Flow Of Viscoelastic Fluid Research Articles

An extremum problem for a linear integro-differential system describing creeping flows of a viscoelastic fluid

We consider an optimal control problem for an integro-differential system (with a quadratic cost functional) modeling a three-dimensional creeping flow of an incompressible viscoelastic fluid in a bounded domain with impermeable solid walls. The fluid flow is controlled by the time-dependent external force. The concept of the control operator is proposed. We prove a theorem on the existence of a unique optimal control under the assumption that the set of admissible controls is convex and that it is closed in a suitable function space. Moreover, we obtain a variational ine-quality for the optimal control. The proof of this theorem is based on the application of the Faedo–Galerkin approximation scheme taking into account energy estimates of approximate solutions and using the lemma on the existence and uniqueness of the metric projection of a point onto a closed convex set in a real Hilbert space.

The Viscoelastic Fluid Flow in a Confusor

The planar flow of a viscoelastic fluid is studied in tapered channel (confusor) with different contraction rates (2:1, 4:1, 6:1). A polymer melt is considered as a specific liquid, rheological behaviour is described using the four mode Giesekus model. The effect of the contraction rate of a confusor on the axial velocity profiles and the components of normal and shear stresses is investigated. Approbation of numerical studies was carried out by means of the example of the planar flow of a viscoelastic fluid in a channel with abrupt contraction.

Large eddy simulations of turbulent planar jets of viscoelastic fluids

Direct numerical simulations and large-eddy simulations of turbulent planar jets are used to assess the distortion similarity (DSIM) model, recently developed by Ferreira et al. [“Large-eddy simulations of forced isotropic turbulence with viscoelastic fluids described by the finitely extensible nonlinear elastic rheological model with Peterlin's closure model,” Phys. Fluids 28, 125104 (2016)] for homogeneous turbulence, in the simulation of turbulent viscoelastic planar jets. Both a priori and a posteriori tests of the DSIM model are used and show that the several assumptions used in the development of the DSIM model hold well in inhomogeneous free turbulent viscoelastic flows, e.g., (i) the scale similarity of the subgrid-scale (SGS) polymer stretching and (ii) the local equilibrium of the elastic energy production and dissipation. The DSIM model for the SGS polymer stretching term, together with the dynamic Smagorinsky model, is able to reproduce well the flow structures and the classical one-point statistics of turbulent viscoelastic planar jets. The model should be equally able to simulate other free shear flows of viscoelastic fluids, e.g., wakes and mixing layers.

New Exact Traveling Wave Solutions for Fractional Order System Describing the Second Grade Fluid through Medium with Heat Transfer

The aim of this paper is to determine the time-dependent MHD fractionalized three-dimensional flow of viscoelastic fluid in porous medium with heat transfer by traveling wave solution. The governing nonlinear partial differential equations are altered by utilizing the wave parameter ξ = lx + my + nz + ωtβ/Γ(β+1) into ordinary differential equations. The exact solutions are attained by applying a traveling wave method for two different cases. Here we discuss some precise cases, the solution of MHD Newtonian fluid in porosity can be obtained by substituting α1 → 0 in general solution. The impact of important governing parameters on the movement of a fluid is examined as well as the comparison of Newtonian and non-viscous fluids have been made by 2D and 3D graphical analysis.

Cilia-assisted flow of viscoelastic fluid in a divergent channel under porosity effects.

Cilia-driven laminar flow of an incompressible viscoelastic fluid in a divergent channel has been conducted numerically using the BVP4C technique. The non-Newtonian Jeffrey rheological model is utilized to characterize the fluid. The flow equations are formulated in a curvilinear coordinate system, and the porosity effects are simulated with a body force term in the Navier-Stokes equation. The flow equations are transformed into a wave frame from a fixed frame of reference using a linear mathematical relationship. A biological approximation of creeping phenomena and the long-wavelength assumption is used in the flow analysis. The flow analysis is carried out by using a complex (wavy) propulsion of cilia beating. The two-dimensional flow is controlled by physical parameters-Darcy's number, curvature parameter, viscoelastic parameter, phase difference, cilia length, and divergent parameter. They also examined the ciliated pumping and bolus trapping in their flow analysis. The boundary layer phenomena in the velocity profile are noticed under more significant porosity and time relaxation effects. The bolus circulations are reduced for a larger porosity medium and larger numeric values of the time relaxation parameter.

Stabilised Variational Multi-scale Finite Element Formulations for Viscoelastic Fluids

The objective of this article is to summarise the work that we have been doing as a group in the context of stabilised finite element formulations for viscoelastic fluid flows. Viscoelastic fluids are complex non-Newtonian fluids, characterised by having an irreducible constitutive equation that needs to be solved coupled with the momentum and continuity equations. The finite element approximation of this kind of fluids presents several numerical difficulties. It inherits obviously the problems associated with the approximation of the incompressible Navier–Stokes equations. But, on top of that, now the constitutive equation is highly non-linear, with an advective term that may lead to both global and local oscillations in the numerical approximation. Moreover, even in the case of smooth solutions, it is necessary to meet some additional compatibility conditions between the velocity and the stress interpolation in order to ensure control over velocity gradients. The stabilised methods detailed in this work allow one to use equal order or even arbitrary interpolation for the problem unknowns ( $${\varvec{\sigma }}$$ - $${\varvec{u}}$$ -p) (elastic deviatoric stress-velocity-pressure) and to stabilise dominant convective terms, and all of them can be framed in the context of variational multi-scale methods. Some additional numerical ingredients that are introduced in this article are the treatment of the non-linearities associated with the problem and the possibility to introduce a discontinuity-capturing technique to prevent local oscillations. Concerning the constitutive equation, both the standard as the logarithmic conformation reformulation are discussed for stationary and time-dependent problems, and different versions of stabilised finite element formulations are presented in both cases.

Well-conditioned ultraspherical and spectral integration methods for resolvent analysis of channel flows of Newtonian and viscoelastic fluids

Modal and nonmodal analyses of fluid flows provide fundamental insight into the early stages of transition to turbulence. Eigenvalues of the dynamical generator govern temporal growth or decay of individual modes, while singular values of the frequency response operator quantify the amplification of disturbances for linearly stable flows. In this paper, we develop well-conditioned ultraspherical and spectral integration methods for frequency response analysis of channel flows of Newtonian and viscoelastic fluids. Even if a discretization method is well-conditioned, we demonstrate that calculations can be erroneous if singular values are computed as the eigenvalues of a cascade connection of the frequency response operator and its adjoint. To address this issue, we utilize a feedback interconnection of the frequency response operator with its adjoint to avoid computation of inverses and facilitate robust singular value decomposition. Specifically, in contrast to conventional spectral collocation methods, the proposed method (i) produces reliable results in channel flows of viscoelastic fluids at high Weissenberg numbers (∼500); and (ii) does not require a staggered grid for the equations in primitive variables.

Localized stress amplification in inertialess channel flows of viscoelastic fluids

Nonmodal analysis typically uses square-integrated quantities to characterize amplification of disturbances. However, such measures may be misleading in viscoelastic fluids, where polymer stresses can be strongly amplified over a small region. Here, we show that when using a localized measure of disturbance amplification, spanwise-constant polymer-stress fluctuations can be more amplified than streamwise-constant polymer-stress fluctuations, which is the opposite of what is observed when a square-integrated measure of disturbance amplification is used. To demonstrate this, we consider a model problem involving two-dimensional pressure-driven inertialess channel flow of an Oldroyd-B fluid subject to a localized time-periodic body force. Nonmodal analysis of the linearized governing equations is performed using recently developed well-conditioned spectral methods that are suitable for resolving sharp stress gradients. It is found that polymer-stress fluctuations can be amplified by an order of magnitude while there is only negligible amplification of velocity fluctuations. The large stress amplification arises from the continuous spectrum of the linearized problem, and may put the flow into a regime where nonlinear terms are no longer negligible, thereby triggering a transition to elastic turbulence. The results suggest an alternate mechanism that may be useful for understanding recent experimental observations of elastic turbulence in microchannel flows of viscoelastic fluids. • Nonmodal amplification of localized time-periodic body forces is examined. • Poiseuille flow of Oldroyd-B fluids is considered. • Stress fluctuations can be amplified much more than velocity fluctuations. • Stress amplification is highly localized in space. • Results suggest a mechanism for transition to elastic turbulence in microchannels.

Electro-Osmotic Propulsion of Jeffrey Fluid in a Ciliated Channel Under the Effect of Nonlinear Radiation and Heat Source/Sink.

Mathematical modeling of mechanical system in microfluidics is an emerging area of interest in microscale engineering. Since microfluidic devices use the hair-like structure of artificial cilia for pumping, mixing, and sensing in different fields, electro-osmotic cilia-driven flow helps to generate the fluid velocity for the Newtonian and viscoelastic fluid. Due to the deployment of artificial ciliated walls, the present research reports the combined effect of an electro-osmotic flow and convective heat transfer on Jeffrey viscoelastic electrolytic fluid flow in a two-dimensional ciliated vertical channel. Heat generation/absorption and nonlinear radiation effects are included in the present mathematical model. After applying Debye-Huckel approximation and small Reynolds number approximation to momentum and energy equation, the system of nonlinear partial differential equation is reduced into nonhomogenous boundary value problem. The problem determines the velocity, pressure, and temperature profiles by the application of semi-analytical technique known as homotopy perturbation method (HPM) with the help of software Mathematica. The graphical results of the study suggest that HPM is a reliable methodology for thermo physical electro-osmotic rheological transport in microchannels.

Elastic instabilities and bifurcations in flows of wormlike micellar solutions past single and two vertically aligned microcylinders: Effect of blockage and gap ratios

This study presents an extensive numerical investigation on the flow characteristics of wormlike micellar (WLM) solutions past a single and vertically aligned two microcylinders placed in a microchannel in the creeping flow regime. The rheological behavior of the micellar solution is realized based on the two-species Vasquez–Cook–McKinley (VCM) constitutive model, which takes into account both the breakage and re-formation dynamics of micelles. For the case of single microcylinder, as the blockage ratio (ratio of the cylinder diameter to that of the channel height) is gradually varied, we find the existence of a flow bifurcation in the system, and also a gradual transition for a range of flow states, for instance, steady and symmetric or Newtonian like, steady and asymmetric, unsteady periodic and asymmetric, unsteady quasi-periodic and asymmetric, and, finally, unsteady quasi-periodic and symmetric. For the case of two microcylinders, we observe the presence of three distinct flow states in the system, namely diverging (D), asymmetric-diverging (AD), and converging (C) states as the intercylinder spacing in between the two cylinders is varied. Similar types of flow states are also observed in the recent experiments dealing with WLM solutions. However, we show that either this transition from one flow state to another in the case of a single microcylinder or the occurrence of any flow state in the case of two microcylinders is strongly dependent upon the values of the Weissenberg number and the nonlinear VCM model parameter ξ, which basically indicates how easy or hard it is to break a micelle. Based on the results and discussion presented herein for the single and two microcylinders, we hope this study will facilitate the understanding behind the formation of preferential paths or lanes during the flow of viscoelastic fluids through a porous media, which was seen in many prior experiments in the creeping flow regime.

Viscoelastic fluid flow driven by non-propagative membrane contraction

A viscoelastic fluid flow model is develoved to study the flow analysis under the application of membrane contraction followed by the contraction and relaxation. This analysis is carried out subject to slip boundary condition. In particular, study of the kinematic of membrane motion in the insect’s tracheal tube that induces the fluid flow. The impact of a single tracheal tube’s membrane contraction has been computed mathematically, which regulated as a wall channel geometrically. Jeffrey fluid model is considered to see the viscoelastic behaviour of the fluids. The lubrication theory is adopted at a low Reynolds number for the non-linear differential equations. The pertinent influences of critical parameters on the pressure distribution have been graphically depicted. It is observed that an increase in the Jeffrey fluid parameter causes the reduction in pressure difference. This model is a highly bioinspired pumping mechanism. This model’s outcomes can be very useful in the designs of smart pumps for various biomedical applications.

Mathematical Analysis of Radiating Viscoelastic Unsteady MHD Fluid Flow through an Absorbent Media between Upstanding Equidistant plates with Joule Heating Impact

Focus of the present study is to explore Joule heating effect on incompressible, radiating, electrically conducting viscoelastic unsteady magnetohydrodynamic fluid and heat transfer through an absorbent media between upstanding equidistant plates with constant suction and slip condition. Lorentz force and Joule heating is discussed by momentum and energy equation respectively. Electric conductivity and the electromagnetic force both are in very small case in fluid flow. A consistent magnetic field is taking to perpendicular to the plane of the plates. The oscillatory time-dependent coupled equations (non-linear) are simplified to provide result for the fluid velocity and heat transfer by using structured perturbation execution. It has been noticed that velocity and heat transfer rise due to rise in Grashof number, Eckert number, heat source parameter or slip parameter while decrease as permeability parameter, radiation parameter or wavelength rises. As Hartmann number rises the fluid velocity diminishes, while opposite behaviour is noted in case of heat transfer effect. The velocity and heat transfer impact of the fluid rise with cross flow Reynolds parameter or frequency of the oscillation. Numerical outcomes for the motion and heat transpose effect profiles for multiples physical numbers also the local skin friction coefficient and spot Nusselt number are argued numerically, presented graphically along with.

Dynamic Modeling of Heated Oscillatory Layer of Non-Newtonian Liquid

In this work, we have effectively used the numerical inversion of the Laplace transform to study the time-dependent thin heated film flow of a viscoelastic fluid flowing on an infinitely long flat substrate. Exact and analytical solutions are obtained in some limiting cases. The model describing this problem is a system of equations, coupling the linearized Navier–Stokes equation of the viscoelastic fluid with regard for gravity as an external force and the temperature relation for the energy profile. By assuming that the fluids are incompressible, we first derive a new system of equations, by taking into account additional terms, due to the insoluble surfactants and the viscoelastic properties. The velocity and temperature profiles are shown and the influence of coupling constant, viscoelastic parameters and the interfacial surfactants on the liquid film are discussed in detail. The validity of our solutions is verified by the numerical results to show the effects of different parameters involved and to show how the fluid flow evolves with time.

Polymer effects on viscoelastic fluid flows in a planar constriction microchannel

A comprehensive understanding of the flow of viscoelastic polymer solutions in a contraction and/or an expansion geometry concerns numerous applications. The effects of polymer type, molecular weight, and concentration are investigated through controlled experiments with polyethylene oxide (PEO), polyvinylpyrrolidone (PVP) and hyaluronic acid (HA) solutions in a planar constriction microchannel in a wide range of Reynolds (Re) and Weissenberg (Wi) numbers. The polymer structure renders drastic differences amongst the contraction flows of 3000 ppm PEO, PVP, and HA solutions despite having similar molecular weights. The expansion flow vortices of these solutions remain analogous to those in the inertial flow of water. Increasing the molecular weight or concentration of PEO polymer promotes the elastic instabilities in both the contraction and expansion flows. It, however, suppresses (and even blocks) the inertial vortices in the expansion flow and makes them start from the salient corners, in contrast to the lips of the expansion walls in water. Interestingly, a sudden decrease in the size of inertial vortices is observed in the expansion flow of 500 ppm, 1 megadalton PEO solution when the elastic disturbances start appearing in the contraction flow. The observed flow regimes and vortex development in the polymer solutions are summarized in the same dimensionless Re−Wi and Re−χL parameter spaces, where χL is the normalized vortex length. The elasticity number, El=Wi/Re, is found to determine and distinguish the contraction and expansion flows.

Threshold for the generalized Non-Fourier heat flux model: Universal closed form analytic solution

The Magnetohydrodynamics (MHD) flow of second grade and viscoelastic fluids due to a linearly stretching sheet in a Darcy porous medium considering Cattaneo-Christov heat flux model (with the thermal relaxation time playing a significant role) is tackled analytically. Closed form analytic solution is presented for the energy equation employing the properties of Appell hypergeometric function of two variables presumably for the 1st time in the literature. It is shown that the associated energy equation yields closed form solution only iff″(0)2 > aλPr; in particular, being a structural constraint for the existence of thermal solutions under the studied model. Here, f″(0)is the normalized shear stress at the wall, λis the relaxation time parameter in Cattaneo-Christov model, Pris Prandtl number and ais the positive stretching ratio.

The Viscoelastic Swirled Flow in the Confusor

A two-dimensional mathematical model for a steady viscoelastic laminar flow in a confusor was developed under the condition of swirled flow imposed at the inlet. Low density polyethylene was considered as a working fluid. Its behavior was described by a two-mode Giesekus model. The proposed mathematical model was tested by comparing it with some special cases presented in the literature. Additionally, we propose a system of equations to find the nonlinear parameters of the multimode Giesekus model (mobility factor) based on experimental measurement. The obtained numerical results showed that in a confusor with the contraction rate of 4:1, an increase in the swirl intensity at Wi < 5.1 affects only the circumferential velocity, while the axial and radial velocities remain constant. The distribution pattern of the first normal stress difference in the confusor is qualitatively similar to the one in a channel with abrupt contraction, i.e., as the viscoelastic fluid flows in the confusor, the value of N1 increases and reaches a maximum at the end of the confusor. Dimensionless damping coefficients of swirl are used to estimate the swirl intensity. The results show that the swirl intensity decreases exponentially.

Zero velocity regions near forward and rare points of circular cylinder: A heat and mass transfer study

This paper examines the unsteady two-dimensional boundary layer heat and mass transfer features of viscoelastic fluid flow near the neighborhood of forward and rare stagnation points. The heated circular cylinder is immersed in a viscoelastic fluid and the forward and rare stagnation points occurs near downward and upward directions of the cylinder respectively. The velocity near the surface of the cylinder is assumed to be zero and at very large distance from cylinder a uniform free stream velocity is spontaneously started in vertical direction. A non-similar transformation reduces the set of partial differential equations into non-similar partial differential equations. The transformed non-linear partial differential system is solved by using Keller box method. This technique shows well-behaved solutions for the steady-state (large time) and transient (small time) flows near the boundary layer region. Moreover, the body force present in this phenomena is in the form of convection which gives assisting (greater than zero) and opposing flows (less than zero). Parameters like the viscoelastic parameter, the convection parameter and the stagnation point parameter (forward and rare stagnation points) are found to control the flow field. Moreover, the Schmidt and the Prandtl numbers are used to control the concentration and temperature distributions. Features of several parameters on the non-dimensional concentration, temperature, local Nusselt and Sherwood numbers are viewed through graphs (2-d and 3-d) and tables. Contour images are plotted in order to visualize the three dimensional surface into two dimensional plane.

Modeling Monophasic Flow of Polymer Solutions in Porous Media: Assessing Relative Impact of Intrinsic Fluid Properties and Pore Microstructure

Polymer solutions are often described as viscoelastic fluids, whose rheological behavior dynamically evolves according to the flow shear rate due to successive contractions and relaxations of inner deformable components. Contrary to shear-thickening and shear-thinning fluids, viscoelastic fluids can undergo an increase in the normal stress-differences even in simple geometries, leading to counterintuitive flow patterns. In this paper, we investigate numerically the monophasic flow of viscoelastic fluids through microfluidic devices. The objective is to qualitatively assess how modification of the rheological properties and the geometry of porous media representative element affect the flow regime at microscale. Deviations from equivalent Newtonian model are analyzed to quantify the contribution of elasticity in the flow. It is demonstrated that depending on the geometry, flow of polymer solution can display anisotropic features and flow resistance at relatively low viscoelastic numbers, which is coherent with previous microscopic experimental and numerical studies. Ensued applications of the formulated microscale model are discussed.

Particle migration induced by hydrodynamic interparticle interaction in the Poiseuille flow of a Giesekus fluid

Particle migration induced by hydrodynamic interparticle interaction in a Poiseuille flow of Giesekus viscoelastic fluid is studied numerically using the direct forcing/fictitious domain method with Weissenberg number 0.1 ≤ Wi ≤ 1.5, mobility parameter 0.1 ≤ α ≤ 0.7, viscosity ratio 0.1 ≤ β ≤ 0.7, block ratio 0.2 ≤ e ≤ 0.4, initial interparticle spacing 0.1 ≤ s ≤ 2.0, and initial vertical position 0.1 ≤ y0 ≤ 0.2. The method is validated by comparing the present results with previous numerical results. The effects of Wi, α, β, e, s and y0 on the particle migration are analyzed. The results showed that a particle tends to move toward the wall with the increases of the elastic effect of the fluid, shear thinning effect, solvent viscosity and block ratio. For three particles in initial parallel arrangement, the trajectory of two particles on the edge is obviously different from that of a single particle. A single particle would move toward the centerline at a definite y0. However, for the case of three particles at a same y0, the upstream particle first migrates a distance toward the wall and then return toward the centerline, while downstream particle migrates quickly to the centerline. This is called abnormal migration. The phenomenon of abnormal migration disappears when the initial interparticle spacing is large enough, and is more obvious when the initial vertical position of particles is close to the wall. The phenomenon of abnormal migration tends to be obvious with the increases of the shear thinning effect, solvent viscosity and the block ratio, but with the decrease of elastic effect of the fluid.

Comparative study on numerical performances of log-conformation representation and standard conformation representation in the simulation of viscoelastic fluid turbulent drag-reducing channel flow

In this paper, a new derivation process of the log-conformation governing equation for viscoelastic fluid flows is presented by using the Taylor series definition of the matrix logarithm. Based on the log-conformation representation (LCR) and standard conformation representation (CR) methods, the turbulent drag-reducing channel flow of viscoelastic fluid described by the Oldroyd-B constitutive model is simulated by the finite difference method. The comparison illustrates that the turbulent drag reduction (DR) effect under the condition of a low Weissenberg number (Wi = 1) or moderate Weissenberg number (Wi = 5) can be successfully reproduced by the CR method but is very difficult to be obtained by the LCR method at the same grid resolution if the commonly used interpolation approaches in the computing domain (i.e., log domain) are employed. Further research reveals that the interpolation method of log-conformation tensor involved is one of the dominant reasons responsible for the disability to obtain a turbulent DR effect by using the LCR method. If the interpolation is performed in a physical domain, the turbulent DR effect can be reproduced by using the LCR method. If the interpolation involved in the CR method is carried out in a log domain, the turbulent DR phenomenon can still be simulated but with a weakened DR effect. In sum, this study demonstrates that the commonly used interpolation approaches in the log domain should be responsible for the poor performance of the LCR method.