Anisotropic, finite-sized particles, common in environmental and industrial flows, exhibit complex dynamics distinct from those of small, spherical particles. Their shape introduces orientation-dependent forces, and their finite size affects how they experience the flow field. While the effects of particle inertia are known to cause preferential sampling, in this study we consider whether preferential sampling can arise due to finite size and shape alone using inertialess rods and slender-body theory. First, we show analytically that preferential sampling can only occur in this limit given the presence of three ingredients: finite particle size, anisotropic particle shape, and a nonlinear flow field. Next, to demonstrate this effect, we simulate rods in a steady two-dimensional cellular flow (the Taylor-Green vortex flow). By analyzing the rod trajectories, we find that rods do indeed preferentially sample areas of both high and low vorticity, corresponding to the fixed points of the flow: the high-vorticity cell centers and the low-vorticity saddle points in the corners of each cell. This preferential sampling increases with increasing rod length. We also analyze the same data with respect to the flow's nonlinearity (i.e., curvature) to find that the rods preferentially sample the linear regions of the flow and undersample the nonlinear regions. Finally, we analyze the nonlinear dynamics of this system, showing that chaotic trajectories appear as the rod length increases and that these chaotic regions in the flow also tend to overlap with higher flow nonlinearity. Overall, we show how finite size and anisotropic shape alone can cause particles to preferentially sample a flow field and that this preferential sampling is highly linked to the flow's nonlinearity.

The multifractal scaling of velocity increments at small scales poses significant challenges to turbulence theories based on structure functions of velocity increments. Velocity circulation has recently emerged as a physically pertinent quantity with simpler inertial-range statistics. It therefore offers a potential route to simplified descriptions of turbulent intermittency. Here, we assess the applicability of circulation-based scaling theories in free-surface turbulence using planar particle image velocimetry below a quasiflat interface. We show that the circulation around closed contours obeys the area rule in this flow. When all loop side lengths lie within the same scaling range, the circulation statistics depend primarily on the enclosed area rather than on loop shape or aspect ratio. For figure-eight contours, the relevant area is the scalar enclosed area rather than the signed algebraic area. We further find that inertial-range circulation moments exhibit bifractal scaling. Low-order moments are consistent with space-filling behavior, whereas high-order moments follow a monofractal branch with an effective dimension close to 2.2. These results indicate that circulation provides a compact description of intermittency in free-surface turbulence, despite the mixed two- and three-dimensional character of the flow.

Small-scale intermittency is a defining feature of fully developed fluid turbulence, marked by rare and extreme fluctuations of velocity increments and gradients that defy mean-field descriptions. Existing multifractal descriptions of intermittency focus primarily on longitudinal increments and gradients, despite mounting evidence that transverse components exhibit distinct and stronger intermittency. Here, we develop a unified multifractal framework that jointly prescribes longitudinal and transverse velocity increments, and extends to gradients. We derive explicit relations linking inertial-range scaling exponents of structure functions to moments of velocity gradients in dissipation range, which are consistent with known exact third moment relations ($4/5\mathrm{th}$ and $4/15\mathrm{th}$ laws) and dissipation anomaly. Our results reveal that longitudinal gradient scaling is solely prescribed by longitudinal structure functions, as traditionally expected; however, transverse gradient scaling is prescribed by mixed longitudinal-transverse structure functions. Validation with high-resolution direct numerical simulations of isotropic turbulence, at Taylor-scale Reynolds number up to 1300 demonstrates excellent agreement, paving the way for a more complete and predictive description of intermittency faithful to the underlying turbulence dynamics.

The effective viscosity of blood in minute vessels highly depends on the vessel radii because of the presence of red blood cell (RBCs), known as the Fåhræus-Lindqvist effect. A single-file flow pattern forms in the narrowest capillaries, with vessel radii below $5\mu \mathrm{m}$, whereas a double-file flow pattern, in which RBCs flow alternately, emerges in wider vessels. Our interest here is to understand how vessel curvature influences these flow patterns, since curved vessels frequently appear both in vivo bodies and in microfluidic devices. We performed direct numerical simulations of pressure-driven RBC flows in toroidal vessels with constant curvature. Our analysis reveals that the transition from single-file to double-file flow is promoted as the effect of vessel curvature increases. The effective viscosity in the double-file regime is higher than in the single-file regime, because wall shear stress is amplified as the gap between RBCs and the vessel wall is reduced. We also show that internal energy dissipation within RBCs is small in single-file flow due to limited membrane rotation, whereas double-file flow exhibits pronounced energy dissipation by enhanced membrane rotations. In contrast, dissipation in the surrounding fluid and in the entire system shows the opposite trend, since the total dissipation is closely related to the flow rate. We further investigate the effects of RBC swelling, which is associated with diseases such as malaria. Our results indicate swollen RBCs are more likely to maintain a single-file arrangement, and that the flow pattern itself—rather than the swelling ratio—is the dominant factor governing the effective viscosity. These findings provide fundamental insights for the design of microfluidic devices.

Solute mixing in porous media results from the interplay between molecular diffusion and the deformation of fluid parcels, which elongates solute elements and steepens concentration gradients in the pore space. In three-dimensional porous media, fluid deformation is asymptotically governed by exponential stretching, arising from the chaotic stretching and folding of fluid elements. However, early-time deformation may be dominated by shear, induced by no-slip boundary conditions at grain surfaces, leading to linear elongation. Yet, the persistence and relative contributions of linear shear and exponential stretching to fluid deformation and solute mixing remain poorly understood. Here, we address this question using numerical simulations of fluid deformation and mixing in body-centered cubic bead packs, where the rate of chaotic stretching can be varied with the direction of flow, while maintaining the same average shear rate. We quantify the mean deformation dynamics of elementary surfaces as a function of their initial orientation with respect to streamlines, and their consequences on mixing through Lagrangian methods. We show that surface elements with certain initial orientations relative to the streamline direction experience a transient phase of shear-dominated deformation at early times, leading to persistently larger deformation compared to those with initial orientations that yield purely exponential growth. By expressing the deformation components in streamline coordinates, we derive approximate analytical expressions linking the different components of fluid deformation to shear, helicity, and chaotic stretching. We discuss consequences for the mixing of solute sheets and blobs, highlighting generic behaviors as well as differences between these two representations.

Feeding by marine invertebrate larvae depends critically on the low-Reynolds-number fluid mechanics of the feeding currents they generate. Yet the relative importance of gravity-driven versus drag-driven mechanisms in small invertebrate larvae has remained unresolved. Here, high-speed microscale imaging, micro-particle image velocimetry (μPIV), and an analytical axisymmetric Stokes-flow model are combined to quantify the feeding currents of free-swimming veliger larvae of the eastern oyster Crassostrea virginica. Despite their small body size ($<330$µm), the larvae possess a high excess density ($\sim 181\mathrm{kg}/{\mathrm{m}}^3$), shifting the force balance such that gravity-driven feeding currents dominate. μPIV measurements reveal flow fields with first-order (Stokeslet-type) spatial decay, yielding ecologically sufficient body-volume-specific maximal clearance rates of $\sim 2.5\times {10}^6/\mathrm{day}$. These values agree with predictions from the Stokes-flow model, which decomposes the clearance into gravity-driven and drag-driven components and shows that the former consistently outweighs the latter across observed swimming speeds. The model further explains how slow swimming modulates clearance by $\pm 44\%$ relative to hovering. Together, these results demonstrate that C. virginica veligers operate in a gravity-dominated feeding regime similar to that of larger calanoid copepods, and that their feeding strategy is shaped by an interplay of excess weight, swimming kinematics, and ciliary propulsion. This work highlights the fundamental role of gravity in larval feeding mechanics and underscores the need to incorporate excess weight when evaluating form-and-function relationships in marine invertebrate larvae, particularly for shelled bivalve and gastropod larvae with high excess density.

We investigate cavitation of a vortex ring in an impulsively starting jet induced by an in-tube expanding bubble, combining experimental, numerical, and theoretical approaches. As the jet cavitation number decreases, the cavitation evolves from discrete cavitation bubbles in the vortex core to a cavitating vortex ring in which the vapor phase fully envelops the vortex core azimuthally. A simplified one-dimensional model that approximates the in-tube bubble as a piston captures the time-dependent velocity of the starting jet, which features a remarkably high initial acceleration. This velocity history is then used as the inlet condition for nozzle-flow simulations, enabling us to address the following two scientific questions: (i) why vortex cavitation occurs at an exceptionally high cavitation number and (ii) why the incipient cavitation number increases with nozzle radius, known as the size-scale effect. To clarify the relationship between flow dynamics and cavitation, we analyze the vorticity flux and identify a previously underexplored contribution—the over-pressure component. This component, which strongly depends on the nozzle radius, dominates vortex ring circulation when the initial nozzle flow experiences an intense and brief acceleration. Notably, it may outweigh the shearing component predicted by conventional slug models. The impulsive nozzle flow provides a plausible mechanism for the unusually high incipient cavitation number and the observed size-scale effect. These results shed new light on the mechanisms governing cavitation onset in submerged jets and offer guidance for promoting or suppressing cavitation in practical applications.

We investigate the effects of horizontal oscillations on the drafting-kissing-tumbling (DKT) dynamics of two monodisperse spherical particles settling under gravity in a viscous fluid. Applying particle-resolved direct numerical simulations, we systematically vary the oscillation frequency and amplitude to assess their impact on the behavior of individual particles, their mutual interaction, and the orientation of the particle arrangement. The results demonstrate that the oscillatory effects on DKT become significant only when the particle Reynolds number ${\text{Re}}_p$, defined as the ratio of oscillation-induced inertial to viscous forces, exceeds unity. In this regime, oscillations alter the temporal characteristics of the DKT process with moderate amplitudes tending to prolong and larger amplitudes to reduce the kissing phase. Moreover, oscillations affect particle reorientation. At low ${\text{Re}}_p$, the particles maintain their initial orientation throughout the interaction, whereas an increasing ${\text{Re}}_p$ promotes a preferential alignment perpendicular to the direction of oscillation. We explain these findings by analyzing the oscillation-induced pressure fields surrounding the individual particles, which develops increasingly pronounced lateral anisotropy with increasing ${\text{Re}}_p$. The corresponding lateral hydrodynamic forcing likewise becomes increasingly anisotropic, providing a consistent physical basis for the observed modification of particle interactions and reorientation. These findings provide a physical framework for understanding how horizontal oscillations govern binary particle-particle interactions and orientation during gravitational settling.

A mathematical model for the evolution of, and deposition from, a thin particle-laden droplet on an infinitely thick, isotropic, fully flooded, porous substrate with interconnected pores undergoing simultaneous evaporation and imbibition is formulated and analyzed. In particular, analytical expressions for the evolution of the droplet, as well as for the flow within the droplet and the substrate, and for the transport and deposition onto the substrate of the particles are obtained for droplets evolving in four different modes. While the physical mechanisms driving evaporation (namely the diffusion-driven flow of vapor in the atmosphere away from the free surface of the droplet) and imbibition (namely the pressure-driven flow of liquid in the substrate away from the base of the droplet) are rather different, perhaps rather unexpectedly, it is found that there are a number of qualitative and quantitative similarities as well as differences in the resulting behavior of the droplet as it loses mass to its environment. For example, it is shown that a droplet undergoing pure imbibition in the constant contact radius mode never completely imbibes (i.e., it has an infinitely long lifetime), but if it evaporates (either with or without imbibition also occurring), then it has a finite lifetime, and increasing the strength of evaporation and/or imbibition shortens its lifetime. It is also shown that in the regime in which diffusion of particles is faster than axial advection but slower than radial advection of particles, the final deposit of particles left behind on the substrate after the droplet has completely evaporated and/or imbibed is independent of both the nature and the strength of the physical mechanism(s) driving the mass loss from the droplet. Not only are these results of theoretical interest, but they are also relevant to a wide variety of practical applications, including ink-jet printing, DNA chip manufacturing, and disease diagnostics, that would benefit from an improved ability to predict and/or control the final deposit pattern from a droplet undergoing simultaneous evaporation and imbibition.

We investigate the thermocapillary and solutocapillary instabilities in a self-rewetting liquid film that has a nonmonotonic variation of surface tension with temperature containing an insoluble surfactant at the interface. The dynamics of such a thin liquid film flowing axisymmetrically on a heated vertical cylinder is examined by deriving a set of coupled evolution equations for the film thickness and the surfactant concentration under the lubrication approximation. We perform a linear stability analysis of the steady film solution with a uniform surfactant concentration at the interface. The effects of solutocapillarity and quadratic thermocapillarity are analyzed in depth using $T_m$, the temperature at which surface tension is nearly minimum as the cutoff point, and $T_{0i,s}$, the interface temperature of the steady film. Beyond a critical value of the solutal Marangoni number ${\text{Ma}}_{\text{s}}$, the dispersion curve, featuring two extrema exhibits midwave instability that is characterized by two cutoff wavenumbers. The conventional thermal Marangoni effect switches to the anomalous/reverse Marangoni effect as $T_{0i,s}$ takes values greater than $T_m$. We perform a detailed parametric study to scrutinize the effect of thermal Marangoni number ${\text{Ma}}_{\text{T}}$, solutal Marangoni number ${\text{Ma}}_{\text{s}}$, Biot number Bi, Bond number Bo, and surface Péclet number ${\text{Pe}}_{\text{s}}$ on the stability of the base state. The Orr-Sommerfeld problem is formulated and solved numerically under the creeping flow condition for arbitrary wavenumbers. A comparison of the neutral curves obtained from the Orr-Sommerfeld problem and the lubrication model reveal excellent agreement for small wavenumber perturbations. The nonlinear simulations of the coupled set of partial differential equations (PDEs) for the interface and surfactant concentration demonstrate the modulation of interface deformation in the presence of temperature gradients and concentration gradients in the regular Marangoni regime, whereas such deformations are suppressed in the anomalous Marangoni regime.

We consider the relationship between Eulerian modal decompositions and Lagrangian coherent structures. The model sensitivity framework developed by Kaszás and Haller [Chaos 30, 113144 (2020)] is used to express data-driven modal representations of fluid flow in a Lagrangian space. The method, based on the computation of the finite-time Lyapunov exponent, computes the amplitude perturbations experienced by fluid particles due to specific modal components of the flow. Demonstrations of the method are presented for both periodic and turbulent flows, including direct numerical simulation (DNS) data of the classical cylinder wake flow, experimental data from the wake past an oscillating foil, and DNS data of a turbulent channel flow. This method provides a way to understand how Eulerian mode structures interact dynamically with features of the Lagrangian coherent structure across scales, offering additional physical insight into modal decompositions.

Turbulent flows can exhibit extreme events, which are characterized by sudden bursts in the system observables. These events pose significant challenges for prediction and control owing to their intermittent, high-dimensional, and strongly nonlinear nature. In this work we present a predominantly data-driven control framework for the suppression of extreme events in turbulent flows, leveraging reduced-order modeling for nonlinear compression of high-dimensional flow fields and a data-driven clustering algorithm for the identification of precursors to extreme events. A control law is defined in the low-dimensional latent space and is optimized to efficiently mitigate extreme events through actuation values associated with the preidentified clusters. To address issues in the latent space structure that arise from invariance transformations present in many nonlinear systems, leading to an inflated latent space, we employ symmetry-aware autoencoders to establish a more structured and compact latent space. By analyzing high-fidelity simulations of a canonical chaotic flow (Kolmogorov flow), we demonstrate how the resulting closed-loop dynamics exhibit a substantial reduction (up to ca. 99.4%) in the frequency and intensity of extreme events. The framework exhibits scalability to a more chaotic flow regime characterized by higher Reynolds number without encountering numerical instabilities, achieving a 96.9% reduction in the occurrence of extreme events. The ability to incorporate controller limitations, including actuator latency relevant to practical applications of the framework, has also been demonstrated. This study highlights the efficacy of data-driven methods that require little to no prior knowledge of the underlying system dynamics to achieve effective flow control, thereby providing a pathway for general real-time suppression of extreme events in turbulent flows.

Gas transport in nanoporous media is fundamentally governed by confinement effects and surface roughness, which challenge the applicability of conventional continuum theories. In this study, a molecular-based apparent permeability model for nanoporous flows is established by incorporating confinement effects, surface roughness, and wettability using a bottom-up approach. The roughness effect is incorporated through a correction function calibrated from molecular simulations, which characterizes the suppression of slip behavior induced by surface roughness. A generalized hydrodynamic model is employed to describe pore-scale transport behavior in smooth nanochannels, from which a semi-empirical scaling law for slip length is developed as a function of fluid-solid interactions, pore size, finite molecular size, and Knudsen number. The resulting slip-length scaling law is then incorporated into a slip-corrected permeability relation to obtain the apparent permeability model for rough nanoporous media. The new model is validated through systematic comparisons with molecular simulation results, experimental data, and existing theoretical models, demonstrating its capability to accurately capture nanoscale transport behavior under varying surface roughness conditions. Parametric analyses further show that the apparent permeability increases with hydrophobicity while decreases with surface roughness, both of which have been largely overlooked in previous apparent permeability models. Overall, the proposed model provides a physically grounded framework linking molecular-scale interactions with fluid transport in nanoporous media.

Fractal flocs are highly porous aggregates whose hydrodynamic response differs fundamentally from that of solid particles of comparable size and shape. In particular, the fluid trapped within the porous structure alters the effective added inertia and rotational coupling experienced by the aggregate, thereby influencing its translational and rotational dynamics in unsteady flows. In this work we investigate the added-mass and added-moment-of-inertia tensors of fractal flocs generated over a broad range of floc sizes and fractal dimensions using a binary-superposition-based computational framework. Large statistical ensembles are used to characterize both the mean behavior and the morphology-induced variability of the inertia tensors. The results show that the principal added-mass and added-moment-of-inertia coefficients exhibit systematic scaling with floc size and fractal morphology while displaying substantial deviations from equivalent solid ellipsoids. Despite significant geometric irregularity at the level of individual particles, the variability among different realizations decreases with increasing aggregate size, revealing an emergent self-averaging behavior for large flocs. Due to their highly anisotropic and porous structure, flocs with smaller fractal dimension exhibit substantially larger added-moment-of-inertia coefficients, which could influence their orientation dynamics in accelerating or turbulent environments. Based on these observations, reduced-order stochastic representations are developed to model the tensor statistics using a small number of physically meaningful parameters. The statistical framework developed here provides a foundation for incorporating morphology-dependent added-inertia effects into large-scale simulations of particle-laden and multiphase flows, including applications involving sediment transport, marine aggregates, aerosols, and industrial particulate systems.

High temporal resolution is essential for resolving the unsteady dynamics of three-dimensional velocity fields in fluid mechanics. However, achieving temporal super-resolution (SR) at large temporal intervals remains challenging for complex turbulent flows. To address this limitation, a residual network that integrates attention mechanisms with convolutional long short-term memory (ConvLSTM) is proposed to reconstruct three-dimensional velocity fields at intermediate time instants between two input flow fields under various temporal SR factors, corresponding to progressively larger temporal separations between the input flow fields. The model is trained and evaluated using direct numerical simulation data of turbulent channel flows spanning different friction Reynolds numbers. The reconstruction accuracy is found to be strongly correlated with the ratio between the temporal separation of the two input snapshots and the Kolmogorov timescale of the flow. A dimensionless temporal parameter ${\mathrm{\Pi }}_t$ is introduced, under which the reconstruction error exhibits an approximately linear dependence. A threshold of ${\mathrm{\Pi }}_t\le 4$ is identified for the proposed model to achieve accurate reconstruction of Reynolds stresses and coherent vortical structures. In contrast, a convolutional neural network based model attains comparable reconstruction accuracy only for ${\mathrm{\Pi }}_t\le 0.8$. To assess applicability to experimental measurements, the noise robustness of the proposed model is examined under different noise intensities and spectral characteristics. The model effectively suppresses noise and extracts physically meaningful flow structures, benefiting from spatial feature extraction enabled by attention mechanisms and temporal dependency modeling provided by ConvLSTM. Nevertheless, the reconstruction error increases monotonically with noise intensity due to the progressive loss of recoverable flow information irreversibly obscured by noise.

Film flow on vertical fibers has been studied for the past 30 years. However, it is only recently that the influence of the nozzle on the film and bead dynamics has been demonstrated. Consequently, similar to the regimes observed with a faucet, both a dripping and a jetting regime have been reported for film flow on fibers. This study investigates the dynamics of thin liquid films on vertical fibers near the nozzle outlet using both experimental and numerical approaches, drawing insights from dripping faucet studies. This allows for a clear definition of the recently observed dripping and jetting regimes, with a particular focus on gaining a deeper understanding of the flow and formation of beads in the dripping regime, as well as the transition to jetting.

We investigate the bursting dynamics of dense granular films punctured at imposed controlled liquid pressure. Using high-speed imaging, we reveal a two-stage opening process. At early times, the hole expands in a liquidlike manner with a nearly circular shape and a constant velocity, though significantly lower than the Taylor-Culick speed. This velocity is modeled by a balance between surface tension and effective surface viscosity, which depends on particle size and liquid depression. We show that as liquid pressure decreases, the initial opening slows down progressively until it ultimately reaches a jamming state. As the opening progresses, the film undergoes a transition to a solidlike regime, where faceting occurs due to fracture propagation within the granular skeleton. We show that the characteristic fracture size follows a $t^{3/4}$ scaling, consistent with a balance between surface energy release and viscous dissipation induced by liquid redistribution within the 2D granular network. These findings advance our understanding of the mechanics of granular films and their transition between fluidlike and solidlike behaviors.

This study examines the effective thermal diffusivity of sheared particle-fluid suspensions with non-Brownian neutrally buoyant particles in a thin gap Taylor-Couette cell. A steady canonical shear flow is generated with Taylor instabilities suppressed by outer cylinder rotation. Spherical acrylic particles of 1.93 mm diameter were used with a density-matched propylene glycol-glycerol solution. Thermal diffusivity of the medium is deduced by applying a thermal pulse to the stationary inner cylinder and monitoring temporal temperature decay on the inner cylinder surface. The study documents the effects of the particle Peclet number $(<1400)$ for four different particle volume fractions of 0.14, 0.22, 0.30, and 0.36. Here the Peclet number is defined as the ratio of timescales of thermal diffusion in the liquid over the particle scale to the timescale of the imposed shear. The applied shear corresponded to particle Reynolds numbers ranging from Stokes flow $({\mathrm{Re}}_p\ll 1)$ to a maximum of 0.85, where weak inertial effects are likely present. At low Peclet number $\mathrm{Pe}<100$, the enhancement due to shear is roughly linear with the Peclet number. At higher $\mathrm{Pe}$ $(100<\mathrm{Pe}<700)$, the enhancement displays power-law behavior with an exponent of 1/2, which has been predicted by some models for concentrated suspensions, but is also consistent with the emergence of a particle-free fluid layer near one wall. At still higher $\mathrm{Pe}$, the 1/2 power-law behavior is retained for all but the highest volume fractions, with $\varphi =0.35$ exhibiting a 1/11 power-law behavior, which has been previously predicted, albeit for dilute suspensions, for the high-$\mathrm{Pe}$ limit. The thermal diffusivity measurements are complemented by limited measurements of particle velocity through particle tracking, which suggest that particle motion does not contribute significantly to the observed enhancement at high Peclet numbers.

This study introduces a vortex gust generation method for isolated vortices impacting a downstream airfoil that is applicable to both numerical simulations and experiments. The vortex gust is generated by a symmetric airfoil undergoing a rapid pitching maneuver during a prescribed heaving motion. The resulting vortices propagate along trajectories nearly parallel to the incoming flow, while the associated wake extends obliquely from the vortex core. Despite differences in Reynolds number, rapid pitching duration, and detailed vortex structure between simulations and experiments, consistent trends are observed in how the vortex rotation orientation, strength, and position vary with the prescribed motion parameters. Analysis of the lift response of the downstream airfoil shows that the aerodynamic influence associated with the wake does not persist over extended timescales. These results demonstrate that the proposed method enables the controlled generation of vortex gusts with prescribed characteristics, providing a flexible approach for systematic studies of vortex-airfoil interaction.

The swirling flow in a Francis type hydropower turbine is known to be susceptible to the formation of a large helical structure, commonly referred to as a vortex rope. This vortex rope can be interpreted as an unstable mode associated with vortex breakdown. This perspective is adopted here in a simplified laminar flow setting. The helical vortex rope mode is shown to bifurcate supercritically from an axisymmetric base flow in a Hopf bifurcation within a turbine draft tube. When wall friction effects are neglected, a large recirculation region at the axis can form and a range of subcritical solutions is identified for a flow regime corresponding to partial load of the turbine. The existence of these subcritical solutions promotes the emergence of a hysteresis loop. We further describe a regular dynamics of a formation of recirculation bubble at the axis and its destruction due to the emergence of a helical vortex rope at its periphery. Increasing the axial flow discharge towards the regime corresponding to nominal turbine load leads to an unfolding of the steady solutions branch in a transcritical bifurcation. This bifurcation takes place at finite Reynolds number and complements existing evidence of transcritical bifurcation of the swirling jet flows, previously reported in the inviscid limit.