We study the two-dimensional creeping flow of a viscoelastic fluid around a cylinder confined between two plates parallel to its axis. First, we solve the governing equations under steady state with our novel stabilized finite-element formulation to obtain converged solutions even at very high Weissenberg numbers. Then, we examine the stability of this solution by perturbing all flow variables and solving the corresponding eigenvalue problem. At critical conditions, a stable asymmetric flow arises, in which more fluid passes from either the upper or the lower gap between the cylinder and the channel wall. Both shear-thinning and elasticity play a crucial role on the onset and subsequent evolution of the instability. Energy analysis shows that the terms of the constitutive equation corresponding to apparent strain-rate thinning and material extensibility are responsible for the flow destabilization. The instability is present at a wider range of flow conditions when the material is more elastic and when the solvent contribution is smaller. The instability is also promoted by increasing the confinement. Beyond the critical conditions, asymmetric flow profiles vanish when the flow is so intense that thinning effects are not important anymore. The critical Weissenberg number for instability inception and cessation depends on material properties and geometry exponentially and linearly, respectively. Furthermore, the instability arises even in a seemingly non-shear-thinning fluid, i.e. one with constant shear viscosity in simple shear, when the solvent contribution is minimal, because of the apparent thinning effect that is created by the convection of the viscoelastic stresses. Finally, models with extension-rate thinning trigger the instability at limited flow conditions, when the shear viscosity decreases with the shear rate, and the normal stresses at the wake of the cylinder are still important. These results agree with previous experiments and simulations, and give new insights on the physical mechanism that triggers this flow instability.

We propose a novel approach for non-Newtonian viscoelastic steady flows based on a decomposition of the rate-of-deformation tensor which here, in a simplified version, leads to an anisotropic generalised Newtonian fluid-like model with separated treatment of kinematics pertaining to shear and extensional flows. Care is taken to assure that the approach is objective and does not introduce spurious effects due to dependence on a specific reference frame. This is done by separating the rate-of-deformation tensor into shear and elongational components, with the local scalar shear and elongation rates being the second invariant of each tensor separately, and by modifying the method used to separate the rate-of-deformation tensor so that it becomes independent of superimposed rigid rotations, thus satisfying the principle of material indifference. We assess the model with two test cases: planar contraction flow, often employed to evaluate numerical methods or constitutive equations and for which experimentally observed corner vortex enhancement and pressure drop increase are seldom found in numerical simulations; and flow past confined or unconfined cylinders, for which experiments indicate a drag increase due to elasticity and most predictions give a drag decrease. With our anisotropic model, incorporating additional elongational-flow-related terms, large vortices and accentuated pressure drop coefficients can be predicted for the contraction problem and enhanced drag coefficient for the cylinder problems. This is the first work where problems of non-Newtonian fluid mechanics are solved numerically with separation of strain rate into shear and elongational components.

A long-wave asymptotic model is developed for a viscoelastic falling film along the inside of a tube; viscoelasticity is incorporated using an upper convected Maxwell model. The dynamics of the resulting model in the inertialess limit is determined by three parameters: Bond number Bo, Weissenberg number We and a film thickness parameter $a$. The free surface is unstable to long waves due to the Plateau–Rayleigh instability; linear stability analysis of the model equation quantifies the degree to which viscoelasticity increases both the rate and wavenumber of maximum growth of instability. Elasticity also affects the classification of instabilities as absolute or convective, with elasticity promoting absolute instability. Numerical solutions of the nonlinear evolution equation demonstrate that elasticity promotes plug formation by reducing the critical film thickness required for plugs to form. Turning points in travelling wave solution families may be used as a proxy for this critical thickness in the model. By continuation of these turning points, it is demonstrated that in contrast to Newtonian films in the inertialess limit, in which plug formation may be suppressed for a film of any thickness so long as the base flow is strong enough relative to surface tension, elasticity introduces a maximum critical thickness past which plug formation occurs regardless of the base flow strength. Attention is also paid to the trade-off of the competing effects introduced by increasing We (which increases growth rate and promotes plug formation) and increasing Bo (which decreases growth rate and inhibits plug formation) simultaneously.

Understanding interfacial instability in a coflow system has relevance in the effective manipulation of small objects in microfluidic applications. We experimentally elucidate interfacial instability in stratified coflow systems of Newtonian and viscoelastic fluid streams in microfluidic confinements. By performing a linear stability analysis, we derive equations that describe the complex wave speed and the dispersion relationship between wavenumber and angular frequency, thus categorizing the behaviour of the systems into two main regimes: stable (with a flat interface) and unstable (with either a wavy interface or droplet formation). We characterize the regimes in terms of the capillary numbers of the phases in a comprehensive regime plot. We decipher the dependence of interfacial instability on fluidic parameters by decoupling the physics into viscous and elastic components. Remarkably, our findings reveal that elastic stratification can both stabilize and destabilize the flow, depending on the fluid and flow parameters. We also examine droplet formation, which is important for microfluidic applications. Our findings suggest that adjusting the viscous and elastic properties of the fluids can control the transition between wavy and droplet-forming unstable regimes. Our investigation uncovers the physics behind the instability involved in interfacial flows of Newtonian and viscoelastic fluids in general, and the unexplored behaviour of interfacial waves in stratified liquid systems. The present study can lead to a better understanding of the manipulation of small objects and production of droplets in microfluidic coflow systems.

Ultrasonic standing wave technology offers an ideal platform for manipulating particles in microfluidics. We study how fluid viscoelasticity and acoustic boundary formation in micro-confinements affect ultrasound-induced perturbations. These perturbations influence acoustic energy density (AED) and consequently particle transport dynamics. Our approach combines theoretical, numerical and experimental methods. Using the Oldroyd-B model for viscoelastic fluids, we advance acoustic radiation force (ARF) formulations of Doinikov et al. (Phys. Rev. E, vol. 104, no. 6, 2021a; Phys. Rev. E, vol. 104, no. 6, 2021b) for particles much smaller than the acoustic wavelength. This improved approach allows us to decouple AED and acoustic contrast factor terms in the ARF expression. It also enables us to examine the effects of viscoelastic parameters: $\mu ^*$ (ratio of the viscosity of the viscoelastic fluid to that of base Newtonian fluid) and $De$ (product of fluid relaxation time and actuation frequency) on AED and particle migration. Remarkably, we show that increasing fluid elasticity or $De$ transitions viscoelastic fluids from the energy dissipation (relaxation) mode to the energy storage (frozen) mode, increasing AED. Conversely, increasing viscosity ($\mu ^*$) reduces AED. Thus, our findings suggest that elastic effects accelerate particle migration, while viscous effects decelerate it. Consequently, a viscoelastic fluid-filled micro-confinement acts as an energy dissipation device at low $De$ and an energy storage device at high $De$. Particle migration can be controlled by adjusting viscoelastic and acoustic parameters, at a fixed power input. Our theoretical and numerical findings are validated with our experimental data. Our study advances the fundamental understanding of particle migration in viscoelastic fluids under ultrasound, and can significantly impact future studies on particle/cell migration in bio-fluids.

Turbulent flow induced by elastorotational instability in viscoelastic Taylor–Couette flow (TCF) with Keplerian rotation is analogous to a turbulent accretion disk destabilized by magnetorotational instability. We examine this novel viscoelastic Keplerian turbulence via direct numerical simulations (DNS) for the shear Reynolds number ($Re$) ranging from $10^2$ to $10^4$. The observed characteristic flow structure consists of penetrating streamwise vortices with axial length scales much smaller than the gap width, distinct from the classic centrifugally induced Taylor vortices, which have axial lengths of the gap width. These intriguing vortices persist for the wide $Re$ range considered and give rise to intriguing scaling behaviour in key flow quantities. Specifically, the characteristic axial length of the penetrating vortices is shown to scale as $Re^{-0.22}$; the angular momentum transport scales as $Re^{0.42}$; the kinetic and elastic boundary-layer thicknesses based on angular velocity and hoop stress near the inner cylinder wall scale as $Re^{-0.48}$ and $Re^{-0.49}$, respectively. This implies that the viscoelastic Keplerian turbulence belongs to the classical turbulent regime of TCF with the Prandtl–Blasius-type boundary layer. Furthermore, we present an analytical relation between the viscous and elastic dissipation rates of kinetic energy and the angular momentum transport and in turn demonstrate its validity using our DNS data. This study has paved the way for future research to explore astrophysics-related Keplerian turbulence and angular momentum transport via the scaling relations of the analogous TCF of dilute polymeric solutions.

Simulations of elastic turbulence, the chaotic flow of highly elastic and inertialess polymer solutions, are plagued by numerical difficulties: the chaotically advected polymer conformation tensor develops extremely large gradients and can lose its positive-definiteness, which triggers numerical instabilities. While efforts to tackle these issues have produced a plethora of specialized techniques – tensor decompositions, artificial diffusion, and shock-capturing advection schemes – we still lack an unambiguous route to accurate and efficient simulations. In this work, we show that even when a simulation is numerically stable, maintaining positive-definiteness and displaying the expected chaotic fluctuations, it can still suffer from errors significant enough to distort the large-scale dynamics and flow structures. We focus on two-dimensional simulations of the Oldroyd-B and FENE-P equations, driven by a large-scale cellular body forcing. We first compare two positivity-preserving decompositions of the conformation tensor: symmetric square root (SSR) and Cholesky with a logarithmic transformation (Cholesky-log). While both simulations yield chaotic flows, only the latter preserves the pattern of the forcing, i.e. its fluctuating vortical cells remain ordered in a lattice. In contrast, the SSR simulation exhibits distorted vortical cells that shrink, expand and reorient constantly. To identify the accurate simulation, we appeal to a hitherto overlooked mathematical bound on the determinant of the conformation tensor, which unequivocally rejects the SSR simulation. Importantly, the accuracy of the Cholesky-log simulation is shown to arise from the logarithmic transformation. We also consider local artificial diffusion, a potential low-cost alternative to high-order advection schemes. Unfortunately, the artificially enhanced diffusive smearing of polymer stress in regions of intense stretching substantially modifies the global dynamics. We then show how the spurious large-scale motions, identified here, contaminate predictions of scalar mixing. Finally, we discuss the effects of spatial resolution, which controls the steepness of gradients in a non-diffusive simulation.

The research on elasto-inertial turbulence (EIT), a new type of turbulent flow, has reached the stage of identifying the minimal flow unit (MFU). On this issue, direct numerical simulations of FENE-P fluid flow in two-dimensional channels with variable sizes are conducted in this study. We demonstrate with the increase of channel length that the simulated flow experiences several different flow patterns, and there exists an MFU for EIT to be self-sustained. At Weissenberg number ($Wi$) higher than the one required to excite EIT, when the channel length is relatively small, a steady arrowhead regime (SAR) flow structure and a laminar-like friction coefficient is achieved. However, as the channel length increases, the flow can fully develop into EIT characterized with high flow drag. Close to the size of the MFU, the simulated flow behaves intermittently between the SAR state with low drag and EIT state with high drag. The flow falling back to ‘laminar flow’ is caused by the insufficient channel size below the MFU. Furthermore, we give the relationship between the value of the MFU and the effective $Wi$, and explain its physical reasons. Moreover, the intermittent flow regime obtained based on the MFU gives us an opportunity to look into the origin and exciting process of EIT. Through capturing the onset process of EIT, we observed that EIT originates from the sheet-like extension structure located near the wall, which is maybe related to the wall mode rather than the centre mode. The fracture and regeneration of this sheet-like structure is the key mechanism for the self-sustaining of EIT.

Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g. sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the swimmer and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here, we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency, defined as a particular flow rate divided by the torque required to spin the motor, over this parameter space. Our findings indicate that motor efficiency (measured by the volumetric flow rate) can be boosted or degraded by relatively moderate or strong shear thinning of the viscoelastic environment.

We investigate theoretically the steady incompressible viscoelastic flow in a rigid axisymmetric cylindrical pipe with varying cross-section. We use the Oldroyd-B viscoelastic constitutive equation to model the fluid viscoelasticity. First, we derive exact general formulae: for the total average pressure-drop as a function of the wall shear rate and the viscoelastic axial normal extra-stress; for the viscoelastic extra-stress tensor and the Trouton ratio as functions of the fluid velocity on the axis of symmetry; and for the viscoelastic extra-stress tensor along the wall in terms of the shear rate at the wall. Then we exploit the classic lubrication approximation, valid for small values of the square of the aspect ratio of the pipe, to simplify the original governing equations. The final equations are solved analytically using a regular perturbation scheme in terms of the Deborah number, De, up to eighth order in De. For a hyperbolically shaped pipe, we reveal that the reduced pressure-drop and the Trouton ratio can be recast in terms of a modified Deborah number, Dem, and the polymer viscosity ratio, η, only. Furthermore, we enhance the convergence and accuracy of the eighth-order solutions by deriving transformed analytical formulae using Padé diagonal approximants. The results show the decrease of the pressure drop and the enhancement of the Trouton ratio with increasing Dem and/or increasing η. Comparison of the transformed solutions with numerical simulations of the lubrication equations using pseudospectral methods shows excellent agreement between the results, even for high values of Dem and all values of η, revealing the robustness, validity and efficiency of the theoretical methods and techniques developed in this work. Last, it is shown that the exact solution for the Trouton ratio gives a well-defined and finite solution for any value of Dem and reveals the reason for the failure of the corresponding high-order perturbation series for Dem > 1/2.

We perform simulations of an impulsively started, axisymmetric viscoelastic jet exiting a nozzle and entering a stagnant gas phase using the open-source code Basilisk. This code allows for efficient computations through an adaptively refined volume-of-fluid technique that can accurately capture the deformation of the liquid–gas interface. We use the FENE-P constitutive equation to describe the viscoelasticity of the liquid, and employ the log-conformation transformation, which provides stable solutions for the evolution of the conformation tensor as the jet thins down under the action of interfacial tension. For the first time, the entire jetting and breakup process of a viscoelastic fluid is simulated, including the pre-shearing flow through the nozzle, which results in an inhomogeneous initial radial stress distribution in the fluid thread that affects the subsequent breakup dynamics. The evolution of the velocity field and the elastic stresses in the nozzle are validated against analytical solutions where possible, and the early-stage dynamics of the jet evolution are compared favourably to the predictions of linear stability theory. We study the effect of the flow inside the nozzle on the thinning dynamics of the viscoelastic jet (which develops distinctive ‘beads-on-a-string’ structures) and on the spatio-temporal evolution of the polymeric stresses in order to systematically explore the dependence of the filament thinning and breakup characteristics on the initial axial momentum of the jet and the extensibility of the dissolved polymer chains.

The peculiar migration and rotational dynamics of non-spherical particles in non-Newtonian flows stem from the interplay between fluid rheology and fluid inertial effects. In this paper, the cross-flow migration of a neutrally buoyant oblate spheroid (aspect ratio $AR = 0.5$) immersed in the elasto-inertial duct flow is investigated by particle-resolved simulations with the immersed boundary method. Different from spherical particles, due to the orientation-dependent lift force, the oblate spheroid migrates in an oscillating manner in the duct. The travelling period for particles reaching the duct centreline undergoes a non-monotonic change with elastic number, revealing the existence of a critical elastic number governing the migrating efficiency of oblate particles within the present flow system. For the particle rotation and orientation, the present results indicate that the particle rotation rate and orbit drifting rate are both hindered by the fluid elasticity. With increasing the fluid elasticity, three different orientation modes – log-rolling mode, kayaking-like mode and longside-flow alignment mode – are observed successively during the elasto-inertial migration of the oblate spheroid. Potentially, the present results could be used to design the rheology-based controlling strategy for guiding particles to achieve optimal focusing and orientation in microfluidic applications without the need for external forcing fields.

The elasto-inertial focusing and rotating characteristics of spheroids in a square channel flow of Oldroyd-B viscoelastic fluids are studied by the direct forcing/fictitious domain method. The rotational behaviours, changes in the equilibrium positions and travel distances are explored to analyse the mechanisms of spheroid migration in viscoelastic fluids. Within the present simulated parameters (1 ≤ Re ≤ 100, 0 ≤ Wi ≤ 2, 0.4 ≤ α ≤3), the results show that there are four kinds of equilibrium positions and six (five) kinds of rotational behaviours for the elasto-inertial migration of prolate (oblate) spheroids. We are the first to identify a new rotational mode for the migration of prolate spheroids. Only when the particles are initially located at a corner and wall bisector, some special initial orientations of the spheroids have an impact on the final equilibrium position and rotational mode. In other general initial positions, the initial orientation of the spheroid has a negligible effect. A higher Weissenberg number means the faster the particles migrate to the equilibrium position. The spheroid gradually changes from the corner (CO), channel centreline (CC), diagonal line (DL) and cross-section midline (CSM) equilibrium positions as the elastic number decreases, depending on the aspect ratio, initial orientation and rotational behaviour of the particles and the elastic number of the fluid. When the elastic number is less than the critical value, the types of rotational modes of the spheroids are reduced. By controlling the elastic number near the critical value, spheroids with different aspect ratios can be efficiently separated.

Many natural and industrial processes involve the flow of fluids made of solid particles suspended in non-Newtonian liquid matrices, which are challenging to control due to the fluid's nonlinear rheology. In the present work, a Taylor–Couette canonical system is used to investigate the flow of dilute to semi-dilute suspensions of neutrally buoyant spherical particles in highly elastic base polymer solutions. Friction measurement synchronized with direct flow visualization are combined to characterize the critical conditions for the onset of elasto-inertial instabilities, expected here as a direct transition to elasto-inertial turbulence (EIT). Adding a low particle volume fraction (${\leq }2\,\%$, dilute regime) does not affect the nature of the primary transition and reduces the critical Weissenberg number for the onset of EIT, despite a significant decrease in the apparent fluid elasticity. However, for particle volume fractions ${\geq }6\,\%$ (semi-dilute regime), EIT is no longer observed in the explored Reynolds range, suggesting an apparent relaminarization with yet not further decrease in fluid elasticity. Instead, a new regime, termed here elasto-inertial dissipative (EID), was uncovered. It originates from particle–particle interactions altering particle–polymer interactions and occurring under elasto-inertial conditions comparable to those of EIT. Increasing particle volume fraction in the semi-dilute regime and, in so, the particle contribution to the overall viscosity, delays the onset of EID similarly to what was observed previously for EIT in lower elasticity fluids. After this onset, a decrease in the pseudo-Nusselt number observed with increasing inertia and particle-to-polymer concentration ratio confirms a particle-induced alteration of energy transfer in the flow.

Cross-stream migration of a deformable fluid particle is investigated computationally in a pressure-driven channel flow of a viscoelastic fluid via interface-resolved simulations. Flow equations are solved fully coupled with the Giesekus model equations using an Eulerian–Lagrangian method and extensive simulations are performed for a wide range of flow parameters to reveal the effects of particle deformability, fluid elasticity, shear thinning and fluid inertia on the particle migration dynamics. Migration rate of a deformable particle is found to be much higher than that of a solid particle under similar flow conditions mainly due to the free-slip condition on its surface. It is observed that the direction of particle migration can be altered by varying shear thinning of the ambient fluid. With a strong shear thinning, the particle migrates towards the wall while it migrates towards the channel centre in a purely elastic fluid without shear thinning. An onset of elastic flow instability is observed beyond a critical Weissenberg number, which in turn causes a path instability even for a nearly spherical particle. An inertial path instability is also observed once particle deformation exceeds a critical value. Shear thinning is found to be suppressing the path instability in a viscoelastic fluid with a high polymer concentration whereas it reverses its role and promotes path instability in a dilute polymer solution. It is found that migration of a deformable particle towards the wall induces a secondary flow with a velocity that is approximately an order of magnitude higher than the one induced by a solid particle under similar flow conditions.

Elastoinertial turbulence (EIT) is a chaotic flow resulting from the interplay between inertia and viscoelasticity in wall-bounded shear flows. Understanding EIT is important because it is thought to set a limit on the effectiveness of turbulent drag reduction in polymer solutions. Here, we analyse simulations of two-dimensional EIT in channel flow using spectral proper orthogonal decomposition (SPOD), discovering a family of travelling wave structures that capture the sheetlike stress fluctuations that characterise EIT. The frequency-dependence of the leading SPOD mode contains distinct peaks and the mode structures corresponding to these peaks exhibit well-defined travelling structures. The structure of the dominant travelling mode exhibits shift–reflect symmetry similar to the viscoelasticity-modified Tollmien–Schlichting (TS) wave, where the velocity fluctuation in the travelling mode is characterised by large-scale regular structures spanning the channel and the polymer stress field is characterised by thin, inclined sheets of high polymer stress localised at the critical layers near the channel walls. The travelling structures corresponding to the higher-frequency modes have a very similar structure, but are nested in a region roughly bounded by the critical layer positions of the next-lower-frequency mode. A simple theory based on the idea that the critical layers of mode $\kappa$ form the ‘walls’ for the structure of mode $\kappa +1$ yields quantitative agreement with the observed wave speeds and critical layer positions, indicating self-similarity between the structures. The physical idea behind this theory is that the sheetlike localised stress fluctuations in the critical layer prevent velocity fluctuations from penetrating them.

Motivated by the recent numerical results of Khalid et al. (Phys. Rev. Lett., vol. 127, 2021, 134502), we consider the large-Weissenberg-number ($W$) asymptotics of the centre mode instability in inertialess viscoelastic channel flow. The instability is of the critical layer type in the distinguished ultra-dilute limit where $W(1-\beta )=O(1)$ as $W \rightarrow \infty$ ($\beta$ is the ratio of solvent-to-total viscosity). In contrast to centre modes in the Orr–Sommerfeld equation, $1-c=O(1)$ as $W \rightarrow \infty$, where $c$ is the phase speed normalised by the centreline speed as a central ‘outer’ region is always needed to adjust the non-zero cross-stream velocity at the critical layer down to zero at the centreline. The critical layer acts as a pair of intense ‘bellows’ which blows the flow streamlines apart locally and then sucks them back together again. This compression/rarefaction amplifies the streamwise-normal polymer stress which in turn drives the streamwise flow through local polymer stresses at the critical layer. The streamwise flow energises the cross-stream flow via continuity which in turn intensifies the critical layer to close the cycle. We also treat the large-Reynolds-number ($Re$) asymptotic structure of the upper (where $1-c=O(Re^{-2/3})$) and lower branches of the $Re$–$W$ neutral curve, confirming the inferred scalings from previous numerical computations. Finally, we remark that the viscoelastic centre-mode instability was actually first observed in viscoelastic Kolmogorov flow by Boffetta et al. (J. Fluid Mech., vol. 523, 2005, pp. 161–170).

We study the spreading of Newtonian viscous (aqueous glycerin solution) and viscoelastic (aqueous polymer solution) drops on solid substrates with different wettabilities. For drops of the same zero-shear viscosity, we find in the early stages of spreading that viscoelastic drops (i) spread faster and (ii) their contact radius shows a different power law vs time than Newtonian drops. We argue that the effect of viscoelasticity is only observable for experimental time scales of the order of or larger than the internal relaxation time of the viscoelastic polymer solution. We attribute this behaviour to the shear thinning of the viscoelastic polymer solution. When approaching the contact line, the shear rate increases and the steady-state viscosity of the viscoelastic drop is lower than that of the Newtonian drop. We support our experimental findings with a simple (first-order) perturbation model that qualitatively agrees with our findings.