FENE-P Model Research Articles

Influence of flexible particle presence on the flow in the channels of microfluidic devices. Problem statement

At the present paper the viscoelastic fluid flow in the element of microfluidic devices is considered. Numerical simulations were made by the FENE-P model. The element of microfluidic devices was considered as a channel with narrowing. The simulation parameters are: Reynolds number Re = 0.01, Weissenberg number We = 0.6, retardation coefficient ß = 0.1 and degree of unraveling of the flexible particle L2=50, 700.

Influence of flexible particle presence on the flow in the channels of microfluidic devices. Results and discussing

At the present paper the distribution of main flow characteristics for the fluid flow in the channel of microfluidic device are considered. Numerical simulations were made by the FENE-P model. Element of microfluidic system is performed as a channel with narrowing. The simulation parameters are: Reynolds number Re = 0.01, Weissenberg number We = 0.6, retardation coefficient ß = 0.1 and degree of unraveling of the flexible particle L2=50, 700. Also, distribution of above mentioned characteristics are performed.

Numerical study on the vortex-induced vibration of a circular cylinder in viscoelastic fluids

The vortex-induced vibration (VIV) in a Newtonian fluid has been widely studied due to its significance in many industrial applications. However, it could be significantly affected by the addition of polymer. In this work, we investigate the viscoelastic effect on the VIV of a circular cylinder whose motion is confined in the cross-flow direction. The Arbitrary Lagrangian–Eulerian (ALE) based finite volume method is used to carry out the two-dimensional simulations at the Reynolds number (Re) range of 30–500. The nonlinear FENE-P model is used to characterize the polymeric viscoelastic fluids. The results show that the addition of polymers in a Newtonian fluid can suppress the VIV amplitude of the cylinder since the polymer can significantly modify the vortex pattern and inhibit the fluctuation in flow. Both higher Weissenberg Number (We) and larger maximum polymer extensibility can reinforce this tendency. Furthermore, the corresponding reduced velocity for the peak of maximum vibration amplitude increases as the increase of Weissenberg number. This is because that the viscoelasticity can reduce the characteristic frequency in the wake flow. Finally, the viscoelasticity can increase the critical Re of the structure vibrations implying a more stable flow.

Effects of polymer/surfactant additives on forced convective heat transfer in vortex shedding flow past a circular cylinder

The unsteady incompressible flow of dilute polymer/surfactant solutions past an unconfined heated circular cylinder is studied to reveal the effects of these additives on convective heat transfer properties in the laminar and transitional vortex shedding regime. Both inelastic and elastic non-Newtonian features are considered and compared using two-dimensional numerical simulations. Inelastic power-law and purely viscoelastic Oldroyd-B as well as Giesekus and FENE-P constitutive models, exhibiting both shear thinning and elastic effects are adopted in the formulation. A finite element based software is used in simulations for the Reynolds number range 80≤Re≤300, power-law index range 0.6≤n≤1 and Weissenberg number Wi=1 for weakly elastic solutions. Heat transfer enhancement is observed under inelastic shear thinning effects through an increase of the local and average Nusselt numbers, however purely viscoelastic effects lead to a decrease in the rate of convective heat transfer for the studied range of parameters. The increase in the Giesekus model mobility factor which is associated with anisotropic Brownian motion effects on polymer/surfactant molecules increases the heat transfer rate, and shifts the location of the maximum local Nusselt number to the front of the cylinder. However, increasing the FENE-P model finite extensibility parameter related to the maximum polymer chain length, decreases the local Nusselt number while the maximum heat transfer rate location moves towards the rear region on the cylinder surface. Temperature gradients decrease in the wake region under elasticity whereas shear thinning leads to an increase in the gradients and to the formation of clustered isotherms.

Effect of polymer-stress diffusion in the numerical simulation of elastic turbulence

Elastic turbulence is a chaotic regime that emerges in polymer solutions at low Reynolds numbers. A common way to ensure stability in numerical simulations of polymer solutions is to add artificially large polymer-stress diffusion. In order to assess the accuracy of this approach in the elastic turbulence regime, we compare numerical simulations of the two-dimensional Oldroyd-B and FENE-P models sustained by a cellular force with and without artificial diffusion. We find that artificial diffusion can have a dramatic effect even on the large-scale properties of the flow and we show some of the spurious phenomena that may arise when artificial diffusion is used.

Turbulent duct flow with polymers

We have performed direct numerical simulation of the turbulent flow of a polymer solution in a square duct, with the FENE-P model used to simulate the presence of polymers. First, a simulation at a fixed moderate Reynolds number is performed and its results compared with those of a Newtonian fluid to understand the mechanism of drag reduction and how the secondary motion, typical of the turbulent flow in non-axisymmetric ducts, is affected by polymer additives. Our study shows that the Prandtl’s secondary flow is modified by the polymers: the circulation of the streamwise main vortices increases and the location of the maximum vorticity moves towards the centre of the duct. In-plane fluctuations are reduced while the streamwise ones are enhanced in the centre of the duct and dumped in the corners due to a substantial modification of the quasi-streamwise vortices and the associated near-wall low- and high-speed streaks; these grow in size and depart from the walls, their streamwise coherence increasing. Finally, we investigated the effect of the parameters defining the viscoelastic behaviour of the flow and found that the Weissenberg number strongly influences the flow, with the cross-stream vortical structures growing in size and the in-plane velocity fluctuations reducing for increasing flow elasticity.

Comparison of inelastic and elastic non-Newtonian effects on the flow around a circular cylinder in periodic vortex shedding

This study focuses on the parametrical investigation and comparison of different non-Newtonian effects due to polymer/surfactant additives on the flow structure around a circular cylinder in periodic vortex shedding regime. For this purpose, inelastic power-law and viscoelastic Oldroyd-B, Giesekus and FENE-P models are adopted in the constitutive formulation. The range of the Reynolds number is 80 ≤ Re ≤ 300, the power-law index range is 0.6 ≤ n ≤ 1.2 and the Weissenberg number range is 0 ≤ Wi ≤ 1.2. A finite-element based solver is used in the two-dimensional flow simulations. The resulting flow structure is compared and discussed in terms of the variation in drag and lift coefficients, vortex shedding frequency, vortex formation length, separation angle and the critical Reynolds number for the onset of vortex shedding. Drag reduction is observed for shear thinning fluids as also observed in previous studies with inelastic models; moreover, the vortex shedding frequency increases, and both the vortex formation length and separation angle decrease under shear thinning. Drag reduction in shear thinning fluids is mainly due to reduction in friction drag. Fluid elasticity leads to an increase in the drag coefficient and vortex formation length, and a decrease in the vortex shedding frequency and separation angle. Also, anisotropy effects through the mobility factor decrease the drag, the vortex formation length, and increases the shedding frequency. The extensibility of polymer molecules increases the drag, the vortex formation length, and decreases the shedding frequency. It is also observed that weakly elastic effects have almost no impact on the onset of periodic vortex shedding.

An N-parallel FENE-P constitutive model and its application in large-eddy simulation of viscoelastic turbulent drag-reducing flow

In this paper, an N-parallel FENE-P constitutive model based on multiple relaxation times is proposed, it can be viewed as a simplified version of the multi-mode FENE-P model under the assumption of identical deformation rate. The proposed model holds the merit of multiple relaxation times to preserve good computational accuracy but could reduce the computational cost, especially in the application of high-fidelity numerical simulation of viscoelastic turbulent drag-reducing flow. Firstly the establishment of N-parallel FENE-P model and the numerical approach to calculate the apparent viscosity are introduced. Then the proposed model is compared with the experimental data and the conventional FENE-P model in estimating rheological properties of two common-used viscoelastic fluids to validate its performance. This work is an extended version of our ICCS conference paper [1]. To further judge the performance of the proposed FENE-P model in complex turbulent flows, the extended application of the proposed model in large-eddy simulation of viscoelastic turbulent drag-reducing channel flow is carried out.

Existence of large-data global-in-time finite-energy weak solutions to a compressible FENE-P model

A compressible FENE-P-type model with stress diffusion is derived from an approximate macroscopic closure of a compressible Navier–Stokes–Fokker–Planck system arising in the kinetic theory of dilute polymeric fluids, where polymer chains immersed in a barotropic, compressible, isothermal, viscous Newtonian solvent are idealised as pairs of massless beads connected with finitely extensible nonlinear elastic (FENE) springs. We develop a priori bounds for the model, including logarithmic bounds, which guarantee the non-negativity of the conformation tensor and a bound on its trace, and we prove the existence of large-data global-in-time finite-energy weak solutions in two and three space dimensions.

CaBER vs ROJER—Different time scales for the thinning of a weakly elastic jet

In this paper, we demonstrate that the capillary thinning dynamics of a weakly viscoelastic jet follow a different timescale than a liquid bridge of the same fluid between two stationary surfaces for similar geometrical scales. The thinning in the latter case observed with capillary breakup extensional rheometry (or CaBER) follows a well established scaling of the radius with time for an elasto-capillary (EC) balance of R∼exp⁡(−t/3λ). However, for the thinning of the filaments between droplets in a jet, it was so far just assumed that the same scaling law holds. In this paper, we experimentally demonstrate that the jet thinning in a Rayleigh–Ohnesorge jetting extensional rheometer (or ROJER) follows a different scaling of R∼exp⁡(−t/2λ). This is demonstrated by a direct comparison of the thinning dynamics of weakly viscoelastic (Oh<0.01) aqueous solutions of polyethylene oxide in the two experimental setups, covering a wide range of jetting velocities or Weber numbers of 1–70. We demonstrate outgoing from a momentum balance that includes inertia and elasticity that this difference in scaling is due to a constant axial tension in the jet arising from the constant creation rate of new surface at the nozzle. Numerical simulations using the FENE-P model support this theory and demonstrate that in the exponential thinning regime of the jet the elastic stresses are indeed balanced by the axial tension (rather than by capillary pressure as in the EC balance regime of the CaBER experiment). It is readily shown from the reduced stress balance that this axial-elastic balance regime in the ROJER experiment leads to a faster exponential thinning, following the new scaling of R∼exp⁡(−t/2λ) that was experimentally observed. Furthermore, we observe both in experiment and simulation that a jet thinning does not exhibit a self-similar structure of the corner region where the thinning filament connects to the drop as it is generally observed for a filament with an axial tension decaying with the filament radius as in the CaBER. The resulting difference of 50% in extensional relaxation time λ extracted from ROJER experiments might require one to revisit previously reported ROJER experiments and is required for the correct evaluation of future jetting rheometry experiments.

Corrigendum to: Finite element approximation of the FENE-P model

Corrigendum to: Finite element approximation of the FENE-P model

Effect of the solvent viscosity on pure electro-osmotic flow of viscoelastic fluids

In this work the fully-developed steady channel flow of the homogeneous polymer solution studied in [4] is revisited and a completely new analytical solution is proposed which is devoid of the limitations of the previous solution (β3ɛDeκ2≤2/27), i.e., it is valid for the complete range of the rheological parameters. The viscoelastic fluid is described by the simplified Phan–Thien and Tanner model with linear stress coefficient function for the polymer contribution plus a Newtonian solvent. The solution is also valid if the polymer contribution is described by the finitely extensive non-linear elastic model with the Peterlin approximation for the average spring force (FENE-P model).

Modeling and numerical simulations of polymer degradation in a drag reducing plane Couette flow

Mechanical molecular scission is the main problem of polymeric drag reducers. The ability to reduce the drag is notably decreased as the molecules break down step by step as time goes on. A number of researchers have given a large part of their time to attempts to further understand the role that some important features play in polymer degradation. Until now, all efforts have been in experimental approaches. This paper is the first attempt to take into account the effect of molecular scission on a drag reducing flow by a direct numerical simulation. We analyse a turbulent plane Couette flow of a FENE-P fluid. Our degradation model is based on the maximum polymer extension length L. Unlike the standard FENE-P model, in which L is a constant, the polymer extension here is a spatio-temporal field L(x, y, z, t). When the molecules are highly stretched, which is measured by the trace of the conformation tensor, their maximum length is locally reduced and, consequently, so is their ability to reduce drag. The degraded L spreads within the domain by means of a transport equation. We show here that with such a simple idea we are able to predict the main aspects of mechanical degradation in the flow, such as the change of the turbulent structures and velocity field, and, consequently, the fall of the drag reduction over time.

Fluctuating viscoelasticity

The smaller the scales on which complex fluids are studied, the more fluctuations become relevant, e.g. in microrheology and nanofluidics. In this paper, a general approach is presented for including fluctuations in conformation-tensor based models for viscoelasticity, in accordance with the fluctuation-dissipation theorem. It is advocated to do this not for the conformation tensor itself, but rather for its so-called contravariant decomposition, in order to circumvent two major numerical complications. These are potential violation of the positive semi-definiteness of the conformation tensor, and numerical instabilities that occur even in the absence of fluctuations. Using the general procedure, fluctuating versions are derived for the upper-convected Maxwell model, the FENE-P model, and the Giesekus model. Finally, it is shown that the fluctuating viscoelasticity proposed here naturally reduces to the fluctuating Newtonian fluid dynamics of Landau and Lifshitz [L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6 of Course of Theoretical Physics, Pergamon Press, Oxford, 1959], in the limit of vanishingly small relaxation time.

Influence of polymer additive on flow past a hydrofoil: A numerical study

Flows of dilute polymer solutions past a hydrofoil (NACA0012) are examined by direct numerical simulation to investigate the modification of the wake pattern due to the addition of polymer. The influence of polymer additive is modeled by the FENE-P model in order to simulate a non-linear modulus of elasticity and a finite extendibility of the polymer macromolecules. Simulations were carried out at a Reynolds number of 1000 with the angle of attack varying from 0° to 20°. The results show that the influence of polymer on the flow behavior of the flow past a hydrofoil exhibits different flow regimes. In general, the addition of polymer modifies the wake patterns for all angles of attack in this study. Consequently, both drag and lift forces are changed as the Weissenberg number increases while the drag of the hydrofoil is enhanced at small angles of attack and reduced at large angles of attack. As the Weissenberg number increases, two attached recirculation bubbles or two columns of shedding vortices downstream tend to be symmetric, and the polymer tends to make the flow less sensitive to the variation of the angle of attack.

Finite element approximation of the FENE-P model

We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the finite element approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain $D ⊂ R d , d = 2 or 3$, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics ($d = 2$) or a reduced version, where the tangential component on each simplicial edge ($d = 2$) or face ($d = 3$) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-element. We show that both of these types of schemes, based on the backward Euler type time discretiza-tion, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case $d = 2$, as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions.

Reduced-stress method for efficient computation of time-dependent viscoelastic flow with stress equations of FENE-P type

Most calculation procedures for time-dependent viscoelastic flows require iteration within the time step used to advance the solution, in order to satisfy simultaneously the momentum and the constitutive equations for each stress component. We have devised a way of reformulating the constitutive equation for the FENE-P model, or models described by similar equations expressed in terms of the stress tensor, which enables iterative methods for simulating time-dependent viscoelastic flows to become much more efficient: the number of iterations to obtain a solution with the reformulated stress equations is substantially smaller (by a factor of 5–10) than with a comparable method applied to the original, non-reformulated, constitutive equations. The proposed reformulation is rather simple and consists in considering as dependent variables the reduced stresses obtained by dividing the stress components by the extensibility function of the model. It is tested with three problems of increasing complexity, start-up of channel and square-duct flows, and start-up of a rotating duct flow.

Numerical simulation of FENE-P viscoelastic fluids flow and heat transfer in grooved channel with rectangular cavities

The forced convection heat transfer for non-Newtonian viscoelastic fluids obeying the FENE-P model in a parallel-plate channel with transverse rectangular cavities is carried out numerically using ANSYS-POLYFLOW code. The flow investigated is assumed to be two-dimensional, incompressible, laminar and steady. The flow behavior and temperature distribution influenced by the re-circulation caused by the variation of cross-section area along the stream wise direction have been studied. The constant heat flux condition has been applied and the effects of the different parameters, such as the aspect ratio of channel cavities (AR = 0.25, 0.5), the Reynolds number (Re = 25, 250, and 500), the fluid elasticity defined by the Weissenberg number (We), and the extensibility parameter of the model (L 2), on heat transfer characteristics have been explored for channels of three successive cavities configuration. Different levels of heat transfer enhancement were obtained and discussed.

Pure axial flow of viscoelastic fluids in rectangular microchannels under combined effects of electro-osmosis and hydrodynamics

This paper presents an analysis of the combined electro-osmotic and pressure-driven axial flows of viscoelastic fluids in a rectangular microchannel with arbitrary aspect ratios. The rheological behavior of the fluid is described by the complete form of Phan-Thien–Tanner (PTT) model with the Gordon–Schowalter convected derivative which covers the upper convected Maxwell, Johnson–Segalman and FENE-P models. Our numerical simulation is based on the computation of 2D Poisson–Boltzmann, Cauchy momentum and PTT constitutive equations. The solution of these governing nonlinear coupled set of equations is obtained by using the second-order central finite difference method in a non-uniform grid system and is verified against 1D analytical solution of the velocity profile with less than 0.06% relative error. Also, a parametric study is carried out to investigate the effect of channel aspect ratio (width to height), wall zeta potential and the Debye–Huckel parameter on 2D velocity profile, volumetric flow rate and the Poiseuille number in the mixed EO/PD flows of viscoelastic fluids with different Weissenberg numbers. Our results show that, for low channel aspect ratios, the previous 1D analytical models underestimate the velocity profile at the channel half-width centerline in the case of favorable pressure gradients and overestimate it in the case of adverse pressure gradients. The results reveal that the inapplicability of the Debye–Huckel approximation at high zeta potentials is more significant for higher Weissenberg number fluids. Also, it is found that, under the specified values of electrokinetic parameters, there is a threshold for velocity scale ratio in which the Poiseuille number is approximately independent of channel aspect ratio.

The influence of fluid - flexible particle interaction on fluid flow optical non-homogeneity in channel bifurcation

In the present paper the peculiar properties of convergent fluid flow in T-junction channel is considered. There is no interaction between flexible particles in the flow. Such kind of situation is described by rheological FENE-P and Oldroyd-B models. The first one predicts viscosity anomaly, dependence of longitudinal viscosity on longitudinal strain rate and elastic properties; the last one – existence of longitudinal viscosity depending on longitudinal strain rate and having a physical sense only for and elastic properties. The model’s governing parameters are the Weissenberg number (We), the Reynolds number (Re), the ability of flexible particle to change its orientation and stretching degree (L2) in the main flow. The bifurcation area is of great importance due to possibility of high stresses and velocities existence not only in central area, but also on the walls and near the corners. The symmetry-loss effect at creeping flows regime (Re≪1) is investigated. It has been showed that at certain set of We and L2 values the symmetrical shape of fluid flow turns to asymmetrical shape.