Simplified Phan-Thien Tanner Research Articles

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.

Forced convection in non-circular tubes with non-linear viscoelastic fluids including viscous dissipation

The steady, laminar, non-isothermal fully developed flow of a class of non-linear viscoelastic fluids in tubes of arbitrary contour is analyzed under constant wall heat flux including viscous dissipation. Equations of motion and energy are solved analytically and velocity and temperature fields are determined through an asymptotic approach in terms of the Weissenberg number coupled with the shape factor method a one-to-one and continuous mapping taking the circular boundary into a large, continuous spectrum of non-circular tube contours. The analysis developed is general and covers all members of the family of constitutive models considered as well as a large array of non-circular tubes. The case of tubes with circular and triangular contours are discussed as specific examples for various numerical combinations of the Weissenberg, Reynolds, Péclet and Brinkman numbers and the Nusselt number variation is computed for fluids abiding by the Modified Phan-Thien-Tanner (MPTT) and Simplified Phan-Thien-Tanner (SPTT) models. Newtonian velocity and temperature fields in a large spectrum of non-circular tubes are recovered at the lowest order of the asymptotic analysis. Through a matching procedure we also extend the computation of the Nusselt number in round tubes to any desired value of the Weissenberg number.

Analysis and experiments on the spreading dynamics of a viscoelastic drop

• Spreading dynamics of a sessile viscoelastic drop on a horizontal surface is studied. • Simplified Phan–Thien–Tanner model represents the rheology of the viscoelastic drop. • Theoretical results are validated with our experimental results. • Elastic nature of the viscoelastic fluids reduces the spreading rate of the drop. We study the spreading dynamics of a sessile viscoelastic drop on a horizontal surface, where a simplified Phan–Thien–Tanner (sPTT) model is considered to represent the rheology of viscoelastic drop. We have adopted a macroscopic approach to obtain the temporal evolution of the spreading drop, while to establish the efficacy of the theoretical model, we have validated the results obtained from the mathematical formulation with the experimental results for both the Newtonian (Si-oil) and viscoelastic (PDMS and aqueous solution of CMC and glycerin) drops. Following the framework of Seaver–Berg approximation, the spherical shape of the drop is assumed as a cylindrical disk here. We observe from this study that an increment in the elasticity of the fluid enhances the velocity gradient and increases the viscous dissipation in the drop volume, leading to a reduction in the spreading rate.

Electroosmotic flow of Phan-Thien–Tanner fluids at high zeta potentials: An exact analytical solution

We present a mathematical model to study the electroosmotic flow of a viscoelastic fluid in a parallel plate microchannel with a high zeta potential, taking hydrodynamic slippage at the walls into account in the underlying analysis. We use the simplified Phan-Thien–Tanner (s-PTT) constitutive relationships to describe the rheological behavior of the viscoelastic fluid, while Navier’s slip law is employed to model the interfacial hydrodynamic slip. Here, we derive analytical solutions for the potential distribution, flow velocity, and volumetric flow rate based on the complete Poisson–Boltzmann equation (without considering the frequently used Debye–Hückel linear approximation). For the underlying electrokinetic transport, this investigation primarily reveals the influence of fluid rheology, wall zeta potential as modulated by the interfacial electrochemistry and interfacial slip on the velocity distribution, volumetric flow rate, and fluid stress, as well as the apparent viscosity. We show that combined with the viscoelasticity of the fluid, a higher wall zeta potential and slip coefficient lead to a phenomenal enhancement in the volumetric flow rate. We believe that this analysis, besides providing a deep theoretical insight to interpret the transport process, will also serve as a fundamental design tool for microfluidic devices/systems under electrokinetic influence.

Estimating Heat Transfer Coefficients and Friction Factors in Non-Newtonian Flows between Parallel Plates

ABSTRACTA simplified method, successfully tested previously for flow in circular pipes, is used in this work to estimate the friction factor and Nusselt number in fully-developed laminar flow between parallel plates of non-Newtonian fluids. Both constant wall temperature and constant wall heat flux cases are considered. The methodology was tested using several constitutive equations, including generalized Newtonian fluids, such as the Herschel–Bulkley, Bingham, Casson, and Carreau–Yasuda models, and also the simplified Phan-Thien–Tanner viscoelastic model. The error of the approximate methodology was found to be small, below 3.4%, except for the fluids with yield stress for which the maximum error increased to 8.4% for the cases analyzed, which cover a wide range of shear viscosity curves.

Entropy Generation Minimization in a Pressure-Driven Microflow of Viscoelastic Fluid With Slippage at the Wall: Effect of Conjugate Heat Transfer

We focus on the entropy generation minimization for the flow of a viscoelastic fluid through a parallel plate microchannel under the combined influences of applied pressure gradient, interfacial slip, and conjugate heat transfer. We use the simplified Phan–Thien–Tanner model (s-PTT) to represent the rheological behavior of the viscoelastic fluid. Using thermal boundary conditions of the third kind, we solve the transport equations analytically to obtain the velocity and temperature distributions in the flow field, which are further used to calculate the entropy generation rate in the analysis. In this study, the influential role of the following dimensionless parameters on entropy generation rate is examined: the viscoelastic parameter (εDe2), slip coefficient (k¯), channel wall thickness (δ), thermal conductivity of the wall (γ), Biot number (Bi) and Peclet number (Pe). We show that there exists a particular value of the abovementioned parameters that lead to a minimum entropy generation rate in the system. We believe the results of this analysis could be of helpful in the optimum design of microfluidic system/devices typically used in thermal management, such as micro-electronic devices, microreactors, and microheat exchangers.

Dispersion coefficient in an electro-osmotic flow of a viscoelastic fluid through a microchannel with a slowly varying wall zeta potential

The dispersion coefficient of a passive solute in a steady-state pure electro-osmotic flow (EOF) of a viscoelastic liquid, whose rheological behaviour follows the simplified Phan-Thien–Tanner (sPTT) model, along a parallel flat plate microchannel, is studied. The walls of the microchannel are assumed to have modulated and low $\unicode[STIX]{x1D701}$ potentials, which vary slowly in the axial direction in a sinusoidal manner. The flow field required to obtain the dispersion coefficient was solved using the lubrication approximation theory (LAT). The solution of the electric potential is based on the Debye–Hückel approximation for a symmetric $(z:z)$ electrolyte. The viscoelasticity of the fluid is observed to notably amplify the axial distribution of the effective dispersion coefficients due to the variation in the $\unicode[STIX]{x1D701}$ potentials of the walls. The problem was formulated for two cases: when the Debye layer thickness (EDL) was on the order of unity (thick EDL) and in the limit where the thickness of the EDL was very small compared with the height of the microchannel (thin EDL limit). Due to the coupling between the nonlinear governing equations and the sPTT fluid model, they were replaced by their approximate linearized forms and solved in the limit of $\unicode[STIX]{x1D700}\ll 1$ using the regular perturbation technique. Here $\unicode[STIX]{x1D700}$ is the amplitude of the sinusoidal function of the $\unicode[STIX]{x1D701}$ potentials. Additionally, the numerical solution of the simplified governing equations was also obtained for $\unicode[STIX]{x1D700}=O(1)$ and compared with the approximate solution, showing excellent agreement for $0\leqslant \unicode[STIX]{x1D700}\leqslant 0.3$. Note that the dispersion coefficient primarily depends on the Deborah number, on the ratio of the half-height of the microchannel to the Debye length, and on the assumed variation in the $\unicode[STIX]{x1D701}$ potentials of the walls.

Towards the minimization of thermodynamic irreversibility in an electrically actuated microflow of a viscoelastic fluid under electrical double layer phenomenon

We discuss the entropy generation minimization for electro-osmotic flow of a viscoelastic fluid through a parallel plate microchannel under the combined influences of interfacial slip and conjugate transport of heat. We use in this study the simplified Phan-Thien–Tanner model to describe the rheological behavior of the viscoelastic fluid. Using Navier’s slip law and thermal boundary conditions of the third kind, we solve the transport equations analytically and evaluate the global entropy generation rate of the system. We examine the influential role of the following parameters on the entropy generation rate of the system, viz., the viscoelastic parameter (εDe2), Debye–Hückel parameter κ¯, channel wall thickness (δ), thermal conductivity of the wall (γ), Biot number (Bi), Peclet number (Pe), and axial temperature gradient (B). This investigation finally establishes the optimum values of the abovementioned parameters, leading to the minimum entropy generation of the system. We believe that results of this analysis could be helpful in optimizing the second-law performance of microscale thermal management devices, including the micro-heat exchangers, micro-reactors, and micro-heat pipes.

Analysis of axial annular flow for viscoelastic fluid with temperature dependent properties

Analytical solutions using Homotopy Perturbation Method (HPM) are obtained for viscoelastic and shear-thinning fluid obeying simplified Phan-Thien–Tanner (SPTT) constitutive equation in concentric annulus under steady, laminar, fully developed and axial flow with variable thermo physical properties. The isoflux boundary conditions are employed at both walls and the Reynold's model is considered for temperature-dependent viscosity and relaxation time. Results show that increasing fluid elasticity (εDe2) causes Nusselt number to rise and the difference between annular fluid temperature and wall temperature to drop while the Brinkman number and α parameter have the opposite effect. Also effects of Brinkman number, fluid elasticity (εDe2) and α parameter on velocity profile are similar, i.e., any increase in all there parameters causes the velocity profile to become flatter. Influence of variable physical properties gets weaker by increasing fluid elasticity.

Interaction of gliding motion of bacteria with rheological properties of the slime

Bacteria which do not have organelles of motility, such as flagella, adopt gliding as a mode of locomotion. In gliding motility bacterium moves under its own power by secreting a layer of slime on the substrate. The exact mechanism by which a glider achieves motility is yet in controversy but there are evidences which support the wave-like undulation on the surface of the organism, as a possible mechanism of motility. Based on this observation, a model of undulating sheet over a layer of slime is examined as a possible model of the gliding motion of a bacterium. Three different non-Newtonian constitutive equations namely, finite extendable nonlinear elastic-peterline (FENE-P), Simplified Phan-Thien-Tanner (SPTT) and Rabinowitsch equations are used to capture the rheological properties of the slime. It is found that the governing equation describing the fluid mechanics of the model under lubrication approximation is same for all the considered three constitutive equations. In fact, it involves a single non-Newtonian parameter which assumes different values for each of the considered constitutive relations. This differential equation is solved using both perturbation and semi-analytic procedure. The perturbation solution is exploited to get an estimate of the speed of the glider for different values of the non-Newtonian parameter. The solution obtained via semi-analytic procedure is used to investigate the important features of the flow field in the layer of the slime beneath the glider when the glider is held fixed. The expression of forces generated by the organism and power required for propulsion are also derived based on the perturbation analysis.

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section

Elasto-viscoplastic fluid flow in tubes of arbitrary cross-section is explored. A constitutive equation superposing elasticity on the viscoplastic behavior through a linear combination of the non-linear simplified Phan–Thien–Tanner constitutive model of viscoelasticity and the non-linear Bingham constitutive equation of viscoplasticity is developed. The equation of motion is solved analytically for the longitudinal field for steady flow in tubes of non-circular cross-section. The evolution of the plug and stagnant zones, hallmarks of viscoplastic behavior, is studied when elasticity is present, and the rate of flow is determined in terms of the Weissenberg and Bingham numbers. Elasticity tends to enhance the rate of flow for given viscoplastic conditions, and stagnant and plug zone configurations are substantially altered.

Electro-osmotic and pressure-driven flow of viscoelastic fluids in microchannels: Analytical and semi-analytical solutions

In this work, we present a series of solutions for combined electro-osmotic and pressure-driven flows of viscoelastic fluids in microchannels. The solutions are semi-analytical, a feature made possible by the use of the Debye–Hückel approximation for the electrokinetic fields, thus restricted to cases with small electric double-layers, in which the distance between the microfluidic device walls is at least one order of magnitude larger than the electric double-layer thickness. To describe the complex fluid rheology, several viscoelastic differential constitutive models were used, namely, the simplified Phan-Thien–Tanner model with linear, quadratic or exponential kernel for the stress coefficient function, the Johnson-Segalman model, and the Giesekus model. The results obtained illustrate the effects of the Weissenberg number, the Johnson-Segalman slip parameter, the Giesekus mobility parameter, and the relative strengths of the electro-osmotic and pressure gradient-driven forcings on the dynamics of these viscoelastic flows.

Electroosmotic flow of a Phan-Thien–Tanner fluid in a wavy-wall microchannel

Abstract In this paper the electroosmotic flow (EOF) of a viscoelastic fluid in a wavy-wall microchannel is asymptotically analyzed. The rheological behavior of the fluid corresponds to the simplified Phan-Thien–Tanner model (sPTT). By using the lubrication theory, the governing equations of the flow are considerably simplified, which are written in dimensionless form. For obtaining the solution for the flow and electric fields, and assuming small amplitudes of the waviness of the microchannel walls, the domain perturbation method is used. The EOF is mainly characterized by the following dimensionless parameters: δ modulates the waviness of the walls; n denotes the number of waves along the microchannel and ɛ De κ 2 , which reflects the viscoelastic character of the fluid. The principal results show that the volumetric flow rate is increased when ɛ De κ 2 is increased, due to this parameter influences in an important manner on the velocity gradient at the microchannel wall. Also, for very small values of δ, the volumetric flow rate increases linearly. In contrast, for increasing values of n, the volumetric flow rate diminishes.

Effect of slip on heat transfer and entropy generation characteristics of simplified Phan-Thien–Tanner fluids with viscous dissipation under uniform heat flux boundary conditions: Exponential formulation

This study concerns the heat transfer and entropy generation characteristics of viscoelastic fluid flow modeled by the exponential formulation of simplified Phan-Thien–Tanner (s-PTT) model. This is the first such study in literature of thermal behavior of viscoelastic fluids modeled by the exponential formulation of s-PTT model. The flow between two parallel plates is laminar, hydrodynamically and thermally fully developed, viscous dissipative and subject to uniform heat flux on the walls. The slip velocity boundary condition is imposed on the fluid–solid interface and the slip is captured by three slip laws, namely, Navier's non-linear slip law, Hatzikiriakos slip law, and asymptotic slip law. The governing equations have been solved analytically. Closed form solutions for the velocity distribution have been derived while the temperature distribution is presented in terms of an infinite but convergent series. The results pertaining to the three slip laws have been presented in detail. Finally, a comparison has been made between the results for exponential formulation and those for the linear formulation of the s-PTT model. The comparison shows that results for linear formulation deviate significantly from those for exponential formulation and thus the accuracy of the exponential formulation justifies the extra mathematical complexity which it entails.

Influence of channel aspect ratio on the onset of purely-elastic flow instabilities in three-dimensional planar cross-slots

In this work, we perform creeping-flow simulations of upper-convected Maxwell and simplified Phan-Thien–Tanner fluids to study the purely-elastic steady bifurcation and transition to time-dependent flow in three-dimensional planar cross-slots. By analysing the flow in geometries with aspect ratios ranging from the near Hele-Shaw flow like limit, up to the very deep, two-dimensional limit, we are able to characterize the mechanism of the cross-slot bifurcation with significant detail. We conclude that the bifurcation mechanism is similar to a buckling instability, by which fluid is redirected via paths of least resistance, resulting in the emergence of peripheral stagnation points, above and below the central stagnation point. The intake of matter at the centre via the inlet axis is thus reduced, being compensated by fluid flowing through low resistance corridors along the central vertical axis, above and below the central point. Furthermore, we propose and locally compute a modified Pakdel–McKinley criterion, thereby producing a scalar stability field and suggesting emergent peripheral stagnation points also indirectly contribute to the onset of time-dependent flow.

A new viscoelastic benchmark flow: Stationary bifurcation in a cross-slot

In this work we propose the cross-slot geometry as a candidate for a numerical benchmark flow problem for viscoelastic fluids. Extensive data of quantified accuracy is provided, obtained via Richardson extrapolation to the limit of infinite refinement using results for three different mesh resolutions, for the upper-convected Maxwell, Oldroyd-B and the linear form of the simplified Phan-Thien–Tanner constitutive models. Furthermore, we consider two types of flow geometry having either sharp or rounded corners, the latter with a radius of curvature equal to 5% of the channel’s width. We show that for all models the inertialess steady symmetric flow may undergo a bifurcation to a steady asymmetric configuration, followed by a second transition to time-dependent flow, which is in qualitative agreement with previous experimental observations for low Reynolds number flows. The critical Deborah number for both transitions is quantified and a set of standard parameters is proposed for benchmarking purposes.

Slip flows of Newtonian and viscoelastic fluids in a 4:1 contraction

This work presents a numerical study of the 4:1 planar contraction flow of a viscoelastic fluid described by the simplified Phan-Thien–Tanner model under the influence of slip boundary conditions at the channel walls. The linear Navier slip law was considered with the dimensionless slip coefficient varying in the range [0;4500]. The simulations were carried out for a small constant Reynolds number of 0.04 and Deborah numbers (De) varying between 0 and 5. Convergence could not be achieved for higher values of the Deborah number, especially for large values of the slip coefficient, due to the large stress gradients near the singularity of the reentrant corner.Increasing the slip coefficient leads to the formation of two vortices, a corner and a lip vortex. The lip vortex grows with increasing slip until it absorbs the corner vortex, creating a single large vortex that continues to increase in size and intensity. In the range De=3–5 no lip vortex was formed. The flow is characterized in detail for De=1 as function of the slip coefficient, while for the remaining De only the main features are shown for specific values of the slip coefficient.

Lubrication theory for electro-osmotic flow in a slit microchannel with the Phan-Thien and Tanner model

AbstractIn this work the purely electro-osmotic flow of a viscoelastic liquid, which obeys the simplified Phan-Thien–Tanner (sPTT) constitutive equation, is solved numerically and asymptotically by using the lubrication approximation. The analysis includes Joule heating effects caused by an imposed electric field, where the viscosity function, relaxation time and electrical conductivity of the liquid are assumed to be temperature-dependent. Owing to Joule heating effects, temperature gradients in the liquid make the fluid properties change within the microchannel, altering the electric potential and flow fields. A consequence of the above is the appearance of an induced pressure gradient along the microchannel, which in turn modifies the normal plug-like electro-osmotic velocity profiles. In addition, it is pointed out that, depending on the fluid rheology and the used values of the dimensionless parameters, the velocity, temperature and pressure profiles in the fluid are substantially modified. Also, the finite thermal conductivity of the microchannel wall was considered in the analysis. The dimensionless temperature profiles in the fluid and the microchannel wall are obtained as function of the dimensionless parameters involved in the analysis, and the interactions between the coupled momentum, thermal energy and potential electric equations are examined in detail. A comparison between the numerical predictions and the asymptotic solutions was made, and reasonable agreement was found.

Graetz problem with non-linear viscoelastic fluids in non-circular tubes

A numerical investigation of the Graetz problem in straight pipes of circular and non-circular cross-sections is carried out to analyze the influence of the rheological parameters on the heat transfer enhancement with negligible axial heat conduction and viscous dissipation for a class of non-linear viscoelastic fluids constitutively represented by the simplified Phan-Thien–Tanner (SPPT) model. The analysis considers both constant wall heat flux and constant wall temperature thermal boundary conditions and concludes that the combined elastic and shear-thinning effects represented by the parameter ɛWe2 lead to heat transfer enhancement for low values of the parameter of order O(1) whereas values of the parameter ɛWe2 > O(10) lead to a decrease in the heat transfer rate in the case of constant wall heat flux. Nusselt number distributions in the entrance region of tubes of equilateral triangular, square and rectangular cross-sections as well as Nusselt numbers Nu = f(ɛWe2) for the fully developed flow in these non-circular tubes are reported for the first time for non-linear viscoelastic fluids of the SPPT type. It is concluded that for small values of elasticity (We), the computations based on the methods included in the Polyflow software are in full agreement with analytical results when available and that discrepancies exist for high values of We. Such limitations may not exist with pseudo-spectral methods.

A Simplified Method for Calculating Heat Transfer Coefficients and Friction Factors in Laminar Pipe Flow of Non-Newtonian Fluids

In this work, we propose an approximate methodology to estimate the Nusselt number and friction factor in fully developed non-Newtonian laminar flow in circular pipes for a constant wall heat flux. The methodology was tested using several constitutive equations, including generalized Newtonian fluids and viscoelastic models such as the simplified Phan-Thien–Tanner (sPTT), Herschel–Bulkley, Bingham, Casson, and Carreau–Yasuda constitutive equations. The error of the approximate methodology was found to be smaller than 3.2%, except for the fluids with yield stress for which the maximum error increased to about 8% for the cases analyzed, which cover a wide range of shear viscosity curves.