Inertialess Flow Research Articles

Sedimentation of a spheroidal particle in an elastoviscoplastic fluid

The sedimentation dynamics of a prolate spheroidal particle in an unbounded elastoviscoplastic (EVP) fluid is studied by direct finite element simulations under inertialess flow conditions. The Saramito–Giesekus constitutive equation is employed to model the suspending liquid. The arbitrary Lagrangian–Eulerian formulation is used to handle the particle motion. The sedimentation, lift, and angular velocities of spheroids with aspect ratio between 1 and 8 are computed as the initial orientation, Bingham, and Weissenberg numbers are varied. Similar to the purely viscoelastic case, a spheroid in an EVP fluid rotates up to align its major axis with the applied force. As the Bingham number increases, the settling rate monotonically reduces, while the angular velocity first increases and then decreases. The initial orientation has a relevant effect on the particle stoppage because of the different drag experienced by the spheroid as its orientation is varied. The yielded and unyielded regions around the spheroid reveal that, for particle oriented transversely to the force, the yielded envelope shrinks near the tips due to the fast spatial decay of the stresses, and unyielded regions appear along the surface of the particle, similar to the solid caps observed at the front and back of a sphere. Fluid plasticity enhances the negative wake phenomenon that is observed at Weissenberg numbers significantly lower than the purely viscoelastic case. The results of the drag correction coefficient for particles aligned with longest axis along the force are presented.

The polymer diffusive instability in highly concentrated polymeric fluids

The extrusion of polymer melts is known to be susceptible to ‘melt fracture’ instabilities, which can deform the extrudate, or cause it to break entirely. Motivated by this, we consider the impact that the recently discovered polymer diffusive instability (PDI) can have on polymer melts and other concentrated polymeric fluids using the Oldroyd-B model with the effects of polymer stress diffusion included. Analytic progress can be made in the concentrated limit (when the solvent-to-total-viscosity ratio β→0), illustrating the boundary layer structure of PDI, and allowing the prediction of its eigenvalues for both plane Couette and channel flow. We draw connections between PDI and the polymer melt ‘sharkskin’ instability, both of which are short wavelength instabilities localised to the extrudate surface. Inertia is shown to have a destabilising effect, reducing the smallest Weissenberg number (W) where PDI exists in a concentrated fluid from W∼8 in inertialess flows, to W∼2 when inertia is significant.

Flow instabilities in circular Couette flow of wormlike micelle solutions with a reentrant flow curve

In this work, we numerically investigate flow instabilities of inertialess circular Couette flow of dilute wormlike micelle solutions. Using the reformulated reactive rod model (RRM-R) (Hommel and Graham, 2021), which treats micelles as rigid Brownian rods undergoing reversible scission and fusion in flow, we study the development and behavior of both vorticity banding and finger-like instabilities. In particular, we focus on solutions that exhibit reentrant constitutive curves, in which there exists some region where the shear stress, τ, has a multivalued relation to shear rate, γ̇. We find that the radial dependence of the shear stress in circular Couette flow allows for solutions in which parts of the domain lie in the region of the flow curve where ∂τ/∂γ̇>0, while others lie in the region where ∂τ/∂γ̇<0; this mixed behavior can lead to complex flow instabilities that manifest as finger-like structures of elongated and anisotropically-oriented micelles. In 3D simulations we find that the initial instability is 2D in origin, and 3D finger-like structures arise through the axial instability of 2D sheets. Finally, we show that the RRM-R can capture vorticity banding in narrow-gap circular Couette flow and that vorticity bands are linearly stable to perturbations.

On the role of viscoelasticity in mucociliary clearance: a hydrodynamic continuum approach

We present numerical and analytical predictions of mucociliary clearance based on the continuum description of a viscoelastic mucus film, where momentum transfer from the beating cilia is represented via a Navier-slip boundary condition introduced by Bottier et al. (PLoS Comput. Biol., vol. 13, issue 7, 2017a, e1005552). Mucus viscoelasticity is represented via the Oldroyd-B model, where the relaxation time and the viscosity ratio have been fitted to experimental data for the storage and loss moduli of different types of real mucus, ranging from healthy to diseased conditions. We solve numerically the fully nonlinear governing equations for inertialess flow, and develop analytical solutions via asymptotic expansion in two limits: (i) weak viscoelasticity, i.e. low Deborah number; (ii) low cilia beat amplitude (CBA). All our approaches predict a drop in the mucus flow rate in relation to the Newtonian reference value, as the cilia beat frequency is increased. This relative drop increases with decreasing CBA and slip length. In diseased conditions, e.g. mucus properties characteristic of cystic fibrosis, the drop reaches 30 % in the physiological frequency range. In the case of healthy mucus, no significant drop is observed, even at very high frequency. This contrasts with the deterioration of microorganism propulsion predicted by the low-amplitude theory of Lauga (Phys. Fluids, vol. 19, issue 8, 2007, 083104), and is due to the larger beat amplitude and slip length associated with mucociliary clearance. In the physiological range of the cilia beat frequency, the low-amplitude prediction is accurate for both healthy and diseased conditions. Finally, we find that shear-thinning, modelled via a multi-mode Giesekus model, does not significantly alter our weakly viscoelastic and low-amplitude predictions based on the Oldroyd-B model.

A catalog of pressure and deformation profile for thin walled hyperelastic tubes conveying inertialess flow and undergoing large deformation

Slender tubes constituted of hyperelastic materials undergoing large deformations and conveying inertialess flow of Newtonian fluids are a model representation of several systems including those in soft robotics, biology and in industry. In this paper, we undertake a systematic study of this system. The tube’s hyperelastic behavior is modeled by five (5) different constitutive laws: Neo-Hookean, Mooney–Rivlin, Fung, Gent and Ogden. Invoking the principles of finite elasticity, we delineate the local pressure — deformation relationship for the tube for each of the hyperelastic models. The structural mechanical field of the tube is then coupled with fluid flow field, which is simplified using the principles of lubrication approximation. The resultant fluid–structure interaction problem then throws up a set of interesting results including the deformation and pressure profile across the tube, and tube laws for five (5) hyperelastic models. Analyzing these results, we posit that the behavior of Neo-Hookean and Mooney–Rivlin tubes is converse of Fung’s and Gent’s tubes; the latter show a strain hardening behavior whilst the former exhibit a strain softening one. A sensitivity analysis which underscores the relative importance of geometrical nonlinearity in the problem then follows.

Dynamics and containment of a viscous liquid atop a granular bed

We consider the dynamics of a gravity current of viscous liquid propagating above a dense granular medium that obeys a $\mu (I)$ -rheology. Initially, the pool of liquid depresses the granular layer to form levees at its edges. Next, these levees are pushed outwards by the gravity-driven slumping of the liquid, but the levees are not surmounted. In the third stage, the top of the levee is pushed out beyond the rest of the levee. This segregates the liquid into a pond trapped by the remnant of the original levees, and a slowly spreading thin film ahead of the levees. The trapped fraction of liquid depends on the extent of the early granular erosion, which in turn is controlled by the initial shape of the deposit and the yield criterion of the granular layer. The key physical ingredients that lead to such dynamics are inertia-less flow and a lower layer with a yield criterion. The latter gives rise to the all-important levees, which lead to the eventual trapping.

Mathematical model of killing an oil horizontal well

The physical picture of the killing fluid and the formation fluid movement in the process of killing a horizontal oil well has been described, a physical and mathematical model of the process has been developed and the limits of the model applicability have been indicated. It is recommended to conduct the killing process in two stages, separated by a technological break. During the first stage, the injected killing fluid fills the tubing, the annulus and the part of the horizontal wellbore. For the duration of the technological break, the denser tubing slowly displaces the formation fluid from the horizontal wellbore into the annulus where the formation fluid floats to the surface in the form of droplets. For the period of the second stage, the remaining amount of tubing is pumped into the well. The model is based on the well-known hydraulic equations for laminar flow of liquids, the solution of the flooded jet problem, as well as the model of one-dimensional vertical inertialess flow of a two-phase medium with incompressible phases. A closed system of equations has been formed taking into account the viscous friction losses on the walls of the horizontal pipe and at the interface between the formation fluid and the killing fluid flows. The solution of this system has been obtained and the values of the main process parameters have been found: velocities and flows of liquid gas and liquid coolant and time of complete replacement of liquid coolant by the killing fluid depending on the inclination angle of the horizontal pipe. It is shown that the character of the movement of the formation fluid drops in the pipe space depends on their volume fraction φ in the mixture with the killing fluid. With the value φ &lt; 1/3 the jelly drops float up with a constant speed, while the jelly droplet radius and the speed of their floating up are unambiguously determined by the jelly stream size, density and viscosity of jelly and liquid hydrocarbon. The graphs of dependence of these parameters on the value of the formation fluid flow have been presented. On the basis of the research the recommendations for the process of killing horizontal oil wells have been formulated.

Mechanism of vorticity amplification by elastic waves in a viscoelastic channel flow

Inertia-less viscoelastic channel flow displays a supercritical nonnormal mode elastic instability due to finite-size perturbations despite its linear stability. The nonnormal mode instability is determined mainly by a direct transition from laminar to chaotic flow, in contrast to normal mode bifurcation leading to a single fastest-growing mode. At higher velocities, transitions to elastic turbulence and further drag reduction flow regimes occur accompanied by elastic waves in three flow regimes. Here, we demonstrate experimentally that the elastic waves play a key role in amplifying wall-normal vorticity fluctuations by pumping energy, withdrawn from the mean flow, into wall-normal fluctuating vortices. Indeed, the flow resistance and rotational part of the wall-normal vorticity fluctuations depend linearly on the elastic wave energy in three chaotic flow regimes. The higher (lower) the elastic wave intensity, the larger (smaller) the flow resistance and rotational vorticity fluctuations. This mechanism was suggested earlier to explain elastically driven Kelvin-Helmholtz-like instability in viscoelastic channel flow. The suggested physical mechanism of vorticity amplification by the elastic waves above the elastic instability onset recalls the Landau damping in magnetized relativistic plasma. The latter occurs due to the resonant interaction of electromagnetic waves with fast electrons in the relativistic plasma when the electron velocity approaches light speed. Moreover, the suggested mechanism could be generally relevant to flows exhibiting both transverse waves and vortices, such as Alfven waves interacting with vortices in turbulent magnetized plasma, and Tollmien-Schlichting waves amplifying vorticity in both Newtonian and elasto-inertial fluids in shear flows.

Universal properties of non-Hermitian viscoelastic channel flows

An addition of long-chain, flexible polymers strongly affects laminar and turbulent Newtonian flows. In laminar inertia-less viscoelastic channel flow, the supercritical elastic instability of non-normal eigenmodes of non-Hermitian equations at finite-size perturbations leads to chaotic flow. Then three chaotic flow regimes: transition, elastic turbulence (ET), and drag reduction (DR), accompanied by elastic waves, are observed and characterized. Here we show that independently of external perturbation strength and structure, chaotic flows above the instability onset in transition, ET, and DR flow regimes reveal similar scaling of flow properties, universal scaling of elastic wave speed with Weissenberg number, Wi, defined the degree of polymer stretching, and the coherent structure of velocity fluctuations, self-organized into cycling self-sustained process, synchronized by elastic waves. These properties persist over the entire channel length above the instability threshold. It means that only an absolute instability exists in inertia-less viscoelastic channel flow, whereas a convective instability, is absent. This unexpected discovery is in sharp contrast with Newtonian flows, where both convective and absolute instabilities are always present in open flows. It occurs due to differences in nonlinear terms in an elastic stress equation, where except for the advective term, two key terms describing polymer stretching along the channel length are present.

Dynamics of non-spherical particles in viscoelastic fluids flowing in a microchannel.

The migration and orientation dynamics of prolate spheroidal particles suspended in a viscoelastic liquid flowing in a square microchannel is experimentally investigated under inertialess flow conditions. The suspending fluid is an aqueous solution of PolyEthylene Oxyde at relatively high concentration characterized by a high level of elasticity and shear-thinning. Fluid viscoelasticity drives the spheroids towards the channel central region at relatively low flow rates when the particles explore the constant viscosity region of the fluid, without showing a preferential orientation. As the flow rate increases and the fluid enters in the shear-thinning region, a smaller fraction of particles migrates at the central channel region, reducing the focusing efficiency. The focused spheroids rotate sufficiently fast to attain a stable orientation with major axis aligned along the flow direction.

Vorticity amplification in wavy viscoelastic channel flow

Surface distortions to an otherwise planar channel flow introduce vorticity perturbations. In Newtonian fluids, the vorticity induced by small surface undulations on the lower wall is advected by the background flow and diffuses into the fluid. When the fluid is viscoelastic, we identify new mechanisms by which significant vorticity perturbations can be generated in both inertialess and elasto-inertial channel flows. We focus on the case where the lengthscale of the surface distortion is much longer than the channel depth, where we find significant departure from plane shear (Page &amp; Zaki, J. Fluid Mech., vol. 901, 2016, pp. 392–429) due to the non-monotonic base-flow streamwise-normal elastic stress. In inertialess flows, a purely elastic response results in streamlines deforming to match the bottom topography in the lower half the channel. However, the vanishing stress at the centreline introduces a blocking effect, and the associated $O(1)$ jump in normal velocity is balanced by a large-amplitude streamwise-oscillating ‘jet’ in a boundary layer, resulting in a localised, chevron-shaped vorticity perturbation field. In elasto-inertial flows, resonance between the frequency of elasto-inertial ‘Alfvén’ waves and the frequency apparent to an observer moving with the fluid results in vorticity amplification in a pair of critical layers on either side of the channel. The vorticity in both layers is equal in magnitude, to leading order in Weissenberg number, and as such the perturbation vorticity field penetrates the full channel depth even when inertia is dominant. The results demonstrate that long-wave distortions, which are relatively innocuous in Newtonian fluids, can drive a significant flow distortion in viscoelastic fluids for a wide range of parameter values.

New direction and perspectives in elastic instability and turbulence in various viscoelastic flow geometries without inertia

We shortly describe the main results on elastically driven instabilities and elastic turbulence in viscoelastic inertialess flows with curved streamlines. Then we describe a theory of elastic turbulence and prediction of elastic waves Re ≪ 1 and Wi ≫ 1, which speed depends on the elastic stress similar to the Alfvén waves in magneto-hydrodynamics and in a contrast to all other, which speed depends on medium elasticity. Since the established and testified mechanism of elastic instability of viscoelastic flows with curvilinear streamlines becomes ineffective at zero curvature, so parallel shear flows are proved linearly stable, similar to Newtonian parallel shear flows. However, the linear stability of parallel shear flows does not imply their global stability. Here we switch to the main subject, namely a recent development in inertialess parallel shear channel flow of polymer solutions. In such flow, we discover an elastically driven instability, elastic turbulence, elastic waves, and drag reduction down to relaminarization that contradict the linear stability prediction. In this regard, we discuss briefly normal versus non-normal bifurcations in such flows, flow resistance, velocity and pressure fluctuations, and coherent structures and spectral properties of a velocity field as a function of Wi at high elasticity number.

Splitting of localized disturbances in viscoelastic channel flow

We examine the response of an inertialess viscoelastic channel flow to localized perturbations. We thus performed an experiment in which we perturbed the flow using a localized velocity pulse and probed the perturbed fluid packet downstream from the perturbation location. While for low Weissenberg numbers the perturbed fluid reaches the measurement location as a single velocity pulse, for sufficiently high Weissenberg numbers and perturbation strengths, a random number of pulses arrive at the measurement location. The average number of pulses observed increases with the Weissenberg number. This observation constitutes a transition to a novel elastic pulse-splitting regime. Our results suggest a possible new direction for studying the elastic instability of viscoelastic channel flows at high elasticity through the growth of localized perturbations.

A Versatile Flow-Profile Engineering Method in the Stokes Flow Regime for Complex-Shaped Flows

Flow profiles are frequently engineered in microfluidic channels for enhanced mixing, reaction control, and material synthesis. Conventionally, flow profiles are engineered by inducing inertial secondary flow to redistribute the streams, which can hardly be reproduced in microfluidic environments with negligible inertial flow. The employed symmetric channel structures also limit the variety of achievable flow profiles. Moreover, each of the flow profiles specifically corresponds to a strictly defined flow condition and cannot be generalized to other flow environments. To address these issues, we present a systematic method to engineer the flow profile using inertialess secondary flow. The flow is manipulated in the Stokes regime by deploying a cascaded series of microsteps with various morphologies inside a microchannel to shape the flow profile. By tuning the shapes of the microsteps, arbitrary outflow profiles can be customized. A numerical profile-transformation program is developed for rapid prediction of the output profiles of arbitrary sequences of predefined microsteps. The proposed method allows the engineering of stable flow profiles, including asymmetric ones, over a wide range of flow conditions for complex microfluidic environmental prediction and design.

Flow of yield stress materials through annular abrupt expansion–contractions

We present an experimental study of the flow of yield stress materials through annular abrupt expansions–contractions, to evaluate the flow invasion into the cavity formed in the larger cross section region. Steady inertialess flows of Carbopol® aqueous dispersions were investigated. The flow pattern reveals yielded and unyielded regions, which were visualized using tracer particles, laser sheets, and a digital camera. The yield surfaces were identified in the experiments by choosing large enough exposure times that allow sufficient particle displacement in the yielded region. To estimate the amount of fluid that remains stagnant in the cavity, we defined the invasion ratio, a quantity that was determined through image processing for different combinations of the governing parameters. The influence of the cavity diameter and axial length, eccentricity, and inlet velocity on the invasion ratio was investigated. Fore-aft asymmetric yield surfaces were observed for all tests, probably due to elastic effects.

Dynamics of phase separation of sheared inertialess binary mixtures

When a viscous binary mixture subject to an applied shear flow is rapidly quenched into the unstable region of its phase diagram, the resulting phase separation is influenced by two competing effects. On one hand, nuclei of the minority phase tend to grow with a constant growth rate, while, on the other hand, they are stretched along the flow direction, forming thinner and thinner layered domains that eventually break. We simulate the dynamics of this system with a thermodynamics-based diffuse interface model, accounting for the full interplay between hydrodynamics (i.e., the Navier–Stokes equations) and species conservation (i.e., the Cahn–Hilliard equation) coupled via the Korteweg body-force. We show that periodic steady-state configurations with stable droplets are obtained for low capillary numbers while phase separation takes place along bands oriented in the direction of the flow in the case of strong shear because, in the long term, diffusion in the cross-flow direction prevails on the convective flow field. The dynamics of phase separation is highly non-linear and diverse even for inertialess flow, featuring multiple coalescence and breakups: although some typical time scaling for the characteristic droplet size in the flow and cross-flow directions can be obtained, the full evolution cannot be characterized only by the capillary number. The wide range of droplet morphologies predicted by the model, from round and elongated shapes to bands and hollow droplets, suggests interesting applications for manufacturing of polymers and soft materials.

A closure model for the drag coefficient of a sphere translating in a viscoelastic fluid

In many large-scale industrial applications dealing with particle-laden viscoelastic fluids, the ensemble-averaged behavior of the mixture is of most interest. The first step to parametrize this behavior is to develop an accurate expression to rapidly evaluate the drag coefficient over a broad range of kinematic parameters. The drag coefficient of a spherical particle translating in a viscoelastic matrix is strongly affected by the viscoelasticity of the fluid. In this study, we aim to parametrize the effects of fluid elasticity, especially the relaxation and retardation times, as well as inertia on the drag coefficient of a sphere translating in a viscoelastic fluid described by the Oldroyd-B model. To this end, we employed three-dimensional direct numerical simulations of viscoelastic flow past a stationary sphere. The accuracy of the numerical formulation is thoroughly tested against a number of benchmark problems consisting of steady flow past a sphere in a bounded circular or square domain filled with either a Newtonian or viscoelastic fluid. Initially, the numerical computations for the drag coefficient over a wide range of geometric and flow parameters are validated by comparison with existing data and drag correction models from the literature. The drag coefficient correction is then evaluated for unconfined flow past a sphere at different Reynolds number, Re, over a wide range of Deborah number, De < 9, and polymer viscosity ratio, 0 < ζ < 1. For small Deborah number (De < 1), the drag coefficient decreases with respect to the Stokes drag coefficient, whereas, at large Deborah number (De > 1), the drag is enhanced due to the large elastic stresses that develop on both the surface and wake of the sphere. These canonical behaviors, observed in the inertia-less flow regime (Re ≤ 1) are amplified as the polymer viscosity ratio approaches unity. At higher Reynolds numbers (Re > 1), the drag coefficient correction arising from viscoelasticity is found to be always bigger than unity, but smaller than the enhancement calculated in the creeping flow limit. In this regime, the formation of an axisymmetric recirculating eddy expands the wake region behind the sphere, and shifts the maximum elastic stresses away from the rear stagnation point towards the flow separation points. Motivated by the self-similarity observed at Re < 1, we propose an approximate model for the viscoelastic drag coefficient based on rational functions, using the lowest possible degree of polynomial functions required to fit the computational data with 95% accuracy, for De ≤ 5 and 0 < ζ < 1. This expression is a key step towards formulating a momentum-exchange model for the flow of dilute suspensions of solid particles in a viscoelastic matrix.

Secondary flows of viscoelastic fluids in serpentine microchannels

Secondary flows are ubiquitous in channel flows, where small velocity components perpendicular to the main velocity appear due to the complexity of the channel geometry and/or that of the flow itself such as from inertial or non-Newtonian effects, etc. We investigate here the inertialess secondary flow of viscoelastic fluids in curved microchannels of rectangular cross-section and constant but alternating curvature: the so-called "serpentine channel" geometry. Numerical calculations (Poole et al, 2013) have shown that in this geometry, in the absence of elastic instabilities, a steady secondary flow develops that takes the shape of two counter-rotating vortices in the plane of the channel cross-section. We present the first experimental visualization evidence and characterization of these steady secondary flows, using a complementarity of microPIV in the plane of the channel, and confocal visualisation of dye-stream transport in the cross-sectional plane. We show that the measured streamlines and the relative velocity magnitude of the secondary flows are in qualitative agreement with the numerical results. In addition to our techniques being broadly applicable to the characterisation of three-dimensional flow structures in microchannels, our results are important for understanding the onset of instability in serpentine viscoelastic flows.

A Model of Incompressible-Fluid Flow with a Free Surface in a Highly Porous Medium

A model is proposed for modeling a slow (inertialess) flow of a thin layer of viscous incompressible fluidwith a free surface inside a highly porousmediumabove an impermeable surface. The fluid flow is described by the Brinkman filtration law, on the bottom boundary the Navier slip condition is prescribed. The equation for the fluid layer thickness is obtained, which has the form of a special case of a nonlinear heat conduction equation. Solutions in the form of traveling waves and self-similar solutions describing the fluid spreading are investigated.

NUMERICAL SIMULATION OF A ELASTO-VISCOPLASTIC FLUID FLOW INSIDE A CAVITY

This article addresses finite element approximations for elasto-viscoplastic flows. Numerical simulations aiming at investigating the role of elasticity for inertialess flows of viscoplastic materials within lid-driven cavity.The mechanical model is made up of the usual governing equations for incompressible fluids coupled with a Oldroyd-B type equation (de Souza Mendes, 2011) modified to incorporate the dependency both of relaxation and retardation time as the viscoplastic viscosity on the strain rate. These parameters depend on the material microstructure, which level is described by an structure parameter . This model is approximated by a multi-field Galerkin least-squares formulation (Behr et al., 1993) in terms of extra-stress tensor, the pressure field and the velocity vector. Results, focused on the determination of yield surface topology, investigate the influence of elastic and viscous governing parameters on the flow pattern.