Phan-Thien Tanner Fluid Research Articles

The effect of asymmetric zeta potentials on the electro-osmotic flow of a generalized Phan–Thien–Tanner fluid

Electrokinetic flows driven by electro-osmotic forces are especially relevant in micro and nano-devices, presenting specific applications in medicine, biochemistry, and miniaturized industrial processes. In this work, we integrate analytical solutions with numerical methodologies to explore the fluid dynamics of viscoelastic electro-osmotic/pressure-driven fluid flows (described by the generalized Phan–Thien–Tanner (gPTT) constitutive equation) in a microchannel under asymmetric zeta potential conditions. The constitutive equation incorporates the Mittag–Leffler function with two parameters (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} and β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}), which regulate the rate of destruction of junctions in a network model. We analyze the impact of the various model parameters on the velocity profile and observe that our newly proposed model provides a more comprehensive depiction of flow behavior compared to traditional models, rendering it suitable for modeling complex viscoelastic flows.

Stagnant rings and uniform film analysis of Phan-Thien Tanner fluid film flow on a vertically upward moving tube

Analyzing Phan-Thien Tanner fluid film behavior on a vertically upward moving tube can help predictive models in engineering, notably in coating and lubrication operations. This paper provides a theoretical analysis of the dynamics of stagnant rings and uniform Phan-Thien Tanner fluid film adhered to a vertically upward moving tube. Formulated ordinary differential equations are solved to get exact analytic expressions for velocity, flow rate, average velocity, shear stress components, and stagnant rings. Highly nonlinear algebraic equations are solved using Newton's method in MAPLE to find the linear and exponential Phan-Thien Tanner film thicknesses. The uniform film thickness widens with increasing constant tube velocity, while it decreases with increasing Deborah number, Stokes number, and elongation parameter. The analysis delineates that stagnant rings shrink around the tube as the Stokes number, Deborah number, and elongation parameter increase. The minimal Stokes condition for the existence of a realistic stagnant ring is also determined. It has been established that whenever the Stokes number is less than the minimal Stokes condition, stagnant rings do not form. A comparison between the exponential Phan-Thien Tanner, linear Phan-Thien Tanner, upper convected Maxwell, and Newtonian fluids is also provided for stagnant rings and fluid film thickness. Following the application of an efficient approximation on tube geometry, plate geometry is approximated, and the outcomes are consistent with existing literature. The results of this research are significant for a wide range of biofluid applications, including agrochemical uses, paint and surface coating flow behavior, thin films on the cornea and lungs, and chemical and nuclear reactor design.

Thermally developing streaming potential-mediated pressure-driven flow of Phan–Thien–Tanner fluid in a microchannel

In this study, we have conducted a semi-analytical investigation into the streaming potential-mediated pressure-driven flow of hydrodynamically fully developed and thermally developing viscoelastic fluid in a parallel plate microchannel. We have utilized a simplified Phan–Thien–Tanner model to describe the rheology of the viscoelastic fluid. Our approach delved into the full Poisson–Boltzmann equation, deriving exact analytical solutions for the electrostatic potential distribution and velocity profile. Furthermore, we concurrently derived semi-analytical solutions for the temperature distribution and Nusselt number, accounting for the effects of heat generation from viscous dissipation and Joule heating in thermally developing flows. We have demonstrated that an increase in the degree of surface charge triggers the streaming potential field, while the volumetric flow rate escalates with the viscoelastic parameter εWik¯2. Moreover, we have observed that the magnitude of the dimensionless temperature decreases with increasing values of the effective Joule heating parameter Speff. Our analysis reveals that the streaming potential effect hampers fluid flow, resulting in an increase in the bulk fluid temperature and consequently reducing the heat transfer rate. We observe that the magnitude of the Nusselt number decreases with increasing Speff. The entropy generation analysis reveals that increasing the Peclet number amplifies flow and temperature gradients, leading to higher fluid irreversibility in microchannels. The Bejan number experiences a significant decrease across the channel, reaching its minimum at a specific axial location before stabilizing further downstream. We find that heat transfer irreversibility predominantly influences system irreversibility, except for the Brinkmann number Br = 0.01, where convective heat transfer dominates at the entrance region, transitioning to friction losses beyond the thermal entry zone.

Flow analysis of temperature-dependent variable viscosity Phan Thien Tanner fluid thin film over a horizontally moving heated plate

Comprehending the temperature-dependent variable viscosity Phan Thien Tanner fluid film flow over a horizontal heated plate with surface tension is essential to enhancing coating and lubrication process prediction models. This paper accords with the flow analysis of a temperature-dependent variable viscosity Phan Thien Tanner fluid film over a horizontal heated plate. The flow over a heated plate occurs as a result of the plate’s motion and the surface tension gradient. The Adomian decomposition method is utilized to solve a system of linear and nonlinear ordinary differential equations, resulting in series-form solutions. The series-form expressions for flow variables such as velocity, temperature, volume flow rate, and surface tension are derived. Moreover, the positions of stationary points are computed using MATHEMATICA. The analysis delineated that when the inverse capillary number, variable viscosity parameter, Deborah number, elongation parameter, and Brinkman number increase, the stationary points move closer to the heated plate. The temperature also rises with an increase in these parameters. The temperature rises with increasing viscous dissipation while it lowers with increasing thermal diffusion. When the Deborah number is high, the Phan Thien Tanner fluid behaves like a solid and the flow is only driven by the motion of the plate. A comparison between Newtonian and Phan Thien Tanner fluids is made for velocity, temperature, and stationary points as a special case.

Numerical simulation of vortex rings in non-Newtonian fluids

The impulsive viscous flow through an orifice produces vortex rings that self-propagate until total dissipation of the vorticity. This work aims to study numerically such vortex rings for different types of non-Newtonian fluids, including the power-law, Carreau and simplified Phan-Thien-Tanner (PTT) rheological models, with particular focus on the post-formation phase. The simulations were carried out with the finite-volume method, considering small stroke ratios (L/D ≤ 4) and laminar flow conditions (ReG ≤ 103), while parametrically varying shear-thinning, inertia and elasticity. The vortex rings generated in power-law fluids revealed some peculiar features compared to Newtonian fluids, such as a faster decay of the total circulation, a reduction of the axial reach and a faster radial expansion. The behavior in Carreau fluids was found to be bounded between that of power-law and Newtonian fluids, with the dimensionless Carreau number controlling the distance to each of these two limits. The vortex rings in PTT fluids showed the most disruptive behavior compared to Newtonian fluids, which resulted from a combined effect between inertia, elasticity and viscous dissipation. Depending on the Reynolds and Deborah numbers, the dye patterns of the vortex rings can either move continuously forward or unwind and invert their trajectory at some point. Elasticity resists the self-induced motion of the vortex rings, lowering the axial reach and creating disperse patterns of vorticity. Overall, this work shows that the particular non-Newtonian rheology of a fluid can modify the vortex ring behavior typically observed in Newtonian fluids, confirming qualitatively some experimental observations.

ELECTRO-OSMOTIC EFFECT ON PERISTALTIC FLOW OF PHAN–THIEN–TANNER FLUID IN A PLANAR CHANNEL

Abstract There are several factors that can cause the excessive accumulation of biofluid in human tissue, such as pregnancy, local traumas, allergic responses or the use of certain therapeutic medications. This study aims to further investigate the shear-dependent peristaltic flow of Phan–Thien–Tanner (PTT) fluid within a planar channel by incorporating the phenomenon of electro-osmosis. This research is driven by the potential biomedical applications of this knowledge. The non-Newtonian fluid features of the PTT fluid model are considered as physiological fluid in a symmetric planar channel. This study is significant, as it demonstrates that the chyme in the small intestine can be modelled as a PTT fluid. The governing equations for the flow of the ionic liquid, thermal radiation and heat transfer, along with the Poisson–Boltzmann equation within the electrical double layer, are discussed. The long-wavelength ( $\delta \ll 1$ ) and low-Reynolds-number approximations ( $Re \to 0$ ) are used to simplify the simultaneous equations. The solutions analyse the Debye electronic length parameter, Helmholtz–Smoluchowski velocity, Prandtl number and thermal radiation. Additionally, streamlines are used to examine the phenomenon of entrapment. Graphs are used to explain the influence of different parameters on the flow and temperature. The findings of the current model have practical implications in the design of microfluidic devices for different particle transport phenomena at the micro level. Additionally, the noteworthy results highlight the advantages of electro-osmosis in controlling both flow and heat transfer. Ultimately, our objective is to use these findings as a guide for the advancement of lab-on-a-chip systems.

Analytical study of the annular flow of a generalised Phan-Thien–Tanner fluid

AbstractThe annular flow of complex viscoelastic fluids, described by the generalised Phan-Thien–Tanner model, is studied. This model considers the Mittag-Leffler function instead of the usual linear or exponential functions of the trace of the stress tensor, and includes two new parameters that provide additional fitting flexibility. We derive a semi-analytical solution that provides a better understanding of the behaviour of this type of fluid in annular flows and also helps to improve the modelling of complex materials.

Thermally developing combined electroosmotic and pressure-driven flow of Phan–Thien–Tanner fluids in a microchannel

We present semi-analytical solutions for the hydrodynamically developed and thermally developing flow of a non-Newtonian fluid through an isothermal rectangular microchannel. The fluid motion is actuated by the combined consequences of the electroosmotic and pressure-gradient forces. For the rheological behavior of the non-Newtonian fluid, we have used the simplified Phan–Thien–Tanner viscoelastic model. Going beyond the Debye Hückel linearization approximation, we have used the full-scale solution for the electrical double-layer potential equation to obtain the exact analytical solutions for the velocity, flow rate, and shear rate parameters. In contrast, the temperature distribution and heat transfer for the thermally developing flow have been obtained by solving the energy equation numerically considering the effects of volumetric heat generation due to Joule heating and viscous dissipation. We find that a larger value of the viscoelastic set ε̃Wĩk2 contributes toward the net gain in flow rate. Both the normal and shear stress increase for increasing ε̃Wĩk2, while the shear viscosity reduces with a degree of surface charging. The average shear viscosity reduces with the degree of surface charging and at higher ε̃Wĩk2 values. The heat transfer is enhanced for augmenting ε̃Wĩk2, although the thermal entrance region gets contracted for a pure electroosmotic flow at higher Peclet numbers. Our study reveals that the heat transfer rate can be amplified by effectively modulating the degree of surface charging and ε̃Wĩk2. We have also carried out an entropy generation analysis, which shows the dominance of heat transfer irreversibility over fluid friction irreversibility. We believe that the present research will offer essential approaches for designing advanced energy-efficient microchannels appropriate to modern industrial applications using viscoelastic fluids.

Metachronal wave form analysis on cilia-driven flow of non-Newtonain Phan–Thien–Tanner fluid model: A physiological mathematical model

The Phan–Thien–Tanner non-Newtonian model of fluid flow is considered to mathematically model the cilia governed flow inside the human body. The collective action of cilia generates a rhythmic motion in form of metachronal wave traveling along the wall of channel. We have utilized the necessary assumptions to non-dimensionalize this complex problem and the exact solutions are obtained for the cilia governed flow problem. The graphical results are also incorporated to further discuss these solutions in detail. The wavy movements developed due to rhythmic actions of cilia are clearly evident in pressure gradient graphs. Further, a parabolic flow pattern is also observed in this ciliated channel flow problem. Streamlines disclose the trapping phenomenon and the multiple pumping regions can be seen in intrigued pressure rise graphical solutions.

Numerical verification of sharp corner behavior for Giesekus and Phan-Thien–Tanner fluids

We verify numerically the theoretical stress singularities for two viscoelastic models that occur at sharp corners. The models considered are the Giesekus and Phan-Thien–Tanner (PTT), both of which are shear thinning and are able to capture realistic polymer behaviors. The theoretical asymptotic behavior of these two models at sharp corners has previously been found to involve an integrable solvent and polymer elastic stress singularity, along with narrow elastic stress boundary layers at the walls of the corner. We demonstrate here the validity of these theoretical results through numerical simulation of the classical contraction flow and analyzing the 270° corner. Numerical results are presented, verifying both the solvent and polymer stress singularities, as well as the dominant terms in the constitutive equations supporting the elastic boundary layer structures. For comparison at Weissenberg order one, we consider both the Cartesian stress formulation and the alternative natural stress formulation of the viscoelastic constitutive equations. Numerically, it is shown that the natural stress formulation gives increased accuracy and convergence behavior at the stress singularity and, moreover, encounters no upper Weissenberg number limitation in the global flow simulation for sufficiently large solvent viscosity fraction. The numerical simulations with the Cartesian stress formulation cannot reach such high Weissenberg numbers and run into convergence failure associated with the so-called high Weissenberg number problem.

Mixed electroosmotic/pressure-driven flow for a generalized Phan–Thien–Tanner fluid in a microchannel with nonlinear Navier slip at the wall

This work derives an approximate solution of the laminar viscoelastic fluid flow through a parallel flat plate microchannel, driven by electroosmotic and external pressure forces. In contrast to other works published in the past, in the present analysis, the fluid’s rheology obeys the generalized Phan–Thien–Tanner model that has been proposed recently, which uses the Mittag-Leffler function instead of linear or exponential functions of the trace for the stress tensor. The hydrodynamic analysis considers a non-linear Navier slip law at the wall that follows a power-law behavior on the shear stress. For the electric potential in the electric double layer, the Debye–Hückel approximation was used, and it is assumed that the wall’s zeta potentials are symmetric. The solution obtained in this work depends on several dimensionless parameters that allow controlling the flow: ɛWi2 characterizes the fluid’s viscoelasticity, the electrokinetic parameter κ̄, the dimensionless slip coefficient L̄, the slip-power exponent m, and the ratio between pressure and electroosmotic forces, G. Besides, the influence of two parameters of the Mittag-Leffler function, α and β, notably affects the flow’s characteristics. Our results evidence the important consequences of considering the generalized Phan–Thien–Tanner model and the slippage effect on the flow field compared to that described by the Phan–Thien–Tanner model.

Thermodynamic of Ion-slip and magnetized peristalsis channel flow of PTT fluid by considering Lorentz force and Joule heating

The main focus of present attempt is model a peristaltic phenomenon of Phan-Thien-Tanner (PTT) fluid with consideration of Joule heating, Hall and ion-slip and dissipation effects. The Lorentz force is further considered to investigate the fluid flow transport in a channel. Electrically conducted fluid is considered in a symmetric channel. Energy conservation principles is employed to inspect the heat transfer analysis. Explicit expressions of stream function, pressure gradient and temperature for small Weissenberg number are constructed. Solutions of modeled problem is computed by employing regular perturbation technique. Graphical views of significant variables on axial velocity, pressure gradient and temperature are interpreted. The phenomena for pumping and trapping are carefully analyzed. Outcomes disclose that velocity of the channel boosts up against Hall current and ion slip variables. Furthermore, the presence of Joule heating greatly effects the behavior of magnetic and Hall current variables on temperature. The outcomes of current model could be utilized in drug delivery and controlling the blood flow systems.

Numerical simulations for electro-osmotic flow of PTT fluids in diverging microchannel

In the current work, the electroosmotic coupled pressure driven flow through a diverging microchannel has been numerically investigated for the simplified Phan Thien Tanner (PTT) fluids. The Debye-Hückel approximation is used for the linearization of electric potential distribution in the electric double layer and the Debye length has been kept fixed (κH = 20). The pressure and electroosmotic forces are coupled and by, Г, the forcing ratio (the ratio of pressure to the electrokinetic gradient in the direction of flow), which is varied in the range of −4 ≤Г≤ 4. The viscoelastic fluid rheology has been modulated by the combined effect of the characteristic Deborah number and the PTT parameter as 0.1 ≤εDe≤ 5. The influence divergent angle (5° ≤α≤ 15°) has also been demonstrated to analyze the role of channel geometry on the resulting flow kinematics. The resulting variations in the flow field have been presented and discussed in the form velocity vectors, surfaces and velocity profiles at the outlet of the channel.

Full linear Phan-Thien–Tanner fluid model: Exact analytical solutions for steady, startup, and cessation regimes of shear and extensional flows

Exact, fully explicit, purely real analytical expressions for the material functions describing steady, startup, and cessation regimes of shear flows and of planar, uniaxial, and biaxial extensional flows of full linear Phan-Thien–Tanner fluids are obtained. These expressions, which have no analogs in the literature, are formulated in compact, beautiful forms, partially due to the unique scaling procedure reducing the number of the model parameters from four to one. The properties of the material functions are investigated in detail. For steady extensional flows, the possible shapes of the extensional viscosity curves are described and the conditions for these shapes to occur are determined. For startup flows, it is found when exactly the stress dynamics is oscillatory, and, in this case, a detailed characterization of oscillations is given, which includes expressions for the position and magnitude of stress overshoots and undershoots.

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.

Fluid model using recursive approach: Application to permeable slit with uniform reabsorption and velocity slip

The present mathematical study discussed the two-dimensional slow flow of linear Phan Thien Tanner (LPTT) fluid by taking into account the permeable slit with slippage walls and reabsorption undergoes uniformly through slit walls. Mathematical modeling of the two-dimensional flow situation is carried out in Cartesian coordinates and as result, a system of nonlinear PDEs along with nonhomogeneous boundary conditions are solved analytically using the recursive approach. The explicit expressions for velocity components in axial and radial directions along with stream function, pressure fields, normal stresses difference, and shear stress are obtained which are strongly dependent on the elongation parameter, Weissenberg number, slip parameter, and porosity parameter. The leakage flux and axial flow rate are calculated. The points are identified for maximum velocity. Graphical analysis is also established and found that an increment in the fluid elasticity enhances the magnitude of axial velocity and a significant change is noticed in the profile of axial velocity due to the extensional characteristic of the LPTT fluid. This theoretical study helps to understand the physical phenomena that describe the 2-dimensional flow through the permeable slit and have significant importance in industry and biosciences.

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.

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.

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.