Phan-Thien Tanner Research Articles

A comparison between the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear

The finitely extensible nonlinear elastic with Peterlin closure (FENE-P) and simplified Phan-Thien–Tanner (sPTT) viscoelastic models are used widely for modelling of complex fluids. Although they are derived from distinct micro-structural theories, these models can become mathematically identical in steady and homogeneous flows with a particular choice of the values of the model parameters. However, even with this choice of parameter values, the model responses are known to differ from each other in transient flows. In this work, we investigate the responses of the FENE-P and sPTT constitutive models in large-amplitude oscillatory shear (LAOS). In steady shear, the shear stress scales with the non-dimensional group $Wi/(aL)$ ( $Wi\,\sqrt {\epsilon }$ ) for the FENE-P (sPTT) model, where $Wi$ is the Weissenberg number, $L^2$ is the limit of extensibility in the FENE-P model ( $a$ being $L^2/(L^2-3)$ ), and $\epsilon$ is the extensibility parameter in the sPTT model. Our numerical and analytical results show that in LAOS, the FENE-P model shows this universality only for large values of $L^2$ , whereas the sPTT model shows it for all values of $\epsilon$ . In the strongly nonlinear region, there is a drastic difference between the responses of the two models, with the FENE-P model exhibiting strong shear stress overshoots that manifest as self-intersecting secondary loops in the viscous Lissajous curves. We quantify the nonlinearity exhibited by each constitutive model using the sequence of physical processes framework. Despite the high degree of nonlinearity exhibited by the FENE-P model, we also show using fully nonlinear one-dimensional simulations that it does not shear band in LAOS within the range of conditions studied.

Viscoelastic vapor bubble collapse near solid walls and corresponding shock wave formation

This study investigates the influence of viscoelasticity on the collapse of aspherical vapor bubbles near a solid boundary through numerical simulations. A fully compressible three-dimensional finite volume method is employed, incorporating a single-fluid homogeneous mixture cavitation model and the simplified linear Phan-Thien Tanner viscoelastic constitutive model. The collapse dynamics, liquid jetting, shock wave formation, and associated pressure impact are analyzed, and the viscous and viscoelastic stress fields are presented. A comparison of viscoelastic to Newtonian dynamics reveals significant differences in collapse behavior and shock wave formation due to viscoelasticity. Viscoelasticity can induce jet piercing, which is not observed in the Newtonian collapse, and increases vapor re-evaporation after the first collapse. The effect of changing the initial standoff distance is examined for both viscoelastic and Newtonian fluids, where a second jet formation is present only for the viscoelastic collapse, and the second collapse's intensity is increased due to increased vapor production during rebound. Additionally, the variation of elasticity in the viscoelastic case demonstrates a correlation between the amount of vapor produced during rebound and the relaxation time for the investigated cases.

Electro-pumping paradigm of non-Newtonian blood with tetra-hybrid nanoparticles infusion in a ciliated artery

The electrokinetics-based pumping mechanism is on emergence to cater to a wide range of applications in medical sciences, bioengineering, microscale transport, biosensing, medical process monitoring, and beyond, owing to its versatile capabilities. This research aims to report the electro-pumping paradigm of Phan–Thien–Tanner (PTT) blood flow infused with tetra-hybrid nanoparticles in the ciliated artery pathway and examine how cilia on the artery’s wall impact the proposed pumping process, taking into account the presence of entropy generation. The devised streaming model is configured by encountering the assembled impact of heat origination, viscous and Joule heating, and thermal slip boundary conditions. Blood serves as the base fluid containing dispersed nanoparticles of gold, silver, alumina, and titania. To acquire solutions for the dimensionless model equations, a novel semi-analytical technique is prosecuted. The computational outcomes unveil significant contributions from key control parameters, shedding light on their substantial impacts and implications. The worthy findings are that electro-osmotic forces have been seen to have a significant impact on the mobility of tetra-hybrid nanoparticles-infused blood through the artery, potentially aiding in maintaining adequate blood pressure during cardiac surgeries for medical practitioners. An uprising in the Joule heating parameter is associated with a significant boost in the blood temperature profile. Longer cilia lengths lead to an uplift in entropy generation within the artery.

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.

Numerical simulations of Phan-Thien-Tanner viscoelastic fluid flows based on the SPH method

Numerical simulations of viscoelastic fluid flows are always a hot issue in the field of computational fluid dynamics and the study of their complex rheological properties has important academic and engineering application value. In the present work, we develop a smoothed particle hydrodynamics (SPH) method for simulating transient viscoelastic fluid flows governed by the Phan-Thien-Tanner (PTT) constitutive equation. To improve the computational accuracy of the SPH method, the mixed symmetric correction algorithm of kernel gradient is implemented. To remove the particle clustering and unphysical fracture in fluid stretching which is named as the tensile instability, the artificial stress model is added into the momentum equation. We firstly apply the proposed SPH method to solve the plane Poiseuille flow of a PTT viscoelastic fluid, in which the effectiveness and advantage of the method are verified by comparing the SPH solution with those obtained by the finite volume method (FVM) and analyzing the l2 norm error of different SPH solutions to the FVM solution. Then, the method is employed to simulate the impact behavior of a PTT viscoelastic droplet with a rigid plate. In particular, we not only investigate the spreading behavior of PTT viscoelastic droplet after impacting the rigid plate, but also for the first time capture and analyze the bouncing behavior of droplet by decreasing the Reynolds number. The influences of the Reynolds number, the Weissenberg number, the solvent viscosity ratio, and the PTT elongational parameter on the droplet dynamics behavior are further deeply studied. Numerical results demonstrate that the SPH method proposed in this paper is a powerful computation tool for simulating PTT viscoelastic fluid flows and is capable of effectively describing their complex rheological properties and free surface variation characteristics.

Induced-charge electroosmosis flow of viscoelastic fluids under different voltage arrangements

Efficient mixing of chemical analysis reagents with laboratory samples at a microscale is a key issue in numerous biomedical and chemical analyses but hardly to implement due to the limited of the low diffusivity in laminar flow. Induced-charge electroosmosis flow, as an innovative mixing method, has been proved to be effective and simple in rapid mixing attributes to its mechanism of vortex generation. This work aims to propose a new strategy for chaotic induced-charge electroosmosis flow based on different voltage arrangements to improve the mixing of viscoelastic fluids. The Phan–Thien–Tanner constitutive model is applied to characterize the flow behavior of viscoelastic fluid in a microfluidic preparation mixer. The direct numerical simulation method is used to solve the fully coupled Navier–Stokes and Poisson–Nernst–Planck equations for a polarizable cylinder in a two-dimensional cavity filled with electrolyte solution. The impact of Weissenberg number (Wi), Debye parameter, voltage strength on the velocity, net charge density, and potential profiles is investigated. The simulation results indicate that a greater Wi leads to the decrease in the maximum velocity, and a large voltage strength can heighten the net charge density and potential, thus improve the peak velocity. Moreover, the classical theoretical prediction that the maximum velocity is proportional to the square of the applied voltage has been authenticated.

Sharp Rates of Decay and Global-in-Time Stability of Large Solutions to The Three-Dimensional Incompressible Phan-Thien–Tanner System of Polymeric Flows

Sharp Rates of Decay and Global-in-Time Stability of Large Solutions to The Three-Dimensional Incompressible Phan-Thien–Tanner System of Polymeric Flows

Experimental validation of two constitutive models for numerical simulation on the extrusion of rubber blends

AbstractNumerical simulation plays a crucial role in polymer processing. This paper analyzes the extrusion of two rubber blends using the Bird–Carreau viscous model and the Phan‐Thien–Tanner (PTT) viscoelastic model. The study found that wall slip phenomena occurred in both rubber blends when the apparent shear stress exceeded the critical value. Under these conditions, the physical parameters of the two constitutive models, along with the wall slip parameters that describe the degree of rubber sliding, were obtained. This information provided a theoretical basis for the numerical simulation of the extrusion process. The PTT model was found to be superior to the Bird–Carreau rheological model in predicting the rheological behavior of die swell, velocity field, and pressure field distribution in the extrusion process of the two rubber blends.Highlights Based on the numerical simulation analysis of the capillary extrusion process, we designed a more feasible method for fitting the physical parameters of the constitutive equation and parameters of the Navier slip model and established the foundation for the subsequent numerical simulation of extrusion processing. Phan‐Thien–Tanner model is proved to be the best candidate for predicting the extrusion of rubber blends with high Mooney viscosity, including die swell, velocity, and pressure distribution. Demonstrates that wall slip is an important factor in rubber extrusion. Combining simulation results with experimental data improves the accuracy and predictability of the simulation.

Flow and heat transfer in Phan-Thien–Tanner viscoelastic fluid around a rotating hot cylinder in a square enclosure

Flow and heat transfer problems in non-Newtonian fluids are considerably more complicated compared to Newtonian fluids due to their non-linear behavior and complex properties. This paper numerically studied the flow and heat transfer of a viscoelastic fluid around a hot rotating cylinder in a square enclosure. The Phan-Thien–Tanner (PTT) model, one of the most accurate models available to describe the behavior of viscoelastic fluids, was used to obtain the constitutive equations of the viscoelastic fluid, which were then solved by using the finite-volume method with the Rheoheatfoam solver of the OpenFOAM software. The log-conformation approach was employed to avoid the divergence problem at high Weissenberg numbers. Simulations were conducted to investigate the effects of the Weissenberg number, Brinkman number, Reynolds number, and the ratio of the polymeric viscosity to total viscosity on the behavior of fluid flow and heat transfer. The results showed that increasing the Weissenberg number led to a higher shear thinning effect, which decreased the fluid's effective viscosity and viscous dissipation near the cylinder. As a result, heat transfer from the cylinder increased, and the frictional torque applied to the cylinder decreased. However, the shear thinning effect was negligible when the Weissenberg number exceeded 100. Furthermore, the tangential normal stress significantly increased as the Weissenberg number increased up to one. However, with further increases in the Weissenberg number, the tangential normal stress reduced, eventually reaching close to zero at high Weissenberg numbers.

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.

Numerical investigation on the bubble rising dynamics in ratchet channels filled with viscoelastic liquids

In this paper, we investigated the dynamics of a bubble rising inside ratchet channels filled with viscoelastic liquids by means of volume-of-fluid-based direct numerical simulations. The exponential Phan–Thien–Tanner constitutive model was used to describe the rheological behaviors of the nonlinear viscoelastic fluid. The effects of fluid elasticity [characterized by the relaxation time (λ)] and ratchet angle (θ) are mainly discussed in respect of bubble dynamics (e.g., rising velocity, flow field, and stress field, etc.). Our results found that the bubble rise velocity increases with fluid elasticity, and the average bubble velocity can be reduced up to 20% at low elasticity in ratchet channels. In addition, the periodic arrangement of the ratchet influences the distribution of the stress field, the vorticity component, and also the deformation of the entangled polymers in the flow. It was observed that the distributions of the stress field and the trace of the conformation tensor change significantly in a dense ratchet channel compared to a sparse one. Interestingly, the bubble velocity gradually increases after the bubble emerges from the convergent section, whereas it decreases on approaching the convergent section. The dynamical bubbles can be manipulated by the surrounding fluid viscoelasticity and ratchet channels, which will be useful in oil extraction and chemical process involving complex non-Newtonian fluids.

On spherical vapor bubble collapse in viscoelastic fluids

Cavitating vapor bubbles, occur in variety of engineering applications, employing complex fluids as operating medium. Especially in biomechanics, biomedical applications as well as in polymer processing, fluids exhibit viscoelastic properties and fundamentally different behavior than Newtonian fluids. To explain the influence of viscoelasticity on cavitation a detailed understanding of the viscoelastic influence on bubble dynamics and of the underlying mechanisms is essential. With this study we provide in-depth numerical investigations of the spherical vapor bubble collapse in viscoelastic fluids. A compressible, density-based, 3D finite-volume solver with explicit time integration is used together with a conservative, compressible formulation for the constitutive equations of three different viscoelastic models, namely the upper convected Maxwell model, the Oldroyd-B model and the simplified linear Phan-Thien Tanner model. 3D simulations of the collapse dynamics are carried out to show the viscous and viscoelastic stress development during the collapse, and its relation to the occurring deformations. Collapse behavior is investigated for various elasticity, viscosity and constitutive models. It is demonstrated that viscoelasticity fundamentally alters the collapse behavior and evolution of stresses. It is observed that viscoelastic stresses develop with a time delay proportional to elasticity and show different spatial distributions as opposed to Newtonian stresses. Viscoelasticity introduces isotropic stress components even though the spherical collapse leads to purely deviatoric (elongational) deformation. Furthermore, the distinct influence of constitutive models is illustrated and the influence of viscoelastic models with solvent contribution is explained. For the upper convected Maxwell model, we show that for increased elasticity shock wave emission can be observed depending on the applied grid resolution.

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.

Joule heating and Soret effects on an electro-osmotic viscoelastic fluid flow considering the generalized Phan-Thien–Tanner model

This work analyzes the non-isothermal electro-osmotic fluid flow in a microchannel considering the Soret effect and temperature-dependent properties. The constitutive equation that models the fluid rheology corresponds to the generalized Phan-Thien–Tanner (gPTT) model. Temperature and pressure gradients are induced due to the interaction between an ionized fluid and the electrical field imposed at the microchannel's ends, resulting in Joule heating. The temperature-dependent physical properties of the fluid modify the ionic distribution in the electric double layer and its thickness change along the microchannel walls. The generalized Phan-Thien–Tanner (gPTT) model is used as a constitutive equation that describes the fluid rheology, where the trace-stress tensor is based on the Mittag–Leffler function, which represents the destruction of physical junctions and entanglements in the Lodge–Yamamoto network of viscoelastic fluids, through the inclusion of two fitting parameters: α and β. The gPTT model allows better fitting and flexibility to experimental data and a wider range of variation in the description of rheological responses of complex fluids. The hydrodynamics and thermodiffusion obtained through the gPTT model are compared against that using the linear form of the Phan-Thien–Tanner model (lPTT).

Vanishing limit for the three-dimensional incompressible Phan-Thien–Tanner system

This paper focuses on the vanishing limit problem for the three-dimensional incompressible Phan-Thien–Tanner (PTT) system, which is commonly used to describe the dynamic properties of polymeric fluids. Our purpose is to show the relation of the PTT system to the well-known Oldroyd-B system (with or without damping mechanism). The suitable a priori estimates and global existence of strong solutions are established for the PTT system with small initial data. Taking advantage of uniform energy and decay estimates for the PTT system with respect to time $t$ and coefficients $a$ and $b$, then allows us to justify in particular the vanishing limit for all time. More precisely, we prove that the solution $(u,\,\tau )$ of PTT system with $0\leq b\leq Ca$ converges globally in time to some limit $(\widetilde {u},\,\widetilde {\tau })$ in a suitable Sobolev space when $a$ and $b$ go to zero simultaneously (or, only $b$ goes to zero). We may check that $(\widetilde {u},\,\widetilde {\tau })$ is indeed a global solution of the corresponding Oldroyd-B system. In addition, a rate of convergence involving explicit norm will be obtained. As a byproduct, similar results are also true for the local a priori estimates in large norm.

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.

Time-dependent oscillating viscoelastic Rayleigh-Bénard convection: Viscoelastic kinetic energy budget analysis

Numerical simulations of viscoelastic Rayleigh-B\'enard convection (VRBC) are conducted to investigate time-dependent oscillating convection leading to the formation of reverse flowing cells. A viscoelastic kinetic-energy budget method is used to analyze the energy transport and a parametric analysis for the reversal flow dynamics. The emergence of the reversal convection can be explained by potential energy transfer between flow and fluid elasticity. The image shows periodic total kinetic energy (${E}_{t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}l}$), energy exchange rate (${\mathrm{\ensuremath{\Phi}}}_{G}$), and buoyancy flux (${\mathrm{\ensuremath{\Phi}}}_{F}$) time evolution in VRBC from the local view, during a half-period of a reverse process for Phan-Thien-Tanner (PTT) fluid.

On the effects of fluid elasticity and gas holdup on Taylor bubble rising dynamics in viscoelastic media

In this work, the rise of Taylor bubbles in a vertical tube filled with viscoelastic media is investigated by means of volume-of-fluid-based direct numerical simulations. The rheological behaviors of the nonlinear viscoelastic liquids are described by the exponential Phan-Thien–Tanner constitutive model. The applicability of our simulations to capture the liquid film around a Taylor bubble has been validated by comparisons with numerical and experimental data in the literature. The effects of fluid elasticity [indicated by the Weissenberg (Wi) number] and gas holdup are mainly discussed in respect of Taylor bubble dynamics (e.g., rising velocity, flow field, stress field, liquid film, and so on). Our results show that the Taylor bubble steady rise velocity is logarithmically correlated with Wi and the slope is nearly constant at low Wi numbers. When the fluid elasticity is moderate, the viscoelastic stress in the liquid film is large, and the rising bubble is stretched to form a thin filament tail with a negative wake. Moreover, the fluid elasticity has negligible effects on the steady bubble rising velocities and the liquid film thickness for large Wi numbers. With increasing fluid elasticity and gas holdup, the long Taylor bubble fluctuates due to the surrounding liquid pulling deformation and gas pressure. The viscoelastic stress profiles exhibit a major non-monotonic dependence on the distance to the walls, which seems to be squeezed as the liquid film gets thinner under high elasticity. The dynamical Taylor bubble is found to be significantly affected by the surrounding fluid viscoelasticity and partially independent of the gas holdup, which will guide the design of microreactors in chemical applications.

Micro-imbibition electro-magneto-hydrodynamics of viscoelastic fluids with interactive streaming potential

• Imbibition electro-magneto-hydrodynamics of ionic viscoelastic fluid explored • Capillarity of a sPTT fluid with induced streaming potential solved semi-analytically • Roles of Hartmann number, Electroviscous number, and Deborah number investigated • The critical Hartmann number decreases with increase in Deborah number • Capillarity with rheology independent ( x ¯ ∼ t ¯ ), and later, rheology modulated regime ( x ¯ ∼ t ¯ ). Conjugated surface tension and electro-magneto-hydrodynamics driven micro-capillary imbibition of an ionic viscoelastic fluid, in presence of interacting streaming potential, has been investigated in this article. The rheology of the fluid is modelled by appealing to the linearized simplified Phan-Thien Tanner (sPTT) model. By taking into account the electrolytic nature of the fluid, additional electroviscous forces due to induced streaming potential is also considered in the model. Transverse electric and magnetic fields are applied externally to the fluid to investigate the possibility of achieving volumetric flow gain and/or fluidic control using body force manipulation. Considering all the participating mechanisms, we deduce a semi-analytical formalism and present a comprehensive picture on the dependence of various parameters on capillary imbibition dynamics. We explore the roles played by the governing non-dimensional parameters, viz. Hartmann number, Electroviscous number, and Deborah number. For low Hartmann number regimes, there is evidence of augmented imbibition, while at higher values the flow is suppressed. We also show that the critical Hartmann number decreases with increase in Deborah number, allowing for such viscoelastic fluids to achieve more efficient imbibition velocity with less strength of applied fields. The flow regime is noted to consist of two parts – a rheology independent regime ( x ¯ ∼ t ¯ ), and later, a rheology modulated regime ( x ¯ ∼ t ¯ ). These findings are expected to provide a significant understanding of the imbibition hydrodynamics of such complex fluids, and at the same time, provide means for flow augmentation and control in microfluidic devices.

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.