symmetric geometry with a periodic structure. Based on the simulation, we propose an appropriate LDPE to modify the processability of PP, for which the strain hardening in the elongational viscosity is required. Transient elongational viscosity for polypropylene (PP)/low-density polyethylene (LDPE) blends was evaluated. Because deformed LDPE droplets act as rigid fibers due to its strain hardening, the blends show a strain hardening behavior. The growth curves are, however, affected by their viscosity ratio. These behaviors are calculated by the Phan–Thien Tanner model by assuming a symmetric geometry with a periodic structure. Based on the simulation, we propose an appropriate LDPE to modify the processability, at which the strain hardening in the elongational viscosity is required.

Numerical study of Phan-Thien–Tanner viscoelastic fluid flow around a two-dimensional circular cylinder at a low Reynolds number: a new classification for drag variations regimes

This study numerically investigates a low Reynolds two-dimensional flow of a viscoelastic fluid over a circular cylinder using the finite volume method. The Phan-Thien–Tanner model, as one of the most accurate non-linear models, for the first time, describes the viscoelastic behavior of a high concentration polymer solution in the flow over a cylinder in high elastic regime in Re = 10. To avoid divergence and to stabilize the numerical process in high elastic cases, the log-conformation approach proposed by previous researchers is used. The convective terms of the equations are discretized using a high-resolution scheme. The physical instability, as observed by some researchers in the creeping regime of high De flow, is very weak in the cases of this research. The numerical results of this research show both drag reduction (for the elasticity number El 0.025). Compared with the dissipative nature of viscoelastic fluid flows, the drag coefficient, resulted from the polymeric portion of stress, in the high elastic flow regime (El > 10), approached to zero due to an increase in the storing nature of viscoelastic fluid. At these elasticity numbers, the drag coefficient remains constant with El, and the material behaves like a Newtonian fluid. Compared with the shear-thinning behavior and unlike the creeping flow, the elasticity is the main cause of drag variation in most cases. Shear stress affects the drag force directly, while normal stress influences the drag force by changing the pressure distribution over the cylinder. The maximum normal stress occurs at the back of the cylinder due to the wake behind it, while in creeping flow it occurs at its half-front. Finally, the effects of elasticity number, retardation ratio and model parameters on the distribution of stress components, drag coefficient, and vortex dimensions are investigated separately, and a physical discussion is presented.

Transient 3D finite element method for predicting extrudate swell of domains containing sharp edges

A new transient 3D finite element method for predicting extrudate swell of domains containing sharp edges is proposed. Here, the sharp edge is maintained over a large distance in the extrudate by describing the corner lines as material lines. The positions of these lines can be used to describe the transverse swelling of the 2D free surfaces and expand the domain over which a 2D height function on the free surfaces is applied. Solving the 2D height functions gives the positions of the free surfaces. First a 2D axisymmetric case was tested for comparison, using three different constitutive models. The Giesekus, linear Phan-Thien Tanner (PTT) and exponential PTT constitutive models all showed convergence upon mesh- and time-step refinement. It was found that convergence remains challenging due to the singularity at the die exit. The new method is validated by comparing the final volume change of the extrudate of a 3D cylinder to the final volume change of a reference mesh of the 2D axisymmetric case. Finally, simulations were performed for different, complex, die shapes for a viscous fluid and for viscoelastic fluids. The results compared favorably with literature. Viscoelastic results, using the Giesekus model and the exponential PTT model, were compared for different Weissenberg numbers and different values for the non-linear parameters of the constitutive models. It was found that the swell is highly dependent on the rheological parameters and the constitutive model used.

Semi-Analytical Solutions for the Poiseuille–Couette Flow of a Generalised Phan-Thien–Tanner Fluid

This work presents new analytical and semi-analytical solutions for the pure Couette and Poiseuille–Couette flows, described by the recently proposed (Ferrás et al., A Generalised Phan-Thien–Tanner Model, JNNFM 2019) viscoelastic model, known as the generalised Phan-Thien–Tanner constitutive equation. This generalised version considers the Mittag–Leffler function instead of the classical linear or exponential functions of the trace of the stress tensor, and provides one or two new fitting constants in order to achieve additional fitting flexibility. The analytical solutions derived in this work allow a better understanding of the model, and therefore contribute to improve the modelling of complex materials, and will provide an interesting challenge to computational rheologists, to benchmarking and to code verification.

A generalised Phan–Thien—Tanner model

In this work we propose a novel generalised form of the Phan–Thien and Tanner (PTT) model by considering a new functional form of the nonlinear expression characterizing the destruction of physical network junctions and entanglements. This new function of the trace of the stress tensor is given by the generalized Mittag–Leffler function, and contains the familiar exponential form of the original Phan–Thien and Tanner model as a limiting case, but affords additional fitting flexibility through the inclusion of one or two additional fitting constants. We perform fits to experimental data in shear and extension and show that this generalized expression allows a better description of the rheological responses for a range of complex materials such as polymer melts and semidilute polymer solutions. By using an appropriate information criterion, we also demonstrate that the resulting generelized model remains parsimonious but provides improved fits of real world data.