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.

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.

Effect of die exit flow conditions on air-gap film dynamics in two-dimensional film casting processes: Short Communication

The impact of the inlet velocity profile at the die exit on the film dynamics within the air-gap region was investigated using numerical simulations for both Newtonian and viscoelastic Phan-Thien and Tanner (PTT) fluids in isothermal two-dimensional (2-D) film casting processes. In an industrial context, intentional adjustments were made to reduce the inlet velocities at the edge of the casting die, effectively mitigating the edge-beads characterized by a higher edge thickness than the center thickness of the final films. By varying the inlet velocity conditions with decreasing edge velocities, the steady film dynamics were correlated with the onsets of draw resonance instability and frequency responses to a disturbance, which were determined using the transfer function data obtained under tension-controlled conditions. The results revealed that decreasing the inlet velocity at the edge improved not only the formation of films but also the process stability for both Newtonian and viscoelastic fluids. In addition, the sensitivity or frequency response to a disturbance was effectively reduced by decreasing the inlet velocity at the edge. This observed impact of the inlet velocity was closely related to the increased tension levels at take-up and a greater portion of the neck-like deformation type in the air-gap region.

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.

Thermal entry flow problem for Rabinowitsch fluid subject to circular tube and flat channel with uniform heat flux boundary conditions

The heat exchangers, chemical reactors, cooling/heating systems and haemodialysis are specific areas due to which Graetz thermal problem is considered as fundamental flow in the duct along with heat and mass transfer aspects. Owing to its importance we examine the thermal entry flow problem for Rabinowitsch liquid in the pipe and channel ducts along with heat transfer aspects. The heat equation with uniform heat flux conditions is tackled via the superposition principle and separation of variable approach due to the involvement of non-homogeneous boundary conditions. The transformed boundary value problem is solved with the MATLAB algorithm for the calculation of eigenvalues and related eigenfunctions. The special cases for simplified Phan-Thien–Tanner (SPTT) fluid and finite extendable non-linear elastic peterline (FENE-P) fluids are also discussed. For SPTT and FENE-P models the controlling parameter can only take positive values and hence these models can only predict shear-thinning effects. The graphical illustrations of fully developed temperature, local and mean Nusselt numbers are displayed through the main effect brought by the controlling parameter.

On the non-Newtonian blood flow in the elliptical multi-stenosed inclined artery: Analytical approach

The analysis of blood flow in the stenotic artery is imperative as the existence and progression of stenosis effectively disturbs the blood flow in the artery, which may cause vascular disease. In this study, we considered the linearized Phan-Thien–Tanner (PTT) fluid model to study the non-Newtonian nature of blood flow in the stenosed artery inclined at an angle [Formula: see text]. The artery is considered to have multiple stenoses and a cross-section of elliptical shape. The PTT model is appropriate for this analysis as viscoelastic properties and shear-thinning characteristics characterize it. The elliptical cross-section of the artery raises the nonlinearity of the mathematical model and makes it effortful to solve the equations analytically. The mathematical equations of the model are processed to non-dimensional form under the assumptions of mild stenosis, which help us to get the exact solutions of the equations in the elliptical domain. The analytically acquired solutions are investigated in detail by their graphical interpretation. It is analyzed that progressive height of stenosis generates the more significant disorder in the narrowed part of the channel as the velocity reverses its behavior in that region. The advancing values of the Weissenberg number, Grashof number and extensional and heat source parameters cause to lessen the axial velocity and slow the fluid flow. The rising Grashof number has nearly no impact on the velocity in the center of the channel. The wall shear stress aggrandizes with the growth of stenosis height, and its behavior for other physical constraints is similar to the flow velocity. It also has a practical enhancement in the stenotic region of the conduit. It is observed that fluid velocity and wall shear stress have larger values for zero angles of inclination than the inclination angle of [Formula: see text]. The streamlined examination indicates the generation of the vortices in the stenotic part of the artery. Moreover, the flow velocity and wall shear stress have lower values for the nonuniform shape than for the uniform form of stenosis.

Non-Newtonian characteristics of blood flow in a multi-stenosed elliptical artery: A case of sensitivity analysis

The occurrence and growth of stenosis effectively interrupt the blood flow in the artery, which may result in vascular disease. It makes the study of blood flow in the artery narrowed with crucial stenosis. This work studies the non-Newtonian nature of blood flow in a diseased artery with an elliptic cross-section. The artery is harmed due to several stenosis, which diminishes its lumen. The Phan-Thein–Tanner fluid is considered to analyze the non-Newtonian characteristics of blood. The Phan-Thein–Tanner fluid model is much suitable for blood flow analysis because of its viscoelastic and shear thinning properties. The governing equations are processed to dimensionless form by employing dimensionless variables and assumptions for a mild stenosis case. The solutions of the nondimensional equations are acquired analytically. The visual examination of the exact solutions is discussed in detail. The fluid velocity is strongly affected by stenosis height, and a more significant disorder is generated in the constricted region with the growing size of stenosis. The flow velocity is found as a decreasing function of the Weissenberg number. The velocity profile is parabolic and axisymmetric as well. The most significant and least influential physical constraints are identified by completing the local sensitivity analysis.

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.

Study of non-Newtonian synovial fluid flow by a recursive approach

This study analyzes the non-Newtonian synovial fluid flow between the joints in a synovitis, which is a diseased condition due to inflammation of synovial membrane. It is assumed in this study that the secretion of synovial fluid through the inflamed synovial membrane is a linear function of the membrane length. The mathematical modeling of synovial fluid through a synovial membrane is made by the non-Newtonian Linear Phan-Thien–Tanner (LPTT) fluid model through a thin conduit having permeable walls. The nonlinear flow of LPTT fluid gives the non-homogeneous complex boundary value problem, and the recursive approach is used to solve the problem. The flow of synovial fluid along and across the membrane is calculated under the inflamed membrane, and results are displayed through graphs. The axial pressure required for the non-Newtonian fluid flow and deformation of synovial fluid that produces the shearing forces near the synovial membrane are also calculated. The purpose of this research is to observe the shear stress on the synovial fluid and inflammation rate on the flow along the membrane at different position and pressure required for the flow of synovial fluid in diseased condition. The mathematical and graphical results for pressure, flow, volume flux, and streamline are calculated and plotted using the software MATHEMATICA. This study is very helpful for the biomedical engineers to measure the compression force and shear stress on the synovial fluid in a diseased condition and can be controlled by the viscosity of the synovial fluid.

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.

The sharp time decay rates and stability of large solutions to the two-dimensional Phan-Thien–Tanner system with magnetic field

In this paper, we consider the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid in two space dimensions. We focus upon the sharp time decay rates (upper and lower bounds) and global-in-time stability of large strong solutions for the PTT system with magnetic field. Firstly, the convergence of large solutions to the equilibrium have been investigated and these convergence rates are shown to be sharp. We then show that two large solutions converge globally in time as long as two initial data are close to each other. One of the main objectives of this paper is to develop a way to capture L 2 -convergence result via auxiliary logarithmic time decay estimates with the initial data in L p ( R 2 ) ∩ L 2 ( R 2 ). Improving time decay rates for the high-order derivatives of large solutions by using interpolation inequalities. In addition, time-weighted energy estimate, Fourier time-splitting method, semigroup method and iterative scheme have also been utilized.

The sharp time decay rates for the incompressible Phan-Thien–Tanner system with magnetic field in [formula omitted

Our focus in this paper is to investigate the sharp time decay rates (upper and lower bounds) for the higher order spatial derivatives of large solutions to the magnetohydrodynamic (MHD) flow of an incompressible Phan-Thien–Tanner (PTT) fluid. In dimension two, the sharp time decay rates of large solution (u,H) and all its derivatives have been shown by Chen et al. (2022). The L2-decay rate of the stress tensor τ was improved to (1+t)−1, however, the sharp time decay rates of any spatial derivatives of τ cannot be achieved. Based on Fourier time-splitting and semigroup methods, we shall obtain the optimal upper and lower bounds of time decay rates for the higher spatial derivatives of order k (k=1,2) of τ. More precisely, the kth order spatial derivatives of τ decay at (1+t)−1−k2-rate, which will be shown to be sharp. In addition, our result is valid for the incompressible Oldroyd-B system.

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.

Bifurcation analysis for a flow of viscoelastic fluid due to peristaltic activity

In this article, bifurcation analysis is performed to study the qualitative nature of stagnation points and various flow regions for a peristaltic transport of viscoelastic fluid through an axisymmetric tube. The rheological behavior of viscoelastic fluid is characterized by the simplified Phan–Than–Tanner fluid model. An analytic solution in a wave frame is obtained subject to the low Reynolds number and long wavelength approximations. The stagnation points and their bifurcations (critical conditions) are explored by developing a system of autonomous differential equations. The dynamical system theory is employed to examine the nature and bifurcations of obtained stagnation points. The ranges of various flow phenomena and their bifurcations are scrutinized graphically through global bifurcation diagrams. This analysis reveals that the bifurcation in the flow is manifested at large flow rate for high extensional parameter and Weissenberg number. Backward flow phenomenon enhances and trapping diminishes with an increase in the Weissenberg number. At the end, the results of present analysis are verified by making a comparison with the existing literature.