The laminar jet of a Newtonian liquid emanating from a long circular nozzle into a gaseous environment is studied. Provided that gravity effects can be neglected, the flow depends solely on the Reynolds number ($\text{Re}$) and the Capillary number ($\text{Ca}$) based on the nozzle bulk velocity ($U_b$). At high $\text{Re}$, the jet contracts downstream of the nozzle, while at low $\text{Re}$, the jet expands, which is known as the Newtonian die-swell phenomenon. To study the influence of $\text{Re}$ on the jet behavior, an integral momentum balance is theoretically derived for the entire flow, both in the nozzle and in the jet. A useful decomposition is made of the viscous force in the integral momentum balance into a contribution from the excess wall pressure drop ($P_{\text{exc}}$) and a contribution from the excess integral wall shear stress ($T_{\text{exc}}$) relative to, respectively, the pressure drop and integral wall shear stress for Poiseuille flow. The results of the numerical simulations for $\text{Re}\in [4.2:47.4]$ show that $P_{\text{exc}}$ is larger than zero for $\text{Re}\lesssim 6$, while negative for $\text{Re}\gtrsim 6$. Furthermore, $T_{\text{exc}}$ is always negative and is the dominant contribution to the overall viscous force for $\text{Re}\lesssim 12$. Normalized by the integral wall shear stress for Poiseuille flow, $T_{\text{exc}}$ appears to scale with ${\text{Re}}^{-1/3}$, implying that axial velocity perturbations near the nozzle wall scale with ${\text{Re}}^{-1/3}$ too when normalized by $U_b$. Self-similar behavior is also found for the axial velocity perturbation at the nozzle centerline, which scales with ${\text{Re}}^{-2/3}$ when normalized by $U_b$. The observed power-law scalings in $\text{Re}$ are explained theoretically from the development of a boundary layer along the jet interface, originating from the stick-slip transition at the nozzle lip, and the constant integral mass flux along the jet. The boundary-layer flow is responsible for axial viscous stresses at the nozzle exit plane that result in a perturbation flow within the nozzle over an axial distance on the order of the nozzle radius. The die-swell phenomenon at low $\text{Re}$ is caused by the radially outward flow at the nozzle exit and the accompanying compressive normal viscous stress that pushes the interface outwards till equilibrium is achieved. Self-similar behavior is also found for the perturbation flow far downstream of the nozzle with respect to the plug flow at equilibrium. With increasing $\text{Re}$, the exponential decay rate of the perturbation in the centerline velocity converges toward a prediction first made by Bohr for a related problem [N. Bohr, Philos. Trans. R. Soc. A 209, 281 (1909)], while the exponential decay rate of the perturbation in the jet radius is approximately twice as high. Finally, a discussion is given of the implications of our findings for applications in practice.

This study investigates the effect of compliant walls on the turbulent heat transfer in channel flows over viscous-hyperelastic walls. We perform direct numerical simulations, fully resolving the mutual fluid-structure interactions between the turbulent flow and the compliant walls, varying the wall elasticity and the thermal diffusivity in a fully turbulent condition. We show that the compliant wall leads to an increase not only of the momentum transfer, but also of the heat transfer. Since the compliant wall can dynamically move, in the near-wall region heat flux is mostly transferred via turbulent convection rather than diffusion, as typically found with rigid walls. Thus, the heat flux can be controlled not only by varying the thermal diffusivity, but also by changing the transverse modulus of elasticity which governs the wall-normal velocity fluctuations and consequently the temperature ones. Finally, we show that the physical mechanism controlling these modifications is the sweep and ejection events.

Understanding the isothermal flow features associated with different vortex breakdown (VB) structures is of great significance to the design of swirl-stabilized gas turbine combustors. A recent experimental study [A. Dolai et al., Phys. Fluids 37, 025212 (2025)] has identified three distinct VB structures (bluff-swirl, bubble, and conical) in the isothermal flow field of a twin-swirl combustor by varying the outer-to-inner jet momentum ratio. The present study employs stress-blended eddy simulations to investigate the recirculation zone structure, pressure field, mixing characteristics, and dynamic stability under these conditions using budget analysis, entrainment mass evaluation, and proper orthogonal decomposition (POD), after reproducing the experimentally measured velocity fields with high fidelity. The recirculation zone shape is governed by the formation of a radial pressure gradient, which is conventionally attributed to centripetal acceleration; here we demonstrate that advective and turbulent contributions play a significant role in its development. The entrainment velocity is another crucial parameter that governs the mixing between two coannular streams in the near field region. It is shown that the entrainment velocity is better approximated by the rms components rather than by the inner or outer mean velocities. Interestingly, the level of mixing is found to be independent of the momentum ratio and the swirling direction. The dynamic stability of the flow field is further assessed using POD analysis. A precessing vortex core (PVC) structure is identified rotating at a nearly constant frequency ($\sim 200\mathrm{Hz}$). The PVC frequency is again found to be independent of both the momentum ratio and the swirling direction. However, the rotation direction of the PVC follows the core swirler. Overall, the present study sheds light on the similarities (mixing, dynamic stability) and the differences (recirculation zone structure, pressure gradient formation mechanisms) between the isothermal co- and counterswirling flow fields under varying operating conditions.

The dynamics of retraction of an initially highly slender liquid sheet of an incompressible Newtonian fluid the surface of which is covered with a monolayer of insoluble surfactant and that is surrounded by a passive gas is analyzed in the Stokes limit. In this limit, the sheets retract without the formation of a rim or capillary waves along the liquid-gas (L-G) interface. The dynamics of retraction is studied in situations in which surface rheological effects are absent as well as when they are present. Initially, the cross sections of liquid sheets are rectangular in shape albeit with semicircular caps at both ends, i.e. the cross sections resemble elongated two-dimensional (2D) drops. The 2D drops begin to retract due to the capillary pressure difference between their tips and their rectangularly shaped interiors. The free surface flow within the 2D drop is governed by the continuity and the Stokes equations which determine the flow, i.e. the velocity and pressure fields, and the convection-diffusion equation which governs surfactant transport, and hence surfactant concentration, along the L-G interface. Surface tension is related to surfactant concentration by the Szyskowski equation and the surface rheology is described by the Boussinesq-Scriven constitutive equation which involves surface shear and dilatational viscosities that are both functions of surfactant concentration. The dynamics is governed by five dimensionless groups: initial aspect ratio $L_0$, initial surfactant concentration ${\mathrm{\Gamma }}_0$, surfactant strength parameter $\beta$, surface Peclet number $\mathrm{Pe}$ (the dimensionless ratio of surface convection to diffusion of a surfactant), and reference Boussinesq-Scriven number $B_0$ which measures the relative importance of surface to bulk viscous stresses. Taking advantage of sheet slenderness, a set of one-dimensional (1D) slender-sheet equations is derived using a control volume analysis. The 1D equations are solved numerically by a finite-element-based method and predictions made with the 1D algorithm are shown to accord well with ones in which the 2D free surface flow within the 2D drop is solved numerically without invoking slenderness. The results reveal that the early time retraction dynamics far from the two tips are nearly independent of $\mathrm{Pe}$. Moreover, since the two surfaces of the sheet in this region remain planar for long times after the initiation of retraction, a control volume analysis is used to analytically calculate the maximum film thickness and retraction velocity that are likewise independent of $\mathrm{Pe}$. In the theoretical analysis, advantage is taken of the fact that away from the tips, surfactant concentration and sheet half-thickness are simply functions of time and proportional to one another. The analytical results are expressed as implicit functions of time in terms of the exponential integral function and explicitly if surface tension varies slightly during retraction. It is shown that sheets with $B_0\ne 0$ retract slower than ones with $B_0=0$, which in turn retract slower than ones with clean interfaces. The role of finite inertia is also investigated briefly and it is demonstrated that rim formation is suppressed so long as $\mathrm{Oh}(1+B_0{\mathrm{\Gamma }}_0)\gg L_0$ where $\mathrm{Oh}$ is the Ohnesorge number $[\text{viscous}\text{force}/\sqrt{\text{(inertia)(surface}\text{tension}\text{force)}}]$.

Freeze-thaw cycles can be regularly observed in nature in water and are essential in industry and science. Objects present in the medium will interact with either an advancing solidification front during freezing or a retracting solidification front, i.e., an advancing melting front, during thawing. It is well known that objects show complex behaviors when interacting with the advancing solidification front, but the extent to which they are displaced during the retraction of the solid-liquid interface is less well understood. To study potential hysteresis effects during freeze-thaw cycles, we exploit experimental model systems of oil-in-water emulsions and polystyrene (PS) particle suspensions, in which a water-ice solidification front advances and retracts over an individual immiscible (and deformable) oil droplet or over a solid PS particle. We record several interesting hysteresis effects, resulting in nonzero relative displacements of the objects between freezing and thawing. PS particles tend to migrate further and further away from their initial position, whereas oil droplets tend to return to their starting positions during thawing. We rationalize our experimental findings by comparing them to our prior theoretical model of Meijer et al. [Phys. Rev. Fluids 10, 034002 (2025)], yielding a qualitatively good agreement. Additionally, we look into the reversibility of how the droplet deforms and reshapes throughout one freeze-thaw cycle, which will turn out to be remarkably robust.

The dissolution of trapped bubbles in porous media plays a key role in subsurface gas storage, groundwater ecology, and multiphase chemical reactors. We demonstrate that the classic Epstein-Plesset model for bubble dissolution in open space fails in porous media, because pore-throat geometry (1) decouples the characteristic diffusion distance and effective mass transfer area from the bubble size and (2) distorts concentration field near bubble surface. We propose kinetic dissolution theories with analytical solutions, validated through numerical simulations and microfluidic experiments. Our results provide physical mechanisms for the predictive modeling of bubble dissolution in diverse porous media.

Understanding the shedding mechanism of partial cavitation is of importance for suppression or utilization of cavitation. To explore the condensation front mechanism of partial cavitation, flows in a three-dimensional axisymmetric Venturi are investigated by combining incompressible numerical simulation and a high-speed photography experiment at a cavitation number of 0.37 and a Reynolds number of $8.94\times {10}^4$. Based on the volume of fluid (VOF) multiphase flow model, the large eddy simulation (LES) method, and Sauer-Schnerr cavitation model are used to predict the generation and propagation process of the condensation front in the cavitating flow. The instantaneous flow topology and flow field, temporal evolution of cavity shedding, condensation front characteristics, and interaction between cavitation and vortex are expatiated. It is clearly shown that a condensation front is induced by the collapse of the cavitation cloud at the Venturi downstream and propagates inside the sheet cavity until triggering the “pinch off” of the cavity. Different from the cavitating flow on wedges or hydrofoils, a portion of vapor at the sheet cavity aft shows a negative axial velocity during the shedding process before the sheet cavity is influenced by the condensation front. This may be because of the strong squeezing effect resulting from the high-pressure downstream in the confined space of the Venturi. The condensation front satisfies the one-dimensional Rankine-Hugoniot jump condition most of the time (except for the initial propagation process) and exhibits supersonic characteristics. The predicted condensation front characteristics, such as propagation velocity and pressure rise, are identical to those found in previous studies. The supersonic regions exist on the condensation front and sheet cavity surface and in the vortices. The good agreement of the incompressible simulation results with the experimental results and the minor difference between the results of incompressible and compressible solvers prove that the condensation front is fundamentally different from traditional shock waves in aerodynamics. The density change resulting from evaporation and condensation of cavitation plays an important role in the formation and propagation of the condensation front, instead of compressibility. The findings of this work may provide a new understanding of the shedding mechanism of partial cavitation.

Swimming dolphins generate fully developed turbulence. In the present study, we demonstrate that the hierarchy of multiscale vortices existing in the turbulence is key to understanding how dolphins swim. For this demonstration, we conduct direct numerical simulations of a self-propelled dolphin using a simple model, and examine the role of multiscale vortices in propulsion. By decomposing the turbulent flow into different scales, we elucidate how the dolphin generates the hierarchy of multiscale vortices. Large vortex rings, which are shed due to the oscillation of the caudal fin, produce significant propulsion, while smaller vortices are created through the energy cascade, contributing little to the propulsion. We also demonstrate that the role of these vortices in the propulsion remains robust regardless of the Reynolds number.

Springs can be found in many applications and biological systems, and when these are soft, they easily deform. At small scales, capillarity can induce a force leading to spring deformations when the elastocapillary number is small. We demonstrate through experiments the nontrivial equilibrium-shape liquid droplets adopt in these soft springs, which form an annulus, eruciform, and spherical shapes. When these droplets are set in motion, they display different flow regimes with significant dissipation generated by the internal rotational flow. The static and dynamics of droplets in such a capillary slinky are also used to demonstrate how surface tension can actuate springs by stretching/compression, while providing a way for active flow control in soft springs.

In an early theoretical work published in 1965, Belen'kii and Fradkin [Tr. Fiz. Inst. Akad. Nauk SSSR im. P. N. Lebedeva 29, 207 (1965)] proposed a turbulent diffusivity model for Rayleigh–Taylor (RT) mixing. We review its derivation and present alternative arguments leading to the same final similarity equation. The original work then introduced an approximation that led to a simplified ordinary differential equation (ODE), which was used primarily to derive the important scaling result, $h\sim (lnR)g{t}^2$. Here, we extend the analysis by examining the solutions to both the full similarity ODE and the simplified ODE in detail. It is shown that the full similarity equation captures many now well-known features of non-Boussinesq RT flows, including asymmetric spike and bubble growth and a systematic shift of velocity statistics toward the light-fluid side. Comparisons of the theoretical model with numerical and experimental studies show reasonable agreement in both spatial profiles and growth trends of mixing layer heights. We further show that a global mass correction applied to the simplified solution closely approximates the full solution, highlighting that, to leading order, RT mixing is governed by the competing dynamics between diffusion of $ln\overline{\rho }$ and mass conservation.

Surface-swimming organisms generate a diversity of wave patterns. By analyzing the wakes produced by two juvenile snakes, we show that their surface wave patterns depart significantly from classical predictions for uniform rectilinear motion. Using a simplified model of a slaloming pressure disturbance, we reproduce these wave patterns and demonstrate, both theoretically and experimentally, that they are composed of the spatiotemporal signature of a uniformly translating pulsating source. Our analysis of the mean wave drag reveals in particular that lateral oscillations of the head can reduce the drag when $U>c_{\min }$, where $c_{\min }$ is the minimal phase velocity. In addition, two secondary wave-drag peaks appear at velocities below and above $c_{\min }$, coinciding with the formation of V-shaped wakes from pulsating sources. We show that these transitions result from changes in the resonance conditions of translating pulsating waves, fundamentally redefining the threshold for the formation of unsteady wakes. This key observation enables the construction of a phase diagram of capillary-gravity waves generated by nonuniformly moving bodies.

Metallic coatings are used to enhance the durability of metal surfaces by protecting them from corrosion. These protective layers are typically deposited in a fluid state via a liquid film. Controlling instabilities in the liquid film is crucial to achieving uniform, high-quality coatings. This study explores the possibility of controlling liquid films on a moving substrate using a combination of gas jets and electromagnetic actuators. To model the three-dimensional liquid film, we extend existing integral models to incorporate the effects of electromagnetic actuators. The control strategy was developed within a reinforcement learning framework, in which the proximal policy optimization (PPO) algorithm interacts with the liquid film via pneumatic and electromagnetic actuators to optimize a reward function that accounts for instability-wave amplitude through a trial-and-error process. The PPO identified an optimal control law that reduced interface instabilities via a novel mechanism: gas jets push crests, and electromagnets raise troughs via the Lorentz force.

In this paper, we investigate the breakup dynamics of an inviscid liquid bridge. The volume of the liquid bridge is reduced through perturbative drainage on one side, which leads to spontaneous breakup and forms a satellite droplet. A two-dimensional model is applied to describe the liquid dynamics, and the whole breakup process up to the second pinchoff is considered. Following the previous experimental work conducted by D. Li et al. [Phys. Fluids 34, 084105 (2022)], we focus on the asymmetric behavior and the momentum of satellite droplets. The process can be sequentially divided into a quasistatic stage, an instability stage, and a pinchoff stage. The key parameter is the length-to-radius ratio of the liquid bridge, denoted $L$. For a short liquid bridge with $L\le 4.1$, the profile remains symmetric throughout the process. The pinchoff occurs simultaneously on both sides, leading to a satellite droplet without axial momentum. For a long liquid bridge with $L>4.1$, it becomes asymmetric at the onset of capillary instability, leading to nonsimultaneous pinchoff and nonzero satellite momentum. The transition from symmetric to asymmetric breakup occurs when the dominant perturbation modes of the critical state shift from even-modes to odd-modes. In contrast to the conclusion by Li $etal.$, we argue that the momentum of the satellite droplet stems primarily from the capillary impulses after the flattening moment before the first pinchoff. This work can help eliminate or utilize satellite droplets in practical applications.

Vegetation canopies in turbulent flows often undergo oscillatory motion that modulates drag and reorganizes turbulence near the canopy top. We perform large-eddy simulations of an open-channel flow in which the canopy is modeled as a continuous drag layer using a time-dependent variable-drag model (VDM). The form-drag coefficient is prescribed to vary sinusoidally about its mean value with amplitude $A$ and frequency $f$. We examine how these controls modify the mean flow, the turbulent kinetic energy (TKE) budget, and coherent structures from within the canopy to the outer region. Compared with a steady-drag canopy, temporally varying drag redistributes TKE production, transport, and dissipation: energetic activity shifts from the canopy-top shear layer into the canopy interior, intensifying in-canopy turbulence while slightly reducing the peak production at the canopy top. Increasing $A$ further enhances the TKE production at the canopy top and promotes the associated downward transport of TKE. In contrast, changes in $f$ have only a weak impact on the mean flow but strongly affect the canopy-top velocity statistics. An intermediate $f$ produces velocity PDFs closest to Gaussian, whereas both lower and higher $f$ yield more intermittent, non-Gaussian fluctuations and decrease the phase lag between the drag and canopy-top velocity. Phase-resolved analyses show that coherent structures transition from hairpin-dominated motions within the canopy during intermediate-drag phases to Kelvin-Helmholtz (KH) and KH-hairpin hybrid structures at peak drag, while low-drag phases resemble an open-channel state. These phase-dependent changes are accompanied by systematic variations in sweeps, ejections, and high-speed/low-speed streaks. Overall, the results clarify how temporally varying drag reorganizes shear-layer instability, TKE transport, and sweep–ejection dynamics, and they demonstrate that the VDM provides an efficient framework for isolating turbulence responses to prescribed canopy drag.

Hitherto, the generation and propagation of mode-2 oceanic internal waves and their mutual transformations with mode-1 waves remain inadequately understood, although their significance for localized energy balance and material transport has only recently been unravelled by observational data. To investigate this topic, the wave equations are advantageous not only for allowing transformations between modes 1 and 2 but also for enabling asymptotic and numerical analyses, among which the coupled Korteweg–de Vries (KdV) equations play an important role in the sparse literature. However, their accuracy has not been formally examined through comparisons with primitive equations or, loosely, other reduced models. To address these concerns, a fully dispersive and weakly nonlinear pseudodifferential equation system, or fully dispersive internal wave (FDIW) equations, for a three-layer fluid system is derived from the full Euler equations, which delineate wave amplitudes and velocity potentials at the two interfaces, accounting for bottom topography and background flow. Then regular perturbation analysis and weakly nonlinear analysis in the long-wave limit are conducted to obtain theoretical predictions of the wave amplitude ratio between two interfaces, six resonant conditions with sinusoidal bottom topography, and the associated reflection and transmission coefficients, as well as the phase speeds, vertical structures, and polarities of internal solitary waves. All these predictions are confirmed by direct numerical simulations of the FDIW equations, and a significant discrepancy with the coupled KdV equations indicates that the latter needs much care in practical usage. After that, the FDIW equations are used to investigate the evolution of initial linear waves past an uneven bottom, mimicking the propagation of internal tides in the ocean, and the generation of mode-1 and mode-2 nonlinear waves induced by a constant background current and barotropic tides passing over topography.

We propose a modified suspension-balance model (MSBM) for the flow of red blood cells (RBCs) and other deformable particle suspensions in confined geometries. Specifically, the method includes the hydrodynamic lift force generated by deformable particles interacting with walls leading to a cell-free layer. The lift force is added to the suspension-balance model to drive RBCs migrating away from the wall. Using the MSBM, we simulate blood flows through microvascular channels and tubes. The method is able to capture the transient development of the cell-free layer (CFL) and the corresponding hematocrit and velocity profiles with the development of the CFL. The CFL thickness and hemorheological hallmarks in microcirculation, such as the Fåhræus effect and the Fåhræus-Linqvist effect, are captured in good agreement with existing experimental and direct numerical results of blood flows. This work establishes a novel continuum computational framework that can efficiently capture the microstructural heterogeneity and non-Newtonian flow behavior of concentrated deformable particle suspensions under confinement.

This study focuses on the stationary characteristics of oblique detonation across various reaction rate distributions by using two-dimensional Euler equations coupled with a two-step kinetic model, focusing on the effects of the reduced activation energy and the exothermic reaction rate constant. The results show that increasing the reduced activation energy while decreasing the rate constant of the exothermic reaction postpones the initiation of oblique detonation and stabilizes the initiation structure. This paper innovatively uses these two parameters to regulate the exothermic reaction rate distribution while ensuring that the lengths of the induction and exothermic reaction zones remain constant. It is found that the exothermic reaction rate distribution significantly affects the stationary characteristics of oblique detonation waves. For a constant exothermic zone length, as the activation energy increases, the initiation position of the oblique detonation shifts downstream until it reaches a specific critical threshold. Beyond this point, the initiation structure of the oblique detonation undergoes a rapid transition to instability, evolving into an unstable oblique detonation configuration. This critical behavior occurs because cases with larger activation energy exhibit heightened sensitivity to overdrivenness.

We investigate leaky dielectric droplet dynamics in a superposed electric field consisting of alternating and constant electric fields using both analytical and numerical approaches. In the present study, an analytical formulation is developed within the framework of small deformation theory to describe the mean and amplitude of the steady-state-time-periodic droplet deformation. In addition, a phase field interface tracking numerical method is employed to accurately capture the droplet interface evolution and to explore the parametric space governing the oscillatory response of the droplet subjected to the superposed electric field. The influence of the mixing ratio of the alternating and constant electric field components on both the mean deformation and oscillation amplitude is systematically examined. It is shown that the frequency of the mechanical response of the droplet is twice the frequency of the alternating electric field and corresponds to the base frequency of the superposed electric field. The droplet deformation exhibits a phase difference to the applied superposed electric field, and the effects of electrical relaxation and mechanical response on this phase difference are established. The dependence of mean droplet deformation on the mixing ratio is analyzed for low forcing frequencies and for frequencies at which surface charge polarization remains unaffected. The present work bridges droplet dynamics in alternating and constant electric fields and identifies conditions where superposed fields produce larger mean deformation and oscillation amplitude than a pure ac electric field.

This study utilized direct numerical simulation to investigate the bypass transition over a flat plate featuring isotropic, protruding rough surface and subjected to a favorable-to-adverse pressure gradient. The rough surface was positioned at one of two locations, upstream of or in proximity to the separation point, while the inlet free-stream turbulence (FST) was characterized by two distinct intensities and fundamental frequencies. Through proper orthogonal decomposition, dynamic mode decomposition, local stability, and transient growth analyses, the effects of rough surfaces and inlet FST with varying intensities and fundamental frequencies on the bypass transition process and its disturbance growth mechanism were elucidated. The results indicate that the intensity and fundamental frequency of the inlet FST, along with the rough surface position, have negligible effects on the quasistreamwise vortices and velocity streak structures in the transition region. The rough surface placed upstream of the separation point interacts with the inlet FST to generate a resonant mode. For an inlet FST with a turbulence intensity of 3% and a fundamental frequency of $2\pi /7$, this interaction produces a novel resonant mode distinguished by a more elongated wave packet and an upstream-shifted resonant onset. This new mode comprises numerous high-frequency components that make a significant collective contribution to the system's nonlinear dynamics. Conversely, a rough surface positioned near the separation point markedly amplifies the inlet FST. The second harmonic of the inlet FST demonstrates insensitivity to the rough surface's position, the FST intensity, and its fundamental frequency. A lower fundamental frequency of the inlet FST results in increased nonlinear structural strength within its third and fourth harmonics, an effect that is further augmented when the rough surface is located near the separation point. The novel resonant mode creates extensive regions of linear stability within the transition zone. Within these regions, disturbances at multiple frequencies undergo optimal transient growth, and the spatial distribution of these optimal disturbances mirrors the fibrous morphology of the corresponding energy structure. Regarding the transient growth of wave disturbances in general resonant modes, a higher fundamental frequency yields the maximum energy gain, albeit within the most spatially confined optimal disturbance distribution. Furthermore, an increase in the streamwise wave number of the travelling wave also serves to enhance the gain.