Gravity currents are a ubiquitous density-driven flow occurring in both the natural environment and in industry. They include: seafloor turbidity currents, primary vectors of sediment, nutrient and pollutant transport; cold fronts; and hazardous gas spills. However, while the energetics are critical for their evolution and particle suspension, they are included in system-scale models only crudely, so we cannot yet predict and explain the dynamics and run-out of such real-world flows. Herein, a novel depth-averaged framework is developed to capture the evolution of volume, concentration, momentum and turbulent kinetic energy from direct integrals of the full governing equations. For the first time, we show the connection between the vertical profiles, the evolution of the depth-averaged flow and the energetics. The viscous dissipation of mean-flow energy near the bed makes a leading-order contribution, and an energetic approach to entrainment captures detrainment of fluid through particle settling. These observations allow a reconsideration of particle suspension, advancing over 50 years of research. We find that the new formulation can describe the full evolution of a shallow dilute current, with the accuracy depending primarily on closures for the profiles and source terms. Critically, this enables accurate and computationally efficient hazard risk analysis and earth surface modelling.

Attenuation of shock waves through dense granular media with varying macro-scale and micro-scale parameters has been numerically studied in this work by a coupled Eulerian–Lagrangian approach. The results elucidate the correlation between the attenuation mechanism and the nature of shock-induced unsteady flows inside the granular media. As the shock transmission becomes trivial relative to the establishment of unsteady interpore flows, giving way to the gas filtration, the shock attenuation mechanism transitions from the shock dynamics and deduction of propagation area associated with the shock transmission, to the drag-related friction dissipation alongside the gas filtration. The ratio between the maximum shock transmission length and the thickness of the particle layer is found to be a proper indicator of the nature of shock-induced flows. More importantly, it is this ratio that successfully collapses the upstream and downstream pressures of shock impacted particle layers with widely ranging thickness and volume fraction, leading to a universal scaling law for the shock attenuation effect. We further propose a correlation between the structure of particle layer and the corresponding maximum shock transmission length, guaranteeing adequate theoretical predictions of the upstream and downstream pressures. These predictions are also necessary for an accurate estimation of the spread rate of shock dispersed particle bed through a pressure-gradient-based scaling method.

We investigate the emergent three-dimensional (3-D) dynamics of a rapidly yawing spheroidal swimmer interacting with a viscous shear flow. We show that the rapid yawing generates non-axisymmetric emergent effects, with the active swimmer behaving as an effective passive particle with two orthogonal planes of symmetry. We also demonstrate that this effective asymmetry generated by the rapid yawing can cause chaotic behaviour in the emergent dynamics, in stark contrast to the emergent dynamics generated by rapidly rotating spheroids, which are equivalent to those of effective passive spheroids. In general, we find that the shape of the equivalent effective particle under rapid yawing is different to the average shape of the active particle. Moreover, despite having two planes of symmetry, the equivalent passive particle is not an ellipsoid in general, except for specific scenarios in which the effective shape is a spheroid. In these scenarios, we calculate analytically the equivalent aspect ratio of the effective spheroid. We use a multiple scales analysis for systems to derive the emergent swimmer behaviour, which requires solving a non-autonomous nonlinear 3-D dynamical system, and we validate our analysis via comparison to numerical simulations.

We investigate the momentum fluxes between a turbulent air boundary layer and a growing–breaking wave field by solving the air–water two-phase Navier–Stokes equations through direct numerical simulations. A fully developed turbulent airflow drives the growth of a narrowbanded wave field, whose amplitude increases until reaching breaking conditions. The breaking events result in a loss of wave energy, transferred to the water column, followed by renewed growth under wind forcing. We revisit the momentum flux analysis in a high-wind-speed regime, characterized by the ratio of the friction velocity to wave speed $u_\ast /c$ in the range $[0.3\,{-}\,0.9]$, through the lens of growing–breaking cycles. The total momentum flux across the interface is dominated by pressure, which increases with $u_\ast /c$ during growth and reduces sharply during breaking. Drag reduction during breaking is linked to airflow separation, a sudden acceleration of the flow, an upward shift of the mean streamwise velocity profile and a reduction in Reynolds shear stress. We characterize the reduction of pressure stress and flow acceleration through an aerodynamic drag coefficient by splitting the analysis between growing and breaking stages, treating them as separate subprocesses. While drag increases with $u_\ast /c$ during growth, it decreases during breaking. Averaging over both stages leads to a saturation of the drag coefficient at high $u_\ast /c$, comparable to what is observed at high wind speeds in laboratory and field conditions. Our analysis suggests that this saturation is controlled by breaking dynamics.

We investigate turbulent flow between two concentric cylinders, oriented either axially or azimuthally. The axial configuration corresponds to a concentric annulus, where curvature is transverse to the flow, while the azimuthal configuration represents a curved channel with longitudinal curvature. Using direct numerical simulations, we examine the effects of both types of curvature on turbulence, varying the inner radius from $r_i=0.025\delta$ to $r_i=95.5\delta$, where $\delta$ is the gap width. The bulk Reynolds number, based on bulk velocity and $\delta$, is set at $R_b\approx 5000$, ensuring fully turbulent conditions. Our results show that transverse curvature, although breaking the symmetry of axial flows, induces limited changes in the flow structure, leading to an increase in friction at the inner wall. In contrast, longitudinal curvature has a significant impact on the structure and statistics of azimuthal flows. For mild to moderate longitudinal curvatures ($r_i\gt 1.5\delta$), the convex wall stabilises the flow, reducing turbulence intensity, wall friction and turbulent kinetic energy (TKE) production. For extreme longitudinal curvatures ($r_i\leqslant 0.25\delta$), spanwise-coherent flow structures develop near the inner wall, leading to a complete redistribution of the TKE budget: production becomes negligible near the inner wall, while pressure–velocity correlations increase substantially. As a result, the mean TKE peaks near the inner wall, thereby weakening the stabilising effect of convex curvature.

Dispersion in spatio-temporal random flows is dominated by the competition between spatial and temporal velocity resets along particle paths. This competition admits a range of normal and anomalous dispersion behaviours characterised by the Kubo number, which compares the relative strength of spatial and temporal velocity resets. To shed light on these behaviours, we develop a Lagrangian stochastic approach for particle motion in spatio-temporally fluctuating flow fields. For space–time separable flows, particle motion is mapped onto a continuous time random walk (CTRW) for steady flow in warped time, which enables the upscaling and prediction of the large-scale dispersion behaviour. For non-separable flows, we measure Lagrangian velocities in terms of a new sampling variable, the average number of velocity transitions (both temporal and spatial) along pathlines, which renders the velocity series Markovian. Based on this, we derive a Lagrangian stochastic model that represents particle motion as a coupled space–time random walk, that is, a CTRW for which the space and time increments are intrinsically coupled. This approach sheds light on the fundamental mechanisms of particle motion in space–time variable flows, and allows for its systematic quantification. Furthermore, these results indicate that alternative strategies for the analysis of Lagrangian velocity data using new sampling variables may facilitate the identification of (hidden) Markov models, and enable the development of reduced-order models for otherwise complex particle dynamics.

The transport of droplets in microfluidic channels is strongly dominated by interfacial properties, which makes it a relevant tool for understanding the mechanisms associated with the presence of more or less soluble surfactants. In this paper, we show that the mobility of an oil droplet pushed by an aqueous carrier phase in a Hele-Shaw cell qualitatively depends on the nature of the surfactants: the drop velocity is an increasing function of the drop radius for highly soluble surfactants, whereas it is a decreasing function for poorly soluble surfactants. These two different behaviours are experimentally observed by using two families of surfactant with a carbon chain of variable length. We first focus on the second regime, observed here for the first time, and we develop a model which takes into account the flux of surfactants on the whole droplet interface, assuming an incompressible surfactant monolayer. This model leads to a quantitative agreement with the experimental data, without any adjustable parameter. We then propose a model for a stress-free interface, i.e. for highly soluble surfactants. In these two limits, the models become independent on the physico-chemical properties of the surfactants, and should be valid for any surfactant complying with the incompressible or stress-free limit. As such, we provide a theoretical framework with two limits for all the experimental physico-chemical configurations, which constitute the bounds for the droplet mobility for intermediate surfactant solubility.

Direct numerical simulations in a low-curvature viscoelastic turbulent Taylor vortex flow, with Reynolds numbers ranging from 1500 to 8000 and maximum chain extensibility ($L$) from 50 to 200, reveal a maximum drag reduction (MDR) asymptote. Compared with the classical MDR observed in planar wall-bounded shear flows, that is, drag reduction (DR) is $\sim -80\, \%$, this MDR state achieves only moderate levels of DR ($\sim -60\,\%$). This is due to the existence of large-scale structures (LSSs). A careful examination of the flow structures reveals that the polymer–turbulence interaction suppresses small-scale vortices and stabilizes the LSSs. These structural changes in turn lead to a reduction of Reynolds stress, and consequently to a DR flow state. Although Reynolds stress does not vanish as observed in classical MDR states, the small-scale vortices that heavily populate the near-wall region are also almost completely eliminated in this flow state. Concurrently, significant polymer stresses develop as a consequence of the interaction between polymer chains and LSSs that partially offset the magnitude of DR, leading to MDR asymptotes with moderate levels of DR. Moreover, we demonstrate that polymer deformation, i.e. deviation from the equilibrium state, is directly correlated with the LSSs dynamics, while the polymer deformation fluctuation displays a universal property in the MDR state. Hence, it is not surprising that the extent of DR exhibits a non-monotonic dependence on the maximum chain extensibility. Specifically, the variation in $L$ alters the incoherent and coherent angular momentum transport by small- and large-scale flow structures, respectively. To that end, the most DR flow state occurs at a moderate value $L=100$. Overall, this study further supports the universal property of polymer-induced asymptotic states in wall-bounded turbulence and paves the way for mechanistic understanding of drag modification that arises from the interaction of polymers with small- and large-scale flow structures.

Surfactant transport is central to a diverse range of natural phenomena with numerous practical applications in physics and engineering. Surprisingly, this process remains relatively poorly understood at the molecular scale. Here, we use non-equilibrium molecular dynamics (NEMD) simulations to study the spreading of sodium dodecyl sulphate on a thin film of liquid water. The molecular form of the control volume is extended to a coordinate system moving with the liquid–vapour interface to track surfactant spreading. We use this to compare the NEMD results to the continuum description of surfactant transport on an interface. By including the molecular details in the continuum model, we establish that the transport equation preserves substantial accuracy in capturing the underlying physics. Moreover, the relative importance of the different mechanisms involved in the transport process is identified. Consequently, we derive a novel exact molecular equation for surfactant transport along a deforming surface. Close agreement between the two conceptually different approaches, i.e. NEMD simulations and the numerical solution of the continuum equation, is found as measured by the surfactant concentration profiles, and the time dependence of the so-called spreading length. The current study focuses on a relatively simple specific solvent–surfactant system, and the observed agreement with the continuum model may not arise for more complicated industrially relevant surfactants and anti-foaming agents. In such cases, the continuum approach may fail to predict accompanying phase transitions, which can still be captured through the NEMD framework.

We investigate the dynamics and the stability of the incompressible flow past a corrugated dragonfly-inspired airfoil in the two-dimensional (2-D) $\alpha {-}Re$ parameter space, where $\alpha$ is the angle of attack and $Re$ is the Reynolds number. The angle of attack is varied in the range of $-5^{\circ } \leqslant \alpha \leqslant 10^{\circ }$, and $Re$ (based on the free stream velocity and the airfoil chord) is increased up to $Re=6000$. The study relies on linear stability analyses and three-dimensional (3-D) nonlinear direct numerical simulations. For all $\alpha$, the primary instability consists of a Hopf bifurcation towards a periodic regime. The linear stability analysis reveals that two distinct modes drive the flow bifurcation for positive and negative $\alpha$, being characterised by a different frequency and a distinct triggering mechanism. The critical $Re$ decreases as $|\alpha |$ increases, and scales as a power law for large positive/negative $\alpha$. At intermediate $Re$, different limit cycles arise depending on $\alpha$, each one characterised by a distinctive vortex interaction, leading thus to secondary instabilities of different nature. For intermediate positive/negative $\alpha$, vortices are shed from both the top/bottom leading- and trailing-edge shear layers, and the two phenomena are frequency locked. By means of Floquet stability analysis, we show that the secondary instability consists of a 2-D subharmonic bifurcation for large negative $\alpha$, of a 2-D Neimark–Sacker bifurcation for small negative $\alpha$, of a 3-D pitchfork bifurcation for small positive $\alpha$ and of a 3-D subharmonic bifurcation for large positive $\alpha$. The aerodynamic performance of the dragonfly-inspired airfoil is discussed in relation to the different flow regimes emerging in the $\alpha {-}Re$ space of parameters.

We investigate the effects of bottom roughness on bottom boundary-layer (BBL) instability beneath internal solitary waves (ISWs) of depression. Applying both two-dimensional (2-D) numerical simulations and linear stability theory, an extensive parametric study explores the effect of the Reynolds number, pressure gradient, roughness (periodic bump) height $h_b$ and roughness wavelength $\lambda _b$ on BBL instability. The simulations show that small $h_b$, comparable to that of laboratory-flume materials ($\sim$100 times less than the thickness of the viscous sublayer $\delta _v$), can destabilize the BBL and trigger vortex shedding at critical Reynolds numbers much lower than what occurs for numerically smooth surfaces. We identify two mechanisms of vortex shedding, depending on $h_b/\delta _v$. For $h_b/\delta _v \gtrapprox 1$, vortices are forced directly by local flow separation in the lee of each bump. Conversely, for $h_b/\delta _v \lessapprox 10^{-1}$ the roughness seeds perturbations in the BBL, which are amplified by the BBL flow. Roughness wavelengths close to those associated with the most unstable BBL mode, as predicted by linear instability theory, are preferentially amplified. This resonant amplification nature of the BBL flow, beneath ISWs, is consistent with what occurs in a BBL driven by surface solitary waves and by periodic monochromatic waves. Using the $N$-factor method for Tollmien–Schlichting waves, we propose an analogy between the roughness height and seed noise required to trigger instability. Including surface roughness, or more generally an appropriate level of seed noise, reconciles the discrepancies between the vortex-shedding threshold observed in the laboratory versus that predicted by otherwise smooth-bottomed 2-D spectral simulations.

We report an anomalous capillary phenomenon that reverses typical capillary trapping via nanoparticle suspension and leads to a counterintuitive self-removal of non-aqueous fluid from dead-end structures under weakly hydrophilic conditions. Fluid interfacial energy drives the trapped liquid out by multiscale surfaces: the nanoscopic structure formed by nanoparticle adsorption transfers the molecular-level adsorption film to hydrodynamic film by capillary condensation, and maintains its robust connectivity, then the capillary pressure gradient in the dead-end structures drives trapped fluid motion out of the pore continuously. The developed mathematical models agree well with the measured evolution dynamics of the released fluid. This reversing capillary trapping phenomenon via nanoparticle suspension can be a general event in a random porous medium and could dramatically increase displacement efficiency. Our findings have implications for manipulating capillary pressure gradient direction via nanoparticle suspensions to trap or release the trapped fluid from complex geometries, especially for site-specific delivery, self-cleaning, or self-recover systems.

We propose a hybrid numerical model for the collective motion of fish groups, which integrates an agent-based model with a computational fluid dynamics (CFD)-based model. In the agent-based model, the fish group is represented by self-propelled particles (SPPs), incorporating social forces with local interactive rules. The CFD-based model treats the fish body with an undulated filament that responds to the hydrodynamic forces imposed by the surrounding fluid flow. These two models are coupled using a central pattern generator controller. We test this hybrid model with groups of 30–50 individuals. The results show that the group exhibits various collective behaviours, including tight schooling, sparse schooling and milling patterns, by adjusting the coefficients in the SPP model. Due to the hydrodynamic interactions, particularly with the obstacle avoidance model, both the individuals’ and the group’s mean speed fluctuate, differentiating it from traditional SPP models that typically consider volumeless particles. More interestingly, our findings indicate that fish benefit from collective motion in terms of energetic consumption in both schooling and milling patterns. It is important to note that the swimming fish are actuated using a very simple mechanism without any optimisation strategy. An additional study investigates the effects of the Reynolds number, demonstrating the capability of the current hybrid model to account for fish groups of varying body lengths or swimming speeds. Future applications of this model are promising, offering potential insights into the energetic advantages of collective motion in large-scale fish groups.

A purely elastic linear instability was recently reported for viscoelastic plane Poiseuille flow in the limit of ultra-dilute (solvent to solution viscosity ratio $\beta \gt 0.99$), highly elastic (Weissenberg number $W \sim 1000$) polymer solutions, within the framework of the Oldroyd-B model (Khalid et al., Phys. Rev. Lett., vol. 127, 2021, pp. 134–502). This is the first instance of a purely elastic instability in a strictly rectilinear shearing flow, with the phase speed of the unstable ‘centre mode’ being close to the base-state maximum velocity at the channel centreline. Subsequently, Buza, Page and Kerswell (J. Fluid Mech., vol. 940, 2022, A11) have shown, using the FENE-P model, that the centre-mode instability persists down to moderate elasticities ($W \sim O (100)$), the reduction in threshold evidently due to the finite extensibility of the polymer molecules. In this work, we augment this latter finding and provide a comprehensive account of the effect of finite extensibility on the centre-mode instability in viscoelastic channel flow, using the FENE-P and FENE-CR models, in both the absence and presence of fluid inertia. In both these models, finite extensibility causes a decrease in the polymer relaxation time at high shear rates, and the resulting weakening of elastic stresses would seem to indicate a stabilising effect. The latter trend has been demonstrated by earlier analyses of hoop-stress-driven instabilities in curvilinear flows, and is indeed borne out for the FENE-CR case, where finite extensibility has a largely stabilising influence on the centre-mode instability. In stark contrast, for the FENE-P model, finite extensibility plays a dual role – a stabilising one at lower values of the elasticity number $E$, but, surprisingly, a destabilising one at higher $E$ values. Further, the centre-mode instability is predicted over a significantly larger domain of the $Re$–$E$–$\beta$ parameter space, compared to the Oldroyd-B model, making it more amenable to experimental observations.

In soft porous media, deformation drives solute transport via the intrinsic coupling between flow of the fluid and rearrangement of the pore structure. Solute transport driven by periodic loading, in particular, can be of great relevance in applications including the geomechanics of contaminants in the subsurface and the biomechanics of nutrient transport in living tissues and scaffolds for tissue engineering. However, the basic features of this process have not previously been systematically investigated. Here, we fill this hole in the context of a one-dimensional model problem. We do so by expanding the results from a companion study, in which we explored the poromechanics of periodic deformations, by introducing and analysing the impact of the resulting fluid and solid motion on solute transport. We first characterise the independent roles of the three main mechanisms of solute transport in porous media – advection, molecular diffusion and hydrodynamic dispersion – by examining their impacts on the solute concentration profile during one loading cycle. We next explore the impact of the transport parameters, showing how these alter the relative importance of diffusion and dispersion. We then explore the loading parameters by considering a range of loading periods – from slow to fast, relative to the poroelastic time scale – and amplitudes – from infinitesimal to large. We show that solute spreading over several loading cycles increases monotonically with amplitude, but is maximised for intermediate periods because of the increasing poromechanical localisation of the flow and deformation near the permeable boundary as the period decreases.

Turbidity currents, which are stratified, sediment-laden bottom flows in the ocean or lakes, can run out for hundreds or thousands of kilometres in submarine channels without losing their stratified structure. Here, we derive a layer-averaged, two-layer model for turbidity currents, specifically designed to capture long-runout. A number of previous models have captured runout of only tens of kilometres, beyond which thickening of the flows becomes excessive, and the models without a lateral overspill mechanism fail. In our framework, a lower layer containing nearly all the sediment is a faster, gravity-driven flow that propels an upper layer, where sediment concentration is nearly zero. The thickness of the lower layer is controlled by competition between interfacial water entrainment due to turbulent mixing and water detrainment due to sediment settling at the interface. The detrainment mechanism, first identified in experiments, is the key feature that prevents excessive thickening of the lower layer and allows long-runout. Under normal flow conditions, we obtain an exact solution to the two-layer formulation, revealing a constant velocity and a constant thickening rate in each of the two layers. Numerical simulations applied

to gradually varied flows on both constant and exponentially declining bed slopes, with boundary conditions mimicking field observations, show that the predicted lower layer thickness after 200 km flow propagation compares with observed submarine channel depths, whereas previous models overestimate this thickness three- to fourfold. This formulation opens new avenues for modelling the fluid mechanics and morphodynamics of long-runout turbidity currents in the submarine setting.

In this study, we tackle the challenge of inferring the initial conditions of a Rayleigh–Taylor mixing zone for modelling purposes by analysing zero-dimensional (0-D) turbulent quantities measured at an unspecified time. This approach assesses the extent to which 0-D observations retain the memory of the flow, evaluating their effectiveness in determining initial conditions and, consequently, in predicting the flow’s evolution. To this end, we generated a comprehensive dataset of direct numerical simulations, focusing on miscible fluids with low density contrasts. The initial interface deformations in these simulations are characterised by an annular spectrum parametrised by four non-dimensional numbers. To study the sensitivity of 0-D turbulent quantities to initial perturbation distributions, we developed a surrogate model using a physics-informed neural network (PINN). This model enables computation of the Sobol indices for the turbulent quantities, disentangling the effects of the initial parameters on the growth of the mixing layer. Within a Bayesian framework, we employ a Markov chain Monte Carlo (MCMC) method to determine the posterior distributions of initial conditions and time, given various state variables. This analysis sheds light on inertial and diffusive trajectories, as well as the progressive loss of initial conditions memory during the transition to turbulence. Furthermore, it identifies which turbulent quantities serve as better predictors of Rayleigh–Taylor mixing zone dynamics by more effectively retaining the memory of the flow. By inferring initial conditions and forward propagating the maximum a posteriori (MAP) estimate, we propose a strategy for modelling the Rayleigh–Taylor transition to turbulence.

When a drop impacts a solid substrate or a thin liquid film, a thin gas disc is entrapped due to surface tension, the gas disc retracting into one or several bubbles. While the evolution of the gas disc for impact on solid substrate or film of the same fluid as the drop has been largely studied, little is known on how it varies when the liquid of the film is different from that of the drop. We study numerically the latter unexplored area, focusing on the contact between the drop and the film, leading to the formation of an air bubble. The volume of fluid method was adapted to three fluids in the framework of the Basilisk solver. The numerical simulations show that the deformation of the liquid film due to air cushioning plays a crucial role in bubble entrapment. A new model for the contact time and the entrapment geometry was deduced from the case of the impact on a solid substrate. This was done by considering the deformation of the thin immiscible liquid layer during impact depending mainly on its thickness and viscosity. The lubrication of the gas layer was found to be the major effect governing bubble entrapment. However, the film viscosity was also identified as having a critical role in bubble formation and evolution; the magnitude of its influence was also quantified.

The study examines supersonic square jets in a twin nozzle configuration with the aim of identifying and characterising emergent instability modes during overexpanded operation. Unlike screeching rectangular jets that undergo strong fluctuations normal to the wider jet dimension, the equilateral nature of the exit geometry in square nozzles leads to multiple instability states dictated by shock–turbulence interactions and nozzle operating conditions. Furthermore, strong coupling modes between the jets were identified that led to either phase locked or out of phase interactions of the inner shear layers. Results from experimental studies were examined using spatial and temporal decomposition techniques based on spectral methods to identify the resultants from triadic shock–turbulence interactions. The primary instability mode across both operating conditions were driven by optimal interactions while the harmonics were found to be associated with the suboptimal shock–turbulence interactions.

We analyse the steady viscoelastic fluid flow in slowly varying contracting channels of arbitrary shape and present a theory based on the lubrication approximation for calculating the flow rate–pressure drop relation at low and high Deborah ($De$) numbers. Unlike most prior theoretical studies leveraging the Oldroyd-B model, we describe the fluid viscoelasticity using a FENE-CR model and examine how the polymer chains’ finite extensibility impacts the pressure drop. We employ the low-Deborah-number lubrication analysis to provide analytical expressions for the pressure drop up to $O(De^4)$. We further consider the ultra-dilute limit and exploit a one-way coupling between the parabolic velocity and elastic stresses to calculate the pressure drop of the FENE-CR fluid for arbitrary values of the Deborah number. Such an approach allows us to elucidate elastic stress contributions governing the pressure drop variations and the effect of finite extensibility for all $De$. We validate our theoretical predictions with two-dimensional numerical simulations and find excellent agreement. We show that, at low Deborah numbers, the pressure drop of the FENE-CR fluid monotonically decreases with $De$, similar to the previous results for the Oldroyd-B and FENE-P fluids. However, at high Deborah numbers, in contrast to a linear decrease for the Oldroyd-B fluid, the pressure drop of the FENE-CR fluid exhibits a non-monotonic variation due to finite extensibility, first decreasing and then increasing with $De$. Nevertheless, even at sufficiently high Deborah numbers, the pressure drop of the FENE-CR fluid in the ultra-dilute and lubrication limits is lower than the corresponding Newtonian pressure drop.