The viscoelastic instability of cross-slot flow was first observed experimentally almost half a century ago and reproduced numerically two decades ago, yet its physical origin remains unresolved. We revisit this problem for two-dimensional creeping flow of Oldroyd-B fluid by combining direct numerical simulations, global stability analysis, structural sensitivity analysis and energy-budget analysis. Our simulations reproduce the canonical pitchfork bifurcation, and the stability analysis consistently predicts the threshold and perturbation growth rates. The leading eigenmode consists of a chiral velocity–stress perturbation that tilts and rotates the birefringent strand generated by the extensional flow. Structural sensitivity and energy-budget analyses identify narrow high-extension-rate ridges within the extensional flow as both the spatial core and energetic source of the instability. In these ridges, the stress-based wavemaker colocalizes with large positive disturbance polymeric stress power density, indicating localized transfer of stored elastic energy to the disturbance flow. Analyses of cross-slot variants with rounded corners and with a centred cylinder further reveal that neither sharp corners nor a free central stagnation point is the essential destabilizing ingredient; rather, the instability originates from elastic-energy release in extension-dominated regions characteristic of cross-slot flow.

We extend the linear stability analysis of electroconvective instability in a quiescent Oldroyd-B viscoelastic fluid subjected to an electric field into the strongly elastic regime, up to Weissenberg numbers ($\textit{Wi}$) of order $10^{3}$. The oscillatory onset is governed by the interplay among destabilising Coulomb forcing, viscous dissipation and elastic stress relaxation. For the quiescent base state, the linearised constitutive relation allows the polymer stress to be represented through a complex effective viscosity. We then apply a single-mode Galerkin projection based on the Newtonian critical eigenfunction, obtaining a reduced dispersion relation and a closed-form prediction for the neutral oscillation frequency. In the limit of large $\textit{Wi}$, this analytical solution yields two asymptotic scaling laws, namely a frequency scaling law $\omega _{c} \propto \sqrt {(1-\beta )/\textit{Wi}}$ ($\omega _{c}$ is the critical frequency and $\beta$ is the solvent to total viscosity ratio) and a threshold scaling law $T_{c} = \beta T_{c}^{N} + a(\beta )/\textit{Wi}$ ($T_{c}$ is the critical electric Rayleigh number, $T_{c}^{N}$ is the Newtonian critical electric Rayleigh number and $a(\beta )$ is the coefficient related to $\beta$), which satisfy a self-consistency relation. Here, $T_{c}$ exhibits a three-stage evolution with $\textit{Wi}$, and a frozen-elasticity limit is identified at large $\textit{Wi}$. With a single geometric constant as input, the values of $\omega _{c}$ and $T_{c}$ obtained from the analytical solution and from high-resolution spectral computations agree to within 0.3 %, providing mutual corroboration.

This study employs a direct-forcing fictitious-domain method combined with the Oldroyd-B model to investigate the dynamics of spherical and oblate spheroidal particles in a viscoelastic shear flow. By varying the particle Reynolds number (0.1 ≤ Rep ≤ 10) and the Weissenberg number (1 ≤ Wi ≤ 5), we systematically examine the changes in particle equilibrium positions and rotational behaviours. For spherical particles, the equilibrium position shifts monotonically towards the wall with increasing Wi when Rep < 1 or Rep > 3, whereas it remains largely insensitive to elasticity for 1 ≤ Rep ≤ 3. For oblate particles, the rotational period increases steadily with Wi across low-to-moderate Rep, demonstrating that enhanced fluid elasticity significantly reduces the rotational speed. Owing to the interplay of fluid inertia and elasticity, oblate particles initialised with two representative initial orientations, denoted as np1 and np2 (specified by Euler angles as np1 = (0, 0, 0) and np2 = (π/2, π/4, π/2)) exhibit six distinct modes of lateral migration and rotational motion. Under weak fluid inertia, increasing elasticity induces a kayak-like rotation, while at moderate inertia, the motion is dominated by rolling within the shear plane. When Rep ≥ 7, the rotation may become locked, and this locked state can be released as Wi increases.

We investigate the presence of intermittency, a well-known phenomenon in Newtonian turbulence, in viscoelastic Taylor–Couette flow at fixed inertia, (Reynolds number, $\textit {Re}$ = 120). Using particle image velocimetry measurements in the r–z plane for a broad range of elasticities, $\textit{El}$ = 0–0.64, we reveal a dual role of viscoelasticity, even at moderate $\textit {Re}$: it can destabilise the flow and trigger turbulence at moderate elasticity but suppress turbulence intensity at higher values. Gradual turbulence suppression is evident through reduced velocity fluctuations, longer integral time scales proportional to the polymer relaxation time and weakened coherent structures as the flow approaches elasto-inertial turbulence. Higher-order statistics indicate that elasticity increases spatial and temporal intermittency while altering the dynamics: spatially, through a reduced viscous range and enhanced large-scale decorrelation; temporally, via a characteristic time scale consistent with polymer-driven state switching rather than a classical cascade. This spatio-temporal intermittency manifests as alternating active/hibernating turbulence cycles. During hibernation, the flow reorganises into a state distinct from active turbulence, characterised by low velocity fluctuations and Reynolds stresses, suppressing not only turbulence but also linearly unstable Newtonian states such as Taylor vortices. Rather than being featureless, this state exhibits weakened coherent structures in the bulk of the flow and sustained near-wall dynamics, where elastic stresses are likely generated. The present results demonstrate that hibernation is not exclusive to Newtonian turbulence, but also emerges in viscoelastic flows under conditions where inertia alone is insufficient to trigger or sustain turbulent motion.

We investigate drag reduction (DR) in a turbulent channel flow lubricated by a thin viscoelastic layer using direct numerical simulations at a shear Reynolds number $\textit{Re}_\tau = 300$, coupling a phase-field description of the interfacial dynamics with a FENE-CR model for the polymeric stresses. Different elasticity levels are considered by varying the shear Weissenberg number from $\textit{Wi}_\tau = 0$ (Newtonian) to $\textit{Wi}_\tau = 25$ (weakly elastic) and up to $\textit{Wi}_\tau = 100$ (highly elastic). The Newtonian case yields a DR of about $12\,\%$, which increases to roughly $24\,\%$ at $\textit{Wi}_\tau = 100$, with progressively smaller increments that reveal a saturation effect. To explain this behaviour, we employ the energy-box framework, which graphically represents the fluxes of mean kinetic energy, turbulent kinetic energy and elastic potential energy in each layer, together with the corresponding inter-layer exchanges. Polymer stretching and relaxation introduce an additional energy-transfer pathway in the lubricating layer, redirecting kinetic energy from viscous dissipation towards elastic storage and irreversible relaxation. As $\textit{Wi}_\tau$ increases, elastic stresses progressively suppress wall-normal fluctuations, thereby reducing both the capillary work and the production of turbulent kinetic energy. The resulting decrease in turbulent kinetic energy supplied to the lubricating layer reduces the kinetic-to-elastic energy transfer, leading the system to a self-limiting regime in which further increases in elasticity produce negligible additional DR.

We develop and analyse a theory for thermal convection in a Darcy porous material when the heat transfer is based on a Jeffreys theory. The memory terms in the Jeffreys heat constitutive law lead to the important result that the structure of the instability hinges on the relative size of two non-dimensional parameters, the Maxwell coefficient and the Jeffreys coefficient. This leads to a fundamentally different theory of thermal convection in a porous material. When the Jeffreys parameter is greater than or equal to the Maxwell one we show the linear operator attached to the stability problem is symmetric and instability will only arise via stationary convection. In this case the linear instability limit is the same as the nonlinear stability threshold and so linear theory completely captures the onset of the physics of convection. On the other hand, when the Maxwell parameter exceeds the Jeffreys one a skew symmetric term is present and this allows oscillatory convection to happen with suitable parameters. Physically, this is to be expected, since a large Maxwell parameter is leading to a hyperbolic theory of heat flow and hence oscillatory instabilities, whereas when the Jeffreys parameter is large this represents a very strong form of heat dissipation. With modern heterogeneous porous materials like ceramics or foams different time scales for heat transfer at the molecular level are likely and thus a Jeffreys theory with relaxation time for both heat flux and temperature gradient is useful. This work shows that the memory effect of the heat flux equation leads to a strong structural change in the instability behaviour.

Heat dissipation is critical in modern engineering and polymer additives offer a potential route to improve fluid-based cooling. We study elasticity-enhanced heat transfer in two-dimensional, thermally stratified Poiseuille flow. At Reynolds numbers $\textit{Re} \lesssim 1000$, we observe two linearly unstable modes: the recently identified elasticity-induced centre mode (Khalid et al. 2021 J. Fluid Mech. 915, A43) and the classical buoyancy-driven convective mode (Kelly 1994 Adv. Appl. Mech. 31, 35–112). Direct numerical simulations show that the centre mode develops into a nonlinear ‘arrowhead’ state but yields negligible heat-transfer enhancement (typically $\approx 0.03\,\%$ above the conductive state). By contrast, polymers can enhance the heat flux associated with the convective mode by up to $1100\,\%$. The nonlinear convective-mode states are periodic orbits or travelling waves, dominated by hook-like polymer-stress structures that can attach to channel walls. Both unattached and attached hooks act as ‘polymer walls’, steering flow along rather than across them. Unattached hooks deflect fluid vertically around them in the channel interior and reduce flow rate, whereas wall-attached hooks travel more slowly, reorganising the flow into strong counter-rotating rolls and triggering extreme enhancement. The elasto-buoyant nature of these states is confirmed by perturbation kinetic energy budgets, showing that polymer and buoyancy sustain the states synergistically. Wall-attached hooks enable rapid thermal equilibration but impose a large hydraulic penalty, making them suitable for process streams requiring fast temperature adjustment. Unattached hooks provide a more thermally efficient regime for heat transport. These results highlight the potential of elastic fluids for future heat-transfer enhancement technologies.

We analyse the stability of viscoelastic Dean flow (flow of an elastic fluid through a curved two-dimensional channel, driven by an azimuthal pressure gradient) in the absence of fluid inertia. This configuration is well known to exhibit a hoop-stress-driven ‘purely elastic’ instability (referred to henceforth as the hoop-stress mode – ‘HSM’) on account of the base-flow streamline curvature. The objective of this study is to demonstrate the existence and importance of a distinct elastic instability in this flow configuration, which is not driven by hoop stresses, but instead is a continuation of a novel ‘centre-mode’ (CM) instability recently identified in rectilinear shear flows. On account of its origins, the CM instability in Dean flow is expected for two-dimensional (azimuthally varying) disturbances with no axial variation, but continues to exist for three-dimensional disturbances. In contrast, the HSM instability is expected primarily in the axisymmetric limit, and again continues to exist for three-dimensional disturbances. We use both the Oldroyd-B and FENE-P models to map out parameter regimes in the $\textit{Wi}$–$\epsilon$–$\beta$ space where the aforementioned instabilities are present. Here, $\textit{Wi}$ is a suitably defined Weissenberg number that characterises fluid elasticity, $\beta$ is the ratio of solvent to total solution viscosity and $\epsilon$ is the ratio of the gap (channel) width to the radius of curvature. While its origin in rectilinear shearing flows might lead one to expect the CM instability to only be present for small $\epsilon$ (the ‘narrow-gap’ limit), we show that it exists even for $O(1)$ values of $\epsilon$, and over a larger range of $\textit{Wi}$. Nevertheless, within the Oldroyd-B framework, the HSM determines the stability threshold for experimentally relevant gap-width ratios, corresponding to $0.1 \leqslant \epsilon \leqslant 1$, with the CM becoming the most unstable mode only for $\epsilon \lt 0.001$. For the more accurate FENE-P model, however, decreasing the finite extensibility parameter $L$ has opposing effects on the HSM and CM instabilities – stabilising the former, but destabilising the latter. In the dilute solution regime ($\beta \gt 0.95$) and for realistic values of $L \sim O(100)$, corresponding to polymer molecular weights of $O(10^{5-6})$ g mol−1, the CM remains the most unstable mode for $\epsilon \leqslant 0.25$, rendering it potentially relevant to the onset of elastic turbulence in the flow of such polymer solutions through curved channels.

We perform a comprehensive linear non-modal stability analysis of plane Poiseuille flow in Oldroyd-B fluids, employing the recently proposed geometric decomposition of the polymer conformation tensor (Hameduddin et al., J. Fluid Mech., 2018, vol. 842, 395–427; J. Fluid Mech., 2019, vol. 858, 377–406), where perturbation elastic energy has an appropriate definition. In the presence of weak shear flow, e.g. Reynolds number Re = 1, the spanwise-uniform disturbance with a large wavenumber exhibits the largest transient growth, and perturbation elastic energy is the most amplified, proportional to W2 (where W denotes the Weissenberg number), which is mainly attributed to the combination of different modes in the continuous spectrum (CS). Elastic energy analysis suggests that this growth is physically due to the energy transfer from the perturbation wall-normal velocity to the normal components of the perturbation polymer conformation tensor. As Re increases to Re = 1000, transient growth has a significant increase, and its maximum appears in the oblique disturbance with perturbation kinetic energy dominating amplification instead of elastic energy, which is still induced by the spanwise-uniform components of the input perturbation conformation tensor, just as in the Re = 1 case. Mathematically, this energy growth is caused by the combination of CS and the remaining discrete modes, and it is remarkably larger than that merely caused by CS. Kinetic energy analysis suggests that this significant kinetic energy growth is physically due to the fact that elastic effects enhance energy transfer from the base flow field and the perturbed conformation tensor field to the perturbed hydrodynamic field.

Complex suspensions composed of solid particles dispersed in non-Newtonian liquid matrices are ubiquitous in environmental and industrial processes, where their nonlinear rheological response directly impacts the efficiency and predictability of momentum, heat and mass transfer. Here, we experimentally investigate the interplay between non-colloidal spherical particles and high-molecular-weight polymer chains in the canonical Taylor–Couette flow, a configuration where recent experiments have revealed the emergence of an elasto-inertial dissipation (EID) behaviour whose underlying mechanisms remain unclear. Using a polyacrylamide Boger fluid loaded with particles at varying concentrations, we perform systematic rheological measurements and long-duration shear experiments to quantify viscosity, elasticity and the degradation dynamics. Torque monitoring shows that particle addition accelerates polymer chain scission and reduces the asymptotic viscosity. Ramp-up protocols reveal torque-drop behaviour characteristic of EID, which can be qualitatively reproduced by a cumulative-shear degradation model derived from steady-shear data. However, the inability of this model to capture the strong dependence on particle volume fraction indicates that additional semi-dilute particle–polymer interaction mechanisms are at play. In particular, by relating average interparticle distance to the maximum polymer chain extension, we identify a critical particle volume fraction ($\varPhi \simeq 4\,\%$) coinciding with the observed transition from elasto-inertial turbulence (EIT) to EID. These findings show, that while polymer degradation is a key contributor to dissipation, confinement of stretched chains between particles is a more likely mechanism for the suppression of EIT. This dual influence of degradation and particle–polymer interactions offers new insight into rheo-hydrodynamic transitions in complex suspensions and their implications for industrial applications.

Elastoinertial turbulence (EIT) is a self-sustaining chaotic state resulting from the interplay between inertia and elasticity in the flow of dilute polymeric solutions, and its emergence is believed to limit the achievable drag reduction in turbulence flow using polymer additives. In the present study, we introduce a viscoelastic variant of spectral proper orthogonal decomposition (VESPOD) that decomposes velocity and polymeric stress fields of EIT together into well-defined orthogonal oscillating modes such that the decomposition is optimal in terms of the total mechanical energy of the flow. Using this technique, we investigate the dominant coherently evolving structures underlying the dynamics of EIT in axisymmetric pipe flow. By analysing distinct peaks in the leading eigenvalue of the VESPOD eigenvalue spectrum, we find that the dynamics of EIT in pipe flow is dominated by three distinct families of travelling waves, where the higher wavenumber structures of each family are simple harmonics of their respective fundamental waves. The radial velocity fields of the travelling waves are characterised by the formation of large-scale structures spanning the pipe radial direction. However, the polymeric stress fields corresponding to them are characterised by the formation of thin inclined sheets of high stress fluctuations at the critical layers of the respective waves, i.e. the locations where the wave speed of the VESPOD mode matches the mean streamwise velocity. Additionally, these sheets exhibit nested structures, where the polymeric sheets of faster waves are confined by those of the immediately slower waves.

The linear stability of a viscoelastic liquid film containing insoluble surfactant on a horizontal quasi-periodic oscillating plane for disturbances with arbitrary wavenumbers is investigated. The combined effects of viscoelasticity and quasi-periodic oscillation on the instability mechanism in the presence of surfactant are described with the Floquet theory. For long-wavelength instability, the solution in the limit of long-wave perturbations is obtained by the asymptotic expansion method. The results indicate that, in the absence of gravity, the stability of the oscillating film is determined by the competition between viscoelastic and surfactant effects. In the presence of gravity, the U-shaped neutral curves with separation bandwidth appear. Due to the existence of viscoelasticity, the effects of different frequency ratios and amplitude ratios on the U-shaped neutral curve in the high-frequency region are significantly different from the results under Newtonian fluids. Furthermore, whether gravity exists or not, the viscoelastic effect exhibits a dual effect of initially destabilising, subsequently stabilising and finally destabilising again in the high-frequency oscillating liquid film. The finite-wavelength instability is solved numerically based on the Chebyshev spectral collocation method. The study demonstrates that viscoelasticity results in a non-monotonic trend in the critical Reynolds number of finite-wavelength modes. In the high-frequency region, a bifurcation phenomenon appears due to competition between different standing-wave modes. Increasing the viscoelastic parameter, frequency ratio or amplitude ratio significantly destabilises the flow under high-frequency oscillations.

We investigate viscoelastic flow past a square cylinder using the Phan-Thien–Tanner model to disentangle the competing effects of elasticity and shear-thinning. Simulations cover Weissenberg numbers ranging from 0.1 to 10 000, at a fixed Reynolds number of 10 and a solvent viscosity ratio of 0.6. The results are compared with the Newtonian, Oldroyd-B and Carreau–Yasuda models to isolate the influence of each rheological mechanism on drag, vortex shedding and wake structure. For Weissenberg numbers less than 50, elastic stresses dominate; drag increases by up to 30 %, vortex shedding is delayed as the critical Reynolds number increases by approximately 18 %, and the wake contracts sharply, with recirculation lengths reduced by up to 85 %. In the intermediate regime, where Weissenberg numbers range from 50 to 500, elasticity generates distinctive features, including asymmetric upstream vortices, triangular vortex cores and persistent upstream vortices, with their lifetimes increasing with the retardation ratio. For Weissenberg numbers greater than 500, shear-thinning becomes dominant, restoring behaviour similar to that of Newtonian fluids. Vortex lengths recover, drag decreases steadily, the critical Reynolds number drops and the shedding frequency increases by 15%–40 %, depending on fluid extensibility. These results reveal that the interaction between elasticity and shear-thinning controls wake dynamics through a complex balance.

Highly stretched capillary jets produced by gravity are central to drop generation, micro-thread formation and extensional-rheometry concepts. For Newtonian fluids, the transition from steady jetting to self-excited oscillations in a gravitationally stretched jet is predicted accurately by one-dimensional slender-jet equations that retain the exact interfacial curvature and admit a global eigenvalue analysis (Rubio-Rubio et al. 2013, J. Fluid Mech., vol. 729, pp. 471–483). Separately, weakly viscoelastic jets governed by Oldroyd-B/Giesekus constitutive laws exhibit elastocapillary regimes and beads-on-a-string dynamics that are well captured by one-dimensional free-surface models (Ardekani et al. 2010 J. Fluid Mech., vol. 665, pp. 46–56). Here, we study the global linear stability of a one-dimensional full-curvature model for gravitationally stretched viscoelastic jets in the Oldroyd-B limit. We first benchmark the Newtonian limit, reproducing marginal spectra and base-flow profiles, and then quantify how elasticity shifts the critical jetting–dripping boundary by tracking the leading global Hopf eigenpair across the rheological parametric space. For experimentally relevant moderate elasticity, characterised by order-unity Deborah numbers, polymeric tension modifies both the critical Weber number and the selected oscillation frequency, and endogeneity, i.e. the local contribution of the unperturbed flow dynamics to the selected global eigenvalue, reveals that marginality results from a balance between capillary/kinematic contributions and an additional elastic-stress feedback pathway. To interpret and predict the onset mechanism, we compute wavemakers and receptivity/structural-sensitivity fields from direct–adjoint eigenfunctions, showing that viscoelasticity broadens the sensitivity region downstream while the adjoint remains strongly localised near the inlet, thereby identifying the near-nozzle region as the dominant receptive location.

Yoghurt is a food product with strong sensory acceptance and health appeal, making its matrix a target for various innovative presentations and modifications. This study aimed to develop a yoghurt bar with the addition of probiotic microcapsules and to evaluate the effect of gelatine concentration on its chemical composition, microbiological stability, rheological properties and texture profile. Three formulations were developed: YB6 (6.66% gelatine dispersion), YB9 (9.33% gelatine dispersion) and YB12 (12% gelatine dispersion) Probiotics were microencapsulated by ionic gelation, achieving 93.46% encapsulation efficiency. Gelatine concentration influenced protein (YB6/YB9: 6.95 g.100 g−1; YB12: 7.65 g.100 g−1) and sodium content (YB6/YB9: 56.00 mg.100 g−1; YB12: 57.50 mg.100 g−1) but did not significantly alter other compositional parameters. Probiotic counts (log CFU.g−1) for Lactobacillus delbrueckii subsp. bulgaricus (YB6 = 7.08; YB9 = 7.34; YB12 = 7.4), Streptococcus thermophilus (YB6 = 9.20; YB9 = 9.08; YB12 = 9.18) and Lactobacillus acidophilus NCFM® (YB6 = 7.32; YB9 = 7.3; YB12 = 7.0) remained stable over 30 days at 5 °C, meeting regulatory standards for yoghurt classification. Gelatine concentration and temperature influenced rheological properties. At 10 °C, formulations exhibited no structural failure up to 1000 Pa of shear stress, whereas at 25 °C, YB9 and YB12 demonstrated higher structural integrity than YB6. YB12 presented the highest values of firmness, adhesiveness, cohesiveness, gumminess and elasticity. The yoghurt bar represents an innovative product that expands the range of fermented dairy options. The use of 9% or 12% gelatine dispersion in yoghurt bar production ensures good storage stability, protein enrichment and a suitable texture for handheld consumption, thereby expanding the scope of yoghurt-based offerings.

The spread of a pulse of solute in a pressure-driven channel flow is well described for a wide range of Newtonian flows for which the viscosity and diffusivity are constants. Over many decades, various extensions have been suggested for the dispersion in pressure-driven non-Newtonian channel flows. While many theoretical studies have examined the effect of shear-rate-dependent viscosity on dispersion for a variety of non-Newtonian constitutive models, the solute diffusivity has invariably been treated as a constant. This assumption, however, is in contrast to the expectation that the diffusivity of a colloidal particle is inversely related to the viscosity, e.g. recall the Stokes–Einstein relation. We account for this coupling of transport coefficients – viscosity and diffusivity – by assuming a generalised form of the Stokes–Einstein equation, inspired by the recognition that the viscosity is now a field, although only transport transverse to the main flow direction is relevant because of the common assumptions of Taylor–Aris dispersion. Thus, we derive a general formula for axial dispersion in steady, pressure-driven shear-rate-dependent flows in uniform channels. In particular, we apply our general relation to calculate the Taylor–Aris dispersion coefficient for steady flows of a shear-thinning Carreau fluid and a viscoelastic Phan-Thien–Tanner fluid. Finally, we highlight new theoretical questions raised by this transport situation, where the underlying diffusivity is also a (tensorial) field related to variations in viscosity.

We investigate the orientation dynamics of a neutrally buoyant spheroid, of an arbitrary aspect ratio ($\kappa$), freely rotating in a weakly viscoelastic fluid undergoing simple shear flow. Weak elasticity is characterized by a small but finite Deborah number (${\textit{De}}$), and the suspending fluid rheology is therefore modelled as a second-order fluid, with the constitutive equation involving a material parameter $\epsilon$ related to the ratio of the first and second normal stress differences; polymer solutions correspond to $\epsilon \in [-0.7,-0.5]$. Employing a reciprocal theorem formulation, along with expressions for the relevant disturbance fields in terms of vector spheroidal harmonics, we obtain the spheroid angular velocity to $O(De)$. In the Newtonian limit, a spheroid rotates along Jeffery orbits parametrized by an orbit constant $C$, although this closed-trajectory topology is structurally unstable, being susceptible to weak perturbations. For ${\textit{De}}$ well below a threshold, ${\textit{De}}_c(\kappa )$, weak viscoelasticity transforms the closed-trajectory topology into a tightly spiralling one. A multiple-scales analysis is used to interpret the resulting orientation dynamics in terms of an $O(De)$ orbital drift. The drift in orbit constant over a Jeffery period $\Delta C$, when plotted as a function of $C$, identifies four different orientation dynamics regimes on the $\kappa -\epsilon$ plane. For $\epsilon$ in the polymeric range, prolate spheroids always drift towards the spinning mode. Oblate spheroids drift towards the tumbling mode for $\kappa \gt \kappa _c(\epsilon )$, but towards an intermediate kayaking mode for $\kappa \lt \kappa _c(\epsilon )$; $\kappa _c$ increases from $0.2$ to $0.454$ as $\epsilon$ varies from $-0.5$ to $-0.7$. The rotation of spheroids of extreme aspect ratios, either slender prolate spheroids ($\kappa \gg 1$) or thin oblate ones ($\kappa \ll 1$), about the vorticity axis, is arrested for ${\textit{De}} \geqslant {\textit{De}}_c(\kappa )$. The resulting rotation-arrested states exist within the flow-gradient plane in both cases, being nearly aligned with the flow direction for the prolate case, and oriented in the vicinity of the gradient direction for the oblate one. While rotation-arrested states are unstable to off-plane perturbations in the former instance, secondary bifurcations beyond the primary rotation-arrest threshold lead eventually to stable arrested states for thin oblate spheroids.

The addition of a small amount of long-chain polymers confers viscoelastic properties to Newtonian flows. The resulting non-Newtonian solution now exhibits a different dynamics, such as enhanced mixing at low Reynolds number, where elastic instabilities can trigger elastic turbulence even though inertial turbulence is absent. Here, we study this phenomenon in viscoelastic planar jets and, in particular, we do it from the perspective of coherent structures to understand how elastic turbulence is triggered and sustained, which remain barely explored in this set-up. We introduce the spatio-temporal Koopman decomposition for extracting the dominant flow patterns, and we compare them with those from Newtonian planar jets at high Reynolds number. Global flow structures are similar between jets, with low-frequency streaks and high-frequency wave packets dominating the turbulent dynamics. However, structures are strikingly different in the near field, where elasticity-driven streaks affect the dynamics in the potential core of the viscoelastic planar jet, modifying the bulk flow and interacting with the flow instability. The analysis of the polymer field reveals stretched polymer filaments and centre-mode structures, which support the implication of the near-field streaks sustaining elastic turbulence in three-dimensional viscoelastic planar jets.

We present a linear stability analysis of plane Poiseuille flow of an Oldroyd-B fluid confined between rigid, isotropic porous layers at the upper and lower boundaries. The study aims to investigate the interplay between elasticity, inertia and wall permeability on the onset of flow instability. The system is characterized by four dimensionless parameters: the Reynolds number ($ \textit{Re}$), Weissenberg number ($ \textit{Wi}$), permeability parameter ($\alpha$) and solvent viscosity ratio ($\beta$), with porosity held fixed at $\epsilon = 0.6$. We focus on the low-permeability regime, where inertial effects within the porous layers are negligible. The governing equations are formulated using a modified Darcy–Brinkman–Oldroyd-B model, following the framework of Tan & Masuoka (2005 Phys. Fluids, vol. 17, p. 023101), and are coupled across the fluid–porous interfaces using appropriate interfacial conditions. The formulation recovers the classical results for impermeable viscoelastic channels and Newtonian flows with porous surfaces in the appropriate limits. At low $ \textit{Wi}$, the dominant mode resembles a viscoelastic Tollmien–Schlichting instability, but porous boundaries significantly alter the spectrum, introducing concentric eigenvalue rings and modal shifts. At higher $ \textit{Wi}$, an elastic wall mode emerges near the interfaces and dominates at low $ \textit{Re}$. The critical Reynolds number $ \textit{Re}_{\textit{cr}}$ depends on $ \textit{Wi}$ and $\beta$, with a secondary instability branch appearing at large $ \textit{Wi}$. For highly permeable walls ($\alpha =50$), $ \textit{Re}_{\textit{cr}}$ varies in a non-uniform trend with $\beta$, in contrast to impermeable cases.

Direct numerical simulations are performed to elucidate the influence of counter- and co-rotation on turbulent viscoelastic Taylor–Couette flow in the Rossby number range $ \textit{Ro}^{-1}=-0.6$ to $ \textit{Ro}^{-1}=1$. A novel polymer-induced transition pathway is discovered that is fundamentally different from Newtonian flows. In the counter-rotation regime, the neutral surface is elastically modified and separates the turbulent inner-wall region containing chaotic vortices from the relaminarised layer adjacent to the outer cylinder. Strikingly, co-rotation triggers an elasto-rotational instability, which leads to the breakdown of large-scale Taylor vortices into small-scale penetrating structures, thereby preventing the relaminarisation at high co-rotation rates. Examination of turbulence dynamics demonstrates that the structural changes with increasing $ \textit{Ro}^{-1}$ are accompanied by a transition from elasto-inertial to elastically dominated turbulence. Specifically, the polymer stress progressively exceeds the Reynolds stress and monotonically enhances angular momentum transport, which eliminates the optimal transport characteristic found in the Newtonian flow. The elastically dominated nature of the turbulent flow under co-rotation is further corroborated by the more significant elastic production of the turbulent kinetic energy, as well as the monotonic enhancement of the polymer elongation as $ \textit{Ro}^{-1}$ increases. Furthermore, it is indicated that the polymer orientation strongly depends on vortical structures, with enhanced radial alignment occurring in the boundary region between adjacent vortices. This vortex-mediated polymer orientation is crucial for generating substantial polymer shear stress, establishing a direct link between coherent structures and polymer dynamics.