Upper Convected Maxwell Model Research Articles

A viscoelastic flow model of Maxwell-type with a symmetric-hyperbolic formulation

Maxwell models for viscoelastic flows are famous for their potential to unify elastic motions of solids with viscous motions of liquids in the continuum mechanics perspective. But the usual Maxwell models allow one to define well motions mostly for one-dimensional flows only. To define unequivocal multi-dimensional viscoelastic flows (as solutions to well-posed initial-value problems) we advocated in [ESAIM:M2AN 55 (2021), p. 807-831] an upper-convected Maxwell model for compressible flows with a symmetric-hyperbolic formulation. Here, that model is derived again, with new details.

Sharp corner singularity of the White–Metzner model

The stress singularity is determined using matched asymptotics for complete flow of a White–Metzner (WM) fluid around a re-entrant corner. The model is considered in the absence of a solvent viscosity, with power-law forms for the relaxation time and polymer viscosity. In this form, the model shares the same stress singularity as the upper convected Maxwell (UCM) model, but its wall boundary layers may be thinner or thicker than those for UCM depending upon the relative difference in the power-law exponents. If the exponent for the relaxation time is greater than that for the polymer viscosity, the boundary layer is narrower, whilst it is thicker if the polymer viscosity exponent exceeds that of the relaxation time. When the exponents are the same, the WM boundary layer thickness is the same size as that for UCM. A self-similar solution is derived for the stress and velocity fields and matched to both upstream and downstream boundary layers. Restrictions on the sizes of the power-law exponents are also given for validity of this solution.

Tricritical state and quasi-periodicity triggered by the non-linear elasticity in an Upper Convected Maxwell fluid confined between two co-oscillating cylinders about zero-mean

The effects of harmonically co-oscillating the inner and outer cylinders about zero mean rotation in a Taylor–Couette flow are examined numerically using Floquet theory, for the case where the fluid confined between the cylinders obeys the upper convected Maxwell model. Although stability diagrams and mode competition involved in the system were clearly elucidated recently by Hayani Choujaa et al. (2021) in weakly elastic fluids, attention is focused, in this paper, on the dynamic of the system at higher elasticity with emphasis on the nature of the primary bifurcation. In this framework, we are dealing with pure inertio-elastic parametric resonant instabilities where the elastic and inertial mechanisms are considered of the same order of magnitude. It turns out, on the one hand, that the fluid elasticity gives rise, at the onset of instability, to the appearance of a family of new harmonic modes having different axial wavelengths and breaking the spatio-temporal symmetry of the base flow: invariance in the axial direction generating the O(2) symmetry group and a half-period-reflection symmetry in the azimuthal direction generating a spatio-temporal Z2 symmetry group. On the other hand, new quasi-periodic flow emerging in the high frequency limit and other interesting bifurcation phenomena including bi and tricritical states are also among the features induced by the fluid elasticity. Lastly, and in comparison with the Newtonian configuration of this system, the fluid elasticity leads to a total suppression of the non-reversing flow besides emergence of instabilities with lower wavelengths. Such a comparison provides insights into the dynamics of elastic hoop stresses in altering the flow reversal in modulated Taylor–Couette flow.

Lubrication approximation of pressure-driven viscoelastic flow in a hyperbolic channel

We investigate theoretically the steady incompressible viscoelastic flow in a hyperbolic contracting channel. The fluid viscoelasticity is modeled using the upper convected Maxwell (UCM), Oldroyd-B, Phan-Thien and Tanner (PTT), Giesekus, and the finite elasticity non-linear elastic dumbbell with the Peterlin approximation (FENE-P) models. We first develop the general governing equations for flow within a non-deformable channel whose cross section varies with the distance from the inlet. We then exploit the classic lubrication approximation, assuming a small aspect ratio of the channel to simplify the original governing equations. The final equations, which we formulate in terms of the stream unction, are then solved analytically using a high-order asymptotic scheme in terms of the Deborah number, De, and the formulas for the average pressure drop are derived up to eight orders in De. The accuracy of the original perturbation solution is enhanced and extended over a wide range of parameters by implementing a convergence acceleration method for truncated series. Furthermore, convergence of the transformed solutions for the average pressure drop is demonstrated. The validity and accuracy of the theoretical results is independently confirmed through comparison with numerical results from simulations performed using high-order finite differences and pseudospectral methods. The results reveal the decrease in the average pressure drop with increasing the Deborah number, the polymer viscosity ratio, and the ratio of the inlet to the outlet height. We also show that the fundamental UCM and Oldroyd-B models can predict the major viscoelastic phenomena for this type of internal and confined lubrication flows, while the effect of the rheological parameters of the PTT, Giesekus, and FENE-P models on the results is minor.

Three-dimensional upper-convected Maxwell nanofluid transport due to cross-diffusion and stratification: Numerical solutions

Cross-diffusion and stratification (thermal/solutal) effects on the transport of non-Newtonian nanofluid are analyzed in this investigation. Upper-convected Maxwell model is chosen to depict the non-Newtonian behavior. The flow is considered to be bounded by a stationary flat plate from one side. This phenomenon is formulated mathematically under boundary layer approximation which is further solved by the numerical procedure of Runge-Kutta-Fehlberg along with Newton-Raphson shooting scheme. Effects of appearing parameters are elaborated with the help of tables and figures. It is expected that the stratification and diffusion processes will help in the storage and transport of energy through the moving non-Newtonian nanofluid. It is found that Soret number and solutal stratification have similar impacts on temperature. Moreover, Dufour number and thermal stratification have adverse impacts on temperature/concentration.

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.

Thermocapillary effects on absolute and convective instability of viscoelastic liquid jets

The effect of heat transmission on the absolute instability (AI) and convective instability (CI) of axisymmetrical disturbances in a viscoelastic liquid jet that falls under gravity is investigated. In general, when heat is included to the interface of a viscoelastic jet, it can be used to process droplet sizes and breakup lengths even more. We describe the jet’s dynamics mathematically using the Upper-Convected Maxwell (UCM) model. On the basis of the jet’s slenderness, an asymptotic approach is used to simplify the problem and obtain solutions of steady basic flow, which are then linearly analyzed for absolute and convective instability. When traveling wave modes are considered, a dispersion relation between the wavenumber and the growth rate of viscoelastic jets is derived, which can then be solved numerically using the Newton–Raphson method. The impact of varying some non-dimensional parameters is shown on absolute and convective instability. In this work, absolute instability is explored by employing a mapping technique called the cusp map method. For a variety of parameter regimes, the convective-to-absolute instability boundary (CAIB) is determined. We have found that absolute-to-convective transitions occur at lower critical Weber numbers when heating is included to the interface of the viscoelastic jets.

Acoustic interaction force between two particles immersed in a viscoelastic fluid

The interaction acoustic radiation force in a standing plane wave applied to each small solid sphere in a two-particle system immersed in a viscoelastic fluid is studied in a framework based on perturbation theory. In this work, the first- and second-order perturbation theories are used in the governing equations with considering the upper-convected Maxwell model to obtain mathematical modeling. We use the finite element method to carry out simulations and describe the behavior of the viscoelastic fluid. The mathematical development is validated from three literature case studies: a one-particle system in a viscous fluid, a two-particle system in a viscous fluid, and a one-particle system in a viscoelastic fluid. The novelty of this study is to establish the acoustic interaction force between two spherical particles immersed in a viscoelastic fluid. The results show that the acoustic interaction force between two spheres is greater in a viscous fluid in comparison with the viscoelastic fluid with the same shear viscosity. This behavior is due to the relaxation time effect. It is also indicated that the acoustic interaction force between the particles decreases by the relaxation time and increases by the fluid's viscosity. A mathematical formula is proposed for the acoustic interaction force between particles located close to each other in a viscoelastic fluid.

Network structure and viscoelastic behavior of high-strength crystal microphase crosslinking hydrogels analyzed from combined model

The design of network structure is vital to the preparation of high-strength hydrogels. Using a mechanical model to analyze the network structure and mechanical behavior of hydrogel is an effective means to study the strengthening mechanism of the hydrogel. In this work, a series of acrylonitrile copolymer hydrogels based on crystal microphase crosslinking are synthesized. On this basis, a combined model that consists of the Upper Convected Maxwell model and an ideal spring model is established, which separates the crystal microphase crosslinking network (elastic) from the physical crosslinking network (viscoelastic) based on dipole–dipole interaction. The good fit of the simulated curves to the experimental curves demonstrates that the mechanical behaviors of these hydrogels could be well expressed by such a combined model. The contributions of the two networks to the nominal stress are calculated respectively, and the crosslink densities of the two networks are also compared. It is found that the crosslink density of the viscoelastic network in the hydrogel is higher than that of the elastic network, which further confirms the fact that there is a microphase crosslinking structure in the hydrogel. In addition, the effects of acrylic acid feed ratio, molecular weight and zinc chloride concentration on the network structure of the hydrogel are discussed.

On a variational principle for the Upper Convected Maxwell model

A variational principle for the Upper Convected Maxwell model is presented. The stationary value of the appropriate functional is the drag on an immersed object. From the principle, a formula is derived for the derivative of the drag with respect to the Deborah number for an arbitrarily shaped particle in a circular duct under creeping flow conditions. The formalism is compared with the conventional reciprocal theorem. Whereas the reciprocal theorem gives the drag as a volume integral involving the Stokesian stress tensor, the variational principle involves the stress from the adjoint equation. For low Deborah numbers both approaches provide the correction to the Stokes drag as a volume integral involving only the Stokesian rate-of-strain tensor, in line with second-order fluid theory. • A variational principle for the Upper Convected Maxwell model is presented. • At stationarity the functional equals the drag on a partcle in a duct. • For low Deborah numbers the drag correction involves only the Stokesian strain rate. • This is in correspondence with the reciprocal theorem and second-order fluid theory.

Modeling and Mathematical Investigation of Blood-Based Flow of Compressible Rate Type Fluid with Compressibility Effects in a Microchannel.

In this investigation, the compressibility effects are visualized on the flow of non-Newtonian fluid, which obeys the stress–strain relationship of an upper convected Maxwell model in a microchannel. The fundamental laws of momentum and mass conservation are used to formulate the problem. The governing nonlinear partial differential equations are reduced to a set of ordinary differential equations and solved with the help of the regular perturbation method assuming the amplitude ratio (wave amplitude/half width of channel) as a flow parameter. The axial component of velocity and flow rate is computed through numerical integration. Graphical results for the mean velocity perturbation function, net flow and axial velocity have been presented and discussed. It is concluded that the net flow rate and increase in case of the linear Maxwell model, while they decrease in case of the convected Maxwell model. The compressibility parameter shows the opposite results for linear and upper convected Maxwell fluid.

On the validity of Squire’s theorem for viscoelastic fluid flows

This work presents a practical methodology to verify the validity of Squire’s theorem for viscoelastic fluid flow stability analysis. Squire’s theorem presents a relationship between two-dimensional and three-dimensional disturbances. The theorem shows that the critical Reynolds number for two-dimensional disturbances is smaller than any value for which unstable three-dimensional disturbances exist. This conclusion simplifies the stability analysis for fluid flows which satisfies this theorem by becoming sufficient for the stability analysis to look only for the two-dimensional disturbances to find the most dangerous condition. In the present investigation, the validity of Squire’s theorem is accessed for viscoelastic fluid flows considering the Upper-Convected Maxwell, Oldroyd-B, Giesekus, LPTT and FENE-type models. The mass, momentum, and viscoelastic constitutive equations are manipulated to arrive at the equivalent two-dimensional equations required for Squire’s Theorem validity. It was verified that Squire’s theorem is valid for the Upper-Convected Maxwell and the Oldroyd-B isotropic models as already known in the literature, showing that the proposed methodology is consistent with previous results. However, for the Giesekus, the LPTT and the FENE-type anisotropic models, the analysis shows that Squire’s theorem is not valid, indicating the relation between the isotropic behavior of the fluids and the validity of Squire’s theorem.

Rate-dependent creasing of a viscoelastic liquid

The free surface of an elastic material develops sharp cusp-like creases when compressed beyond a critical strain. We show experimentally that a viscoelastic liquid undergoes a similar, but rate-dependent, creasing instability such that the strain required for creasing increases as rate decreases. A model is developed wherein the creasing criterion known previously for neo-Hookean elastic solids is applied to the elastic portion of the deformation of a viscoelastic liquid. Using the upper-convected Maxwell model for viscoelasticity, we derive an analytical criterion for viscoelastic creasing which is in good agreement with experiments. It predicts that the strain for creasing increases with decreasing Weissenberg number, and creasing is not possible below a critical Weissenberg number.

Steady sphere translation in weakly viscoelastic UCM/Oldroyd-B fluids with perfect slip on the sphere

We study analytically the influence of perfect slip boundary conditions to the steady translation of a spherical particle in a viscoelastic fluid which consists of a Newtonian solvent and a polymer solute. The viscoelastic fluid is modeled with the Oldroyd-B constitutive equation with the ratio of polymer zero shear rate viscosity to the total viscosity, η, considered as a fluid parameter and varying from zero (Newtonian limit) to 1 (Upper Convected Maxwell (UCM) limit). The solution for all the dependent flow variables is expanded as asymptotic power series with the small parameter being the Weissenberg number, Wi=λU/R, where λ is the single relaxation time of the fluid, R is the radius of the particle and U is its steady translational velocity. The resulting sequence of equations is solved analytically up to eight-order in Wi. Hence, the current work is a substantial expansion of the previous work by Gkormpatsis et al., (2020) which has examined the most general case of Navier slip, but the perturbation series has been found only up to fourth order in Wi. The current work confirms the results obtained in that study in the limit of perfect slip conditions. Moreover, the higher-order analytical solution allows for the better investigation of the convergence of the results with techniques that accelerate the convergence of series. Under perfect slip conditions, it is shown that the relative drag force on the sphere increases monotonically with viscoelasticity, as opposed to the cases of no-slip or partial slip where the drag can also decrease or attain a local minimum. This difference in behavior is attributed to the higher dominance of extensional deformations close to the stagnation points of the flow, while the shear deformations prevail away from these points. Finally, evidence is provided that the UCM/Oldroyd-B models become singular at a finite Weissenberg number, Wiu, which depends weakly on η. Although the exact value of Wiu could not be calculated, a sequence of approximations which results from the eight-order perturbation solution reveals that 0.5<Wiu<1. Wiu appears to decrease as η increases, with its minimum value being observed for the UCM model.

Chemically reactive Maxwell nanoliquid flow by a stretching surface in the frames of Newtonian heating, nonlinear convection and radiative flux: Nanopolymer flow processing simulation

AbstractThe effects of a chemical reaction and radiative heat flux in a nonlinear mixed thermo-solutal convection flow of a viscoelastic nanoliquid from a stretchable surface are investigated theoretically. Newtonian heating is also considered. The upper-convected Maxwell (UCM) model is deployed to represent the non-Newtonian characteristics. The model also includes the influence of thermal radiation that is simulatedviaan algebraic flux model. Buongiorno’s two-component nanofluid model is implemented for thermophoretic and Brownian motion effects. Convective thermal and solutal boundary conditions are utilized to provide a more comprehensive evaluation of temperature and concentration distributions. Dimensionless equations are used to create the flow model by utilizing the appropriate parameters. The computed models are presented through a convergent homotopic analysis method (HAM) approach with the help of Mathematica-12 symbolic software. Authentication of HAM solutions with special cases from the literature is presented. The impact of various thermophysical, nanoscale and rheological parameters on transport characteristics is visualized graphically and interpreted in detail. Temperatures are strongly enhanced with Brownian motion and thermophoresis parameters. Velocity is boosted with the increment in the Deborah viscoelastic number and mixed convection parameter, and the hydrodynamic boundary layer thickness is reduced. A stronger generative chemical reaction enhances concentration magnitudes, whereas an increment in the destructive chemical reaction reduces them and also depletes the concentration boundary layer thickness. Temperature and concentration are also strongly modified by the conjugate thermal and solutal parameters. Greater radiative flux also enhances the thermal boundary layer thickness. Increasing the Schmidt number and the Brownian motion parameter diminish the concentration values, whereas they elevate the Sherwood number magnitudes,i.e.enhance the nanoparticle mass transfer rate to the wall.

An Extended Reynolds Equation for Non-Newtonian Lubrication With the Upper Convected Maxwell Model

Abstract The upper convected Maxwell (UCM) model is one of the ways to describe lubricant’s viscoelasticity. In this paper, a new modified Reynolds equation for the UCM model is derived. This equation may be called extended as compared to its analogs because it incorporates the complete structure of the upper convected derivative without the explicit omitting of any term or using the perturbation technique. Then, a numerical scheme for solving the viscoelastic lubrication problems with the employment of the derived equation is described. A mixed Euler–Lagrange approach is used here: the constitutive rheological equations are resolved by a semi-Lagrangian technique and the extended Reynolds equation is discretized by the finite volume method. A constant surface slope problem is considered as a test case. Excellent agreement is achieved between the numerical solution at low Deborah number and one of the approximate solutions. The results of simulations with different types of time derivative used in Maxwell model both for two- and three-dimensional cases are also discussed.

A compressible 3D finite volume approach for the simulation of unsteady viscoelastic cavitating flows

We present a fully compressible, density-based finite volume solver for the simulation of 3D cavitating flows in viscoelastic Maxwell-/Oldroyd-like fluids. The upper-convected Maxwell model, the Oldroyd-B model and the linear and exponential simplified Phan-Thien Tanner models are implemented as viscoelastic constitutive equations in conservative formulation, and we identified the Truesdell rate as appropriate objective time derivative for compressible flows. Cavitation is modeled by a single-fluid homogeneous mixture equilibrium approach considering condensation and evaporation assuming volume averaged mixture quantities. The corresponding simplified quasilinear system is analyzed, and the wave speeds are calculated in order to adapt the employed four step Runge–Kutta explicit time stepping concerning the viscoelastic transport equations. We introduce a novel ghost cell boundary condition for the viscoelastic stress tensor. The approach is tested against (semi-)analytical unsteady and steady-state references and shows very good agreement. 3D simulations of the spherical vapor bubble collapse are performed for all implemented viscoelastic models and show a distinct influence compared to the Newtonian case. For the upper-convected Maxwell fluid a variation of the relaxation time exhibits its perspicuous influence on the dynamics of the collapse.

UNSTEADY STAGNATION FLOW OF VISCOELASTIC FLUID THROUGH POROUS MEDIUM OVER STRETCHING/SHRINKING SURFACE USING UCM MODEL

In the present paper, Upper-Convected Maxwell model is used for formulation of the problem of two-dimensional unsteady stagnation point flow of viscoelastic fluid which passes through a porous medium over a stretching/shrinking surface. The effect of magnetic field on flow is also considered in the presence of time dependent heat source/sink. Using similarity parameters, we convert the governing non-linear system of partial differential equations into non dimensional system of ordinary differential equations. This system of equations is solved by using Runge-Kutta fourth order method with shooting technique. Effect of different physical parameters e.g. Maxwell parameter(β), permeability parameter(K), unsteadiness parameter(γ), velocity ratio parameter(λ) etc. on flow and heat transfer characteristics are analyzed and discussed graphically. It is observed that for some values of λ, dual solution also exists for both velocity and temperature, and existence and uniqueness of solution also depends upon unsteadiness parameter. For the validation of present study, the results are compared to previous investigations and found in good agreement.

波状壁の上を通過する粘弾性流体のせん断流れの数値解析

A two-dimensional shear flow of a viscoelastic fluid over a wavy wall was numerically analyzed by the finite difference method. The upper convected Maxwell (UCM) model was used as the constitutive equation, and the unsteady flow problem was solved numerically using the stream function-vorticity method. Variables are discretized using a staggered grid with uniform mesh width. The volume penalization method was used to represent the moving upper wall and the fixed lower wavy wall that induces a vorticity perturbation. The flow in an infinitely long channel is computed using periodic boundary conditions in the flow direction. The results show that the vorticity intensifies over the wavy surface with positive and negative vorticity stripes, at which extreme wall-normal velocity fluctuations are also observed. At relatively high Deborah numbers, the lower ends of these extreme fluctuation regions are close to the critical line associated with a change of type of the governing partial differential equations in a Couette flow. In contrast, the upper ends of the extreme fluctuation region are close to the critical layer of the shear flow, where the flow velocity coincides with the viscoelastic Alfvén wave speed.

Linear stability of a contaminated shear-imposed viscoelastic liquid flowing down an inclined plane

A linear stability analysis is carried out for a contaminated viscoelastic liquid flowing down an inclined plane in the presence of an imposed shear stress, where the elastic behavior of the liquid follows the upper-convected Maxwell model. The earlier work [Wei, “Stability of a viscoelastic falling film with surfactant subjected to an interfacial shear,” Phys. Rev. E 71, 066306 (2005)] conducted analytically in the long-wave regime is revisited again in exploring the results in the arbitrary wavenumber regime. An Orr–Sommerfeld-type eigenvalue problem is formed for the viscoelastic liquid and solved both analytically and numerically by using the long-wave expansion and Chebyshev spectral collocation technique, respectively. It is found that with increase in the value of the Weissenberg number, the critical Reynolds number for the interface mode reduces, but the stable region enhances in the finite wavenumber regime. Furthermore, the unstable domain induced by the interface mode reduces in the presence of insoluble surfactant but enhances in the presence of applied shear stress. If the Reynolds number is high, but the inclination angle is small, the shear mode arises in the numerical simulation, which becomes weaker in the presence of the Weissenberg number and Marangoni number but becomes stronger in the presence of applied shear stress. In a special case, it is demonstrated that the present study recovers the results of Walters's liquid B″ in the limit of low viscoelastic parameter.