Phan-Thien Tanner Research Articles

Heat transfer analysis of peristaltic flow of a Phan-Thien-Tanner fluid model due to metachronal wave of cilia.

In the present investigation, we have studied the effects of heat transfer on the peristaltic flow considering the Phan-Thien-Tanner fluid model. The fluid is flowing in a uniform circular tube in the form of wave motion. The inner walls of the tube are considered to be ciliated with small hair-like structures. Exact solutions have been derived for velocity, temperature and pressure gradient. Mechanical properties of the fluid, such as velocity, temperature, pressure rise and pressure gradient, have been discussed graphically. Trapping phenomena due to the variation of physical parameters have been deliberated. It has been observed that when the viscous forces are greater than the elastic forces, the velocity of the fluid flow significantly decreases, thermal conductivity of the fluid improves and the pressure gradient along the tube increases.

An approximation to the PTT viscoelastic model for gas assisted injection moulding simulation

An approximation to the Phan-Thien Tanner (PTT) constitutive model is developed with the aim of giving low-cost simulation of Gas Assisted Injection Moulding (GAIM) while incorporating important viscoelastic characteristics. It is shown that the developed model gives a response typical of full viscoelastic models in transient and steady-state uniaxial and constant shear rate deformations. The model is incorporated into a 3D finite element GAIM simulation which uses the ‘pseudo-concentration’ method to predict residual polymer, and applied to published experimental results for a Boger fluid and a shear-thinning polystyrene melt.It is shown that the simulation gives a very good match to published results for the Boger fluid which show increasing Residual Wall Thickness (RWT) with increasing Deborah number. Against the shear-thinning polymer, the quality of match depends upon which of two ‘plausible’ relaxation times is chosen; qualitatively different results arise from two different means of estimating a single relaxation time. A ‘multi-mode’ approach is developed to avoid this uncertainty. It is shown that the multi-mode approach gives decreasing RWT with increasing Deborah number in agreement with the published experimental results, and avoids the issues that arise from estimating a single relaxation time for a molten polymer.

Steady flow of a viscoelastic film over an inclined plane featuring periodic slits

We consider the steady flow of a viscoelastic film over an inclined plane featuring a periodic arrangement of slits, which are oriented normal to the main direction of flow. The film creates a second gas-liquid interface connecting the two hydrophilic sidewalls of a slit. This interface forms two three-phase contact lines and supports a widely varying amount of liquid under different physical and geometrical conditions. We develop a computational model and carry out detailed numerical simulations, based on the finite element method to investigate this flow. To this end, we solve the two-dimensional momentum and mass conservation equations, while employing the Phan-Thien-Tanner (PTT) constitutive model to account for the rheology of the viscoelastic material. An elliptic grid generation scheme is used to follow the large deformation of the film shape. We perform a thorough parametric analysis to investigate the combined effects of elastic, inertia, capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that increasing fluid elasticity decreases the two wetting lengths along each side wall of a slit. On the contrary, the wetting of the slit is enhanced by the shear-thinning of the fluid. Multiple steady solutions connected by turning points forming a hysteresis loop and transcritical bifurcations as well as isolated solution branches are revealed by pseudo-arc-length continuation. In particular, it is predicted that under certain conditions, the transition from the capillary to the inertia regime is not smooth; instead a hysteresis loop arises. This is the signature of an abrupt decrease of the film penetration with increasing flow rate since higher deformations cannot be sustained. Additionally, we have performed calculations for a wide range of the geometrical characteristics of the substrate. We find that the elastic effects of the fluid become more pronounced when the slits are closer together in their periodic arrangement and of width decreasing to become almost comparable to the capillary length of the liquid. These can lead to almost no wetting of the slit, i.e. the Cassie-Baxter state, even in a hydrophilic surface.

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.

Electroosmotic Flow Behavior of Viscoelastic LPTT Fluid in a Microchannel.

In many research works, the fluid medium in electroosmosis is considered to be a Newtonian fluid, while the polymer solutions and biological fluids used in biomedical fields mostly belong to the non-Newtonian category. Based on the finite volume method (FVM), the electroosmotic flow (EOF) of viscoelastic fluids in near-neutral (pH = 7.5) solution considering four ions (K+, Cl−, H+, OH−) is numerically studied, as well as the viscoelastic fluids’ flow characteristics in a microchannel described by the Linear Phan-Thien–Tanner (LPTT) constitutive model under different conditions, including the electrical double layer (EDL) thickness, the Weissenberg number (Wi), the viscosity ratio and the polymer extensibility parameters. When the EDL does not overlap, the velocity profiles for both Newtonian and viscoelastic fluids are plug-like and increase sharply near the charged wall. Compared with Newtonian fluid at Wi = 3, the viscoelastic fluid velocity increases by 5 times and 9 times, respectively, under the EDL conditions of kH = 15 and kH = 250, indicating the shear thinning behavior of LPTT fluid. Shear stress obviously depends on the viscosity ratio and different Wi number conditions. The EOF is also enhanced by the increase (decrease) in polymer extensibility parameters (viscosity ratio). When the extensibility parameters are large, the contribution to velocity is gradually weakened.

Exact solutions for n-layer concentric flow of PTT fluids through a cylindrical pipe

Flows of multiple layers of fluids are encountered in many industrial and manufacturing processes. This paper investigates the concentric n-layer flow for Phan–Thien–Tanner (PTT) fluids through a cylindrical pipe. Finitely many immiscible non-Newtonian fluids are considered to be flowing concentrically in a tube. The flow is modelled using the exponential PTT fluid model and exact solutions for velocity fields and volume flow rates are computed. It has been shown that the corresponding results for linear PTT fluid model as well as Newtonian fluids can be deduced from the obtained expressions, and that they match with the present literature. It has also been observed that for such layered flow, the non-Newtonian parameters significantly affect the flow of fluids in adjacent layers. The effects of involved parameters on the velocity profiles are also shown graphically. We show that a unique velocity maximum exists along the axis of the pipe. Moreover, it is observed with the help of an example that layer thickness can be adjusted to obtain maximal flow rate with a given pressure gradient.

Forced convective heat transfer of nonlinear viscoelastic flows over a circular cylinder at low Reynolds inertia regime

In this paper, the forced convective heat transfer of viscoelastic fluid flow around a circular cylinder at a low Reynolds number is studied numerically. The effect of viscous dissipation on the problem is modeld which is the main innovative aspect of present study. The Phan–Thien–Tanner (PTT) model is used as the nonlinear constitutive equation. To avoid divergence and stabilize the numerical process in high elastic cases, the log-conformation approach is used. The results indicated that the Nusselt number (Nu) monotonically increases with increasing elasticity number (El), retardation ratio (β), Prandtl number (Pr), and Brinkman number (Br) in a wide range. Both drag reduction (for El < 0.025) and drag enhancement (for El > 0.025) were seen in the numerical results. In high elastic flow regime (El > 10), due to the increased storing nature of viscoelastic fluid compared to the dissipative nature, drag coefficient (CD) and Nu become constant, which causes the fluid to behave like a Newtonian fluid. The temperature-dependent properties were also investigated for all of fluid property, and the large effects of viscosity, and relaxation time were seen in hydrodynamic and thermal parameters. The contribution of normal stresses in viscous dissipation (VD) is much lower than shear stress, due to the lower values of corresponding components in deformation tensor. The elasticity influences the heat transfer by two mchanisms of convection empowering as an indirect way and viscous dissipation as a direct way. Increasing El, β, and ξ increases Nu via the convection mechanism (even at Br = 0), while Nu is decreased with ε. However, Nu increases with the variations of all the parameters, in presence of viscous dissipation. It was also seen that the heat transfer direction changes for Br > 90. Finally, the effects of El, β, ε, ξ, Pr, and Br on the distribution of stress components, velocity, temperature, Nu, and CD were investigated. Two accurate correlations were proposed for the variations of CD and Nu against El, which were validated for a wide range of El.

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.

Influence of non-Newtonian blood flow models on drug deposition in the arterial wall

In this paper, we investigate the influence of non-Newtonian blood flow models on drug diffusion from a coronary drug-eluting-stent (DES). We consider the Oldroyd-B, Phan-Thien-Tanner (PTT) and Giesekus viscoelastic models for the description of fluid dynamics of blood. The model for blood flow is coupled with plasma filtration and mass transport from a DES. The model for the transport problem takes into consideration non-Fickian diffusion, drug dissolution, polymer degradation and binding. We propose an Implicit-Explicit (IMEX) finite element method and show numerical experiments that confirm the effectiveness and order of convergence of the employed methodology. The numerical results indicate that an increase in the elastic properties of the blood leads to an increase of the concentration of free drug in the arterial wall. Finally, we present unsteady simulations comparing the Newtonian and non-Newtonian blood flow models.

The extrudate swell singularity of Phan-Thien–Tanner and Giesekus fluids

The stress singularity for Phan-Thien–Tanner (PTT) and Giesekus viscoelastic fluids is determined for extrudate swell (commonly termed die swell). In the presence of a Newtonian solvent viscosity, the solvent stress dominates the polymer stresses local to the contact point between the solid (no-slip) surface inside the die and the free (slip) surface outside the die. The velocity field thus vanishes like rλ0, where r is the radial distance from the contact point and λ0 is the smallest Newtonian eigenvalue (dependent upon the angle of separation between the solid and free surfaces). The solvent stress thus behaves like r−(1−λ0) and dominates the polymer stresses, which are like r−4(1−λ0)/(5+λ0) for PTT and r−(1−λ0)(3−λ0)/4 for Giesekus. The polymer stresses require boundary layers at both the solid and free surfaces, the thicknesses of which are derived. These results do not hold for the Oldroyd-B fluid.

Electroosmotic Flow of Viscoelastic Fluid in a Nanochannel Connecting Two Reservoirs.

Electroosmotic flow (EOF) of viscoelastic fluid with Linear Phan-Thien–Tanner (LPTT) constitutive model in a nanochannel connecting two reservoirs is numerically studied. For the first time, the influence of viscoelasticity on the EOF and the ionic conductance in the micro-nanofluidic interconnect system, with consideration of the electrical double layers (EDLs), is investigated. Regardless of the bulk salt concentration, significant enhancement of the flow rate is observed for viscoelastic fluid compared to the Newtonian fluid, due to the shear thinning effect. An increase in the ionic conductance of the nanochannel occurs for the viscoelastic fluid. The enhancement of the ionic conductance is significant under the overlapping EDLs condition.

Viscoelastic modeling and simulation for polymer melt flow in injection/compression molding

To describe the characteristics of compressing flow, we propose a new constitutive equation based on the Phan–Thien–Tanner (PTT) model to account for the additional stresses induced by rapid shear rate change. Analytical stress solutions are derived for shear and extensional flows to make this model describe shear thinning and strain hardening of polymers. We establish the uniform viscoelastic flow problem for injection/compression in terms of a non-isothermal compressible fluid, and construct the corresponding variational equations using the finite element method. In order to make the numerical scheme stable and efficient, a pseudo Poisson equation for pressure in weak form is founded. In addition, a two-fold iterative scheme is proposed to decouple velocity, pressure, temperature and stress. We simulate the benchmark 4:1 contraction flow to validate the proposed method and compare with the published results. Furthermore, real injection/compression molding experiments are conducted. The agreements with published and measured data shows that the proposed constitutive model accurately describes compressing flow.

Analysis of forced convection of Phan–Thien–Tanner fluid in slits and tubes of constant wall temperature with viscous dissipation

In the present work, analytical solutions are presented for thermal convection of the linear Phan–Thien–Tanner fluid (LPTT) in slits and tubes of constant wall temperature by taking account of the viscous dissipation term. Unlike the similar previous studies in which the advection term was neglected in the heat transfer equation, it is considered in this investigation. A continuous relation between the Nusselt number and the Brinkman number is obtained. Expressions for the temperature distribution are derived in closed form and in terms of a Frobenius series for the slit and tube flows, respectively. Based on these solutions, the effects of fluid elasticity and Brinkman number on thermal convection of LPTT fluid flows are studied in detail. It is shown that at negative Brinkman numbers (fluid cooling), increasing the Deborah number leads to a decrease in the Nusselt number, but an increase in the centerline temperature. Nonetheless, this trend is opposite for positive Brinkman numbers (fluid heating), i.e., an increase in the Nusselt number and a decrease in the centerline temperature. Also, there is a Brinkman number beyond which the Nusselt number is smaller than zero, meaning that there is weak heat convection in the flow. Also, the results confirm that the extensibility parameter affects the temperature profile in the same way as the Deborah number.

Numerical study of the stress singularity in stick-slip flow of the Phan-Thien Tanner and Giesekus fluids

Stick-slip flow is a challenging viscoelastic benchmark problem due to the presence of a separation or transition point at the die exit where a sudden change in flow boundary conditions occurs. We present numerical simulations of transient planar stick-slip flow of the Phan-Thien–Tanner (PTT) and Giesekus fluids, investigating the polymer stress behavior around the stress singularity at the stick-slip point, confirming the asymptotic results presented by Evans et al. [“Stresses of the Oldroyd-B, PTT and Giesekus fluids in a Newtonian velocity field near the stick-slip singularity,” Phys. Fluids 29, 1–33 (2017)]. In order to improve the numerical knowledge about this viscoelastic benchmark problem, two distinct mathematical methodologies are used for comparison in the computational simulations: the Cartesian and natural stress formulations. The former is widely applied in computational rheology, while the latter is used for the first time in the context of this problem. The natural stress formulation gives improved convergence results both temporally and spatially near to the singularity while maintaining the same global flow characteristics as the Cartesian.

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.

Exact Solution for PTT Fluid on a Vertical Moving Belt for Lift with Slip Condition

Objectives: This attempt is made to investigate the problem of a study of thin film flow for lift while considering a steady, incompressible, non-isothermal, Phan-Thien Tanner fluid on a vertical belt with slip condition. Method/Comparative analysis: The nonlinear differential equation has been derived from the continuity and momentum equations. Exact solution has been obtained, which gives us velocity-profile of fluid, flow rate, temperature, average velocity of fluid and net upward flow. The special cases such as the linear PTT (LPTT), quadratic PTT (QPTT), cubic PTT (CPTT), exponential PTT (EPTT) and upper convected Maxwell (UCM) models are considered, from the same model of PTT fluid. The behavior the velocity and depth is discussed, in effect of various parameters. The velocity and temperature are compared for those special cases. Findings: The analogy of the EPTT, CPTT, QPTT, LPTT and UCM fluid models for the velocity profile and temperature distribution reveals that the velocity profile of the UCM fluid model increases quickly as compared to the velocity of the EPTT fluid. The temperature distribution for EPTT fluid is higher than the temperature for its special case: UCM fluid model. The velocity of PTT fluid is observed to be increased with incorporation of slip condition on vertical belt. Improvements: None of the previous work has been done using no-slip boundary conditions, however slip conditions are used in piece of work. Also, a thorough discussion on cases is novelty in this work. Keywords: Exact Solution, Heat Transfer, Lift Problem, Phan-Thien Tanner, Thin Film Flow

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.

Peristaltic flow of Phan-Thien-Tanner fluid: effects of peripheral layer and electro-osmotic force

The two-layered electro-osmotic peristaltic flow of Phan-Thien-Tanner (PTT) fluid in a flexible cylindrical tube is analyzed. The core (inner) layer fluid satisfies the constitutive equation of PTT fluid model and the peripheral (outer) layer is characterized as a Newtonian fluid. For each region, the two-dimensional conservation equations for mass and momentum with electro-osmotic body forces are transformed from the fixed frame to the moving frame of reference. These equations are further simplified by invoking the constraints of long wavelength and low Reynolds number. Closed-form expressions for velocity and stream function are derived and then employed to investigate the pressure variations, trapping, interface region, and reflux for a variety of the involved parameters. The analysis reveals that the reflux and trapping can be restrained by appropriately tuning the electro-kinetic slip parameter and Deborah number. Further, the pumping efficacy can also be improved by adjusting the rheological and the electro-kinetic effects. These results may be helpful for improving the performance of the microfluidic peristaltic pump.

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.

Stress behaviours of viscoelastic flow around square cylinder

In this study, it is aimed the numerically investigation of the flow of liner PTT (Phan-Thien-Tanner) fluid, which is a viscoelastic fluid model over limited square obstacle by finite volume method. The finite volume method has been used for simultaneous solution of continuity, momentum and fluid model equations with appropriate boundary conditions. The effects of the inertia in terms of Reynolds number, Re, (0 &amp;lt; Re &amp;lt; 20) and the of elasticity in terms of Weissenberg number, We, (1 &amp;lt; We &amp;lt; 15) of PTT flow on vertical and shear stress areas are examined in detail.