The mixing of binary fluids by stirrers is a commonplace procedure in many industrial and natural settings, and mixing efficiency directly translates into more homogeneous final products, more enriched compounds, and often substantial economic savings in energy and input ingredients. Enhancements in mixing efficiency can be accomplished by unorthodox stirring protocols as well as modified stirrer shapes that utilize unsteady hydrodynamics and vortex-shedding features to instigate the formation of fluid filaments which ultimately succumb to diffusion and produce a homogeneous mixture. We propose a PDE-constrained optimization approach to address the problem of mixing enhancement for binary fluids. Within a gradient-based framework, we target the stirring strategy as well as the cross-sectional shape of the stirrers to achieve improved mixedness over a given time horizon and within a prescribed energy budget. The optimization produces a significant enhancement in homogeneity in the initially separated fluids, suggesting promising modifications to traditional stirring protocols.

Studies of particle motion in vortical flows have mainly focused on pointlike particles. This approximation assumes that the velocity field that surrounds the particle is linear. We consider an inertialess rigid dumbbell in a two-dimensional steady vortex. While the system remains analytically tractable, the particle experiences the nonlinearity of the surrounding velocity field. By exploiting the rotational symmetry of the flow, we reduce the problem to that of a two-dimensional dynamical system, whose fixed points and periodic orbits can be used to explain the motion of the dumbbell. For all vortices in which the fluid angular velocity decreases with radial distance, the center of mass of the dumbbell follows a spirographic-like trajectory around the vortex center. This results from a periodic oscillation in the radial direction combined with revolution around the center. The shape of the trajectory depends strongly on the initial position and orientation of the dumbbell, but the dynamics is qualitatively the same irrespective of the form of the vortex. If the fluid angular velocity is not monotonic, the spirographic-like motion is altered by the existence of transport barriers, whose shape is now sensitive to the details of the vortex. This analysis may be extended to study the dynamics of inertial and/or self-propelled dumbbells in a two-dimensional vortex.

We investigate gunwale bobbing, a phenomenon in which a person jumping on the gunwales of a canoe achieves horizontal propulsion by forcing it with vertical oscillations. The canoe moves forward by surfing the resulting wave field. After an initial transient, the canoe achieves a cruising velocity which satisfies a balance between the thrust generated from pushing downwards into the surface gradients of the wave field and the resistance due to a combination of profile drag and wave drag. By superposing the linear wave theories of Havelock [Proc. R. Soc. A 95, 354 (1919)] for steady cruising and Helmholtz for an oscillating source, we demonstrate that such a balance can be sustained. We calculate the optimal parameter values to achieve maximum canoe velocity. We compare our theoretical result to accelerometer data taken from an enthusiastic gunwale bobber and to estimates from videos of other canoeing aficionados. Finally, we discuss the similarities and differences with other examples of macroscopic wave-driven bodies and comment on possible applications to competitive sports.

Unforced shoaling waves experience growth and changes to wave shape. Similarly, wind-forced waves on a flat-bottom likewise experience growth/decay and changes to wave shape. However, the combined effect of shoaling and wind-forcing on wave shape, particularly relevant in the near-shore environment, has not yet been investigated theoretically. Here, we consider small-amplitude, shallow-water solitary waves propagating up a gentle, planar bathymetry forced by a weak, Jeffreys-type wind-induced surface pressure. We derive a variable-coefficient Korteweg–de Vries–Burgers (vKdV–B) equation governing the wave profile's evolution and solve it numerically using a Runge-Kutta third-order finite difference solver. The simulations run until convective prebreaking—a Froude number limit appropriate to the order of the vKdV–B equation. Offshore winds weakly enhance the ratio of prebreaking height to depth as well as prebreaking wave slope. Onshore winds have a strong impact on narrowing the wave peak, and wind also modulates the rear shelf formed behind the wave. Furthermore, wind strongly affects the width of the prebreaking zone, with larger effects for smaller beach slopes. After converting our pressure magnitudes to physically realistic wind speeds, we observe qualitative agreement with prior laboratory and numerical experiments on breakpoint location. Additionally, our numerical results have qualitatively similar temporal wave shape to shoaling and wind-forced laboratory observations, suggesting that the vKdV–B equation captures the essential aspects of wind-induced effects on shoaling wave shape. Finally, we isolate the wind's effect by comparing the wave profiles to the unforced case. This reveals that the numerical results are approximately a superposition of a solitary wave, a shoaling-induced shelf, and a wind-induced, bound, dispersive, and decaying tail.

In swimming microorganisms and the cell cytoskeleton, inextensible fibers resist bending and twisting, and interact with the surrounding fluid to cause or resist large-scale fluid motion. In this paper, we develop a numerical method for the simulation of cylindrical fibers by extending our previous work on inextensible bending fibers [O. Maxian et al., Phys. Rev. Fluids 6, 014102 (2021)] to fibers with twist elasticity. In our “Euler” model, twist is a scalar function that measures the deviation of the fiber cross section relative to a twist-free frame, the fiber exerts only torque parallel to the centerline on the fluid, and the perpendicular components of the rotational fluid velocity are discarded in favor of the translational velocity. In the first part of this paper, we justify this model by comparing it to another commonly used “Kirchhoff” formulation where the fiber exerts both perpendicular and parallel torque on the fluid, and the perpendicular angular fluid velocity is required to be consistent with the translational fluid velocity. Through asymptotics and numerics, we show that the perpendicular torques in the Kirchhoff model are small, thereby establishing the asymptotic equivalence of the two models for slender fibers. We then develop a spectral numerical method for the hydrodynamics of the Euler model. We define hydrodynamic mobility operators using integrals of the Rotne-Prager-Yamakawa tensor, and evaluate these integrals through a slender-body quadrature, which requires on the order of 10 points along a smooth fiber to obtain several digits of accuracy. We demonstrate that this choice of mobility removes the unphysical negative eigenvalues in the translation-translation mobility associated with asymptotic slender-body theories, and ensures strong convergence of the fiber velocity and weak convergence of the fiber constraint forces. We pair the spatial discretization with a semi-implicit temporal integrator to confirm the negligible contribution of twist elasticity to the relaxation dynamics of a bent fiber. We also study the instability of a twirling fiber, and demonstrate that rotation-translation and translation-rotation coupling change the critical twirling frequency at which a whirling instability onsets by $\sim$20%, which explains some (but not all) of the deviation from theory observed experimentally. When twirling is unstable, we show that the dynamics transition to a steady overwhirling state, and quantify the amplitude and frequency of the resulting steady crank-shafting motion.

Two-dimensional channel flow simulations of FENE-P (finitely extensible nonlinear elastic-Peterlin) fluid in the elasto-inertial turbulence (EIT) regime reveal distinct regimes ranging from chaos to a steady traveling wave which takes the form of an arrowhead structure. This coherent structure provides insights into the polymer/flow interactions driving EIT. A set of controlled numerical experiments and the study of transfer between elastic and turbulent kinetic energies highlight the role of small- and large-scale dynamics in the self-sustaining cycle of chaos in EIT flows.

Hydraulic falls on the interface of a two-layer density stratified fluid flow in the presence of bottom topography are considered. We extend the previous work [Philos. Trans. R. Soc. London A 360, 2137 (2002)] to two successive bottom obstructions of arbitrary shape. The forced Korteweg-de Vries and modified Korteweg-de Vries equations are derived in different asymptotic limits to understand the existence and classification of fall solutions. The full Euler equations are numerically solved by a boundary integral equation method. New solutions characterized by a train of trapped waves are found for interfacial flows past two obstacles. The wavelength of the trapped waves agrees well with the prediction of the linear dispersion relation. In addition, the effects of the relative location, aspect ratio, and convexity-concavity property of the obstacles on interface profiles are investigated.

Saffman and Turner [P. G. Saffman and J. S. Turner, J. Fluid Mech. 1, 16 (1956)] argued that the collision rate for droplets in turbulence increases as the turbulent strain rate increases. However, the numerical simulations of Dhanasekaran et al. [J. Dhanasekaran et al., J. Fluid Mech. 910, A10 (2021)] in a steady straining flow showed that the Saffman-Turner model is oversimplified because it neglects droplet-droplet interactions. These result in a complex dependence of the collision rate on the strain rate and on the differential settling speed. Here we show that this dependence is explained by a sequence of bifurcations in the collision dynamics. We compute the bifurcation diagram when strain is aligned with gravity and show that it yields important insights into the collision dynamics. First, the steady-state collision rate remains nonzero in the limit $\mathrm{Kn}\to 0$, contrary to the common assumption that the collision rate tends to zero in this limit (Kn is a nondimensional measure of the mean free path of air). Second, the nonmonotonic dependence of the collision rate on the differential settling speed is explained by a grazing bifurcation. Third, the bifurcation analysis explains why so-called closed trajectories appear and disappear. Fourth, our analysis predicts strong spatial clustering near certain saddle points, where the effects of strain and differential settling cancel.

Barnacle-type roughness can be considered as a hybrid form of roughness that combines some of the topographical features of traditional regular rough surfaces composed of discrete roughness elements of uniform size and shape, and of irregular rough surfaces that exhibit features with a wide distribution of sizes and shapes, covering the surface with a random areal distribution. In this study, the influence of barnacle-type rough surfaces with increasing coverage on wall-bounded turbulence is investigated using direct numerical simulations of turbulent channel flow. The barnacle-type rough surfaces were generated with an algorithm that mimics the settlement behavior of barnacles, and the set of seven different surfaces describe the evolution of barnacle fouling with increasing coverage ranging from a lightly fouled surface with small isolated barnacle clusters (10% coverage) to a surface that is fully covered by barnacles (85% coverage). The roughness function recovers the expected trend with frontal solidity, attaining its maximum at a frontal solidity of approximately 0.2. Mean flow, Reynolds stress, and dispersive stresses show signatures of the clustering of roughness features, i.e., the barnacle colonies, at low coverage. This is most distinct for the streamwise Reynolds stresses where a double peak is observed at the lowest coverage; the inner peak can be interpreted as a partial recovery of smooth-wall behavior over the large connected unfouled sections of this surface. The flow over the rough surfaces is further investigated by a topographical characterization of the blanketing layer, i.e., the effective shape of the rough surface “perceived” by the outer flow. A linear relationship between the roughness function and the effective slope of the blanketing layer is observed.

Liquid atomization processes generating sprays and aerosols of droplets are used in many delivery and coating systems involving pure solvents, solutions, and suspensions. In our recent experimental work, we introduced a novel liquid atomization process generating microsprays and aerosols of submicron-diameter droplets for pure solvents, solutions, and suspensions: gas jets disintegrate thin liquid films that are formed as bubbles approach a liquid surface. Here, we develop a theoretical description of droplet sizes and flow rates, using the first principles of conservation of mass and energy, and employing dimensional and scale analyses. We introduce atomization diagrams as a graphical tool to determine possible, impossible, and expected droplet diameters and specific flow rates in aerosols and sprays produced under various conditions. We find a reasonable agreement between the theory and experiments for five different liquids converted into aerosols of submicron-diameter droplets by an atomization process where gas jets disintegrate thin liquid films that are formed as bubbles approach a liquid surface, and also for five traditional pressure nozzles that produce sprays of droplets of hundreds of microns in diameter. Our study explores the overall range of mean droplet diameters between 0.1 and 100 μm and Ohnesorge numbers between 0.01 and 100, and the analysis and atomization diagrams contribute to understanding of liquid atomization and can serve as a theoretical framework for comparing different liquid atomization techniques.

We introduce a closure model for coarse-grained kinetic theories of polar active fluids. Based on a quasiequilibrium approximation of the particle distribution function, the model closely captures important analytical properties of the kinetic theory, including its linear stability and the balance of conformational entropy production and dissipation. Nonlinear simulations show the model reproduces the qualitative behavior and nonequilibrium statistics of the kinetic theory, unlike commonly used closure models. We use the closure model to simulate highly turbulent suspensions in both two and three dimensions in which we observe complex multiscale dynamics, including large concentration fluctuations and a proliferation of polar and nematic defects.

While classic hydrodynamic models predict ordered formations for fish schools, observations show that schools are seemingly disordered. Our experiments on robotic swimmers may help to reconcile this discrepancy by showing that many different formations all emerge spontaneously and are stabilized due to flow interactions. Surprisingly, these locked states extend almost twice as far downstream for laterally displaced swimmers as for those in line. We also observe significant boosts in swimming speed—up to 60% faster than an isolated swimmer—for side-by-side formations. These findings demonstrate that benefits such as group cohesion and speed enhancement arise naturally via flow interactions and for the diverse relative arrangements seen in schools.

Water-absorbing materials consisting of hydrogel particles are widely used for agricultural substrates, hygienic products, drug delivery systems, and microfluidic devices. Hydrogel particles absorb water with expansion, leading to dynamics involving complicated interactions of intergranular liquid flow and hydrogel deformation. It is thus difficult to predict the water absorption rate of these materials. Water flow in hydrogel-coated capillary tubes also occurs with the deformation of flow passages, so it can be analogous to capillary flow through porous structures made of hydrogels. Here we report a combined experimental and theoretical investigation of water flow through capillaries whose inner surface is covered with a hydrogel layer. We experimentally measured the flow rate of water through hydrogel-coated capillary tubes and the volume of water absorbed by the hydrogel layer. We developed a mathematical model for capillary flow including the effects of the water absorption and swelling of the hydrogel layer. Our model is in good agreement with the measurements of the water flow rate and elucidates how the absorption and swelling of hydrogel regulate capillary flow. This study provides a theoretical framework for understanding the dynamics of capillary flow through porous structures composed of hydrogels, leading to insights into the engineering applications of porous hydrogel materials.

We study entrainment in dry thermals in neutrally and unstably stratified ambients, and moist thermals in dry-neutrally stratified ambients using direct numerical simulations. We find, in agreement with results of Lecoanet and Jeevanjee [J. Atmos. Sci. 76, 3785 (2019)], that turbulence plays a minor role in entrainment in dry thermals in a neutral ambient for Reynolds numbers $\text{Re}\lesssim {10}^4$. We then show that the net entrainment rate increases when the buoyancy of the thermals increases, either by condensation heating or because of an unstably stratified ambient. This is in contrast with the findings of Morrison et al. [J. Atmos. Sci. 78, 797 (2021)]. We also show that the role of turbulence is greater in these cases than in dry thermals and, significantly, that the combined action of condensation heating and turbulence creates intense small-scale vorticity, destroying the coherent vortex ring that is seen in dry and moist laminar thermals. These findings suggest that fully resolved simulations at Reynolds numbers significantly larger than the mixing transition Reynolds number $\text{Re}={10}^4$ are necessary to understand the role of turbulence in the entrainment in growing cumulus clouds, which consist of a series of thermals rising and decaying in succession.

Cilia and flagella are organelles that play central roles in unicellular locomotion, embryonic development, and fluid transport around tissues. In these examples, multiple cilia are often found in close proximity and exhibit coordinated motion. Inspired by the flagellar motion of biflagellate cells, we examine the synchrony exhibited by a filament pair surrounded by a viscous fluid and tethered to a rigid planar surface. A geometrically switching base moment drives filament motion, and we characterize how the stability of synchonized states depends on the base torque magnitude. In particular, we study the emergence of bistability that occurs when the antiphase, breast-stroke branch becomes unstable. Using a bisection algorithm, we find the unstable edge state that exists between the two basins of attraction when the system exhibits bistability. We establish a bifurcation diagram, study the nature of the bifurcation points, and find that the observed dynamical system can be captured by a modified version of Adler's equation. The bifurcation diagram and presence of bistability reveal a simple mechanism by which the antiphase breast stroke can be modulated, or switched entirely to in-phase undulations through the variation of a single bifurcation parameter.

The Peregrine soliton (PS) is widely regarded as a prototype nonlinear structure capturing properties of rogue waves that emerge in the nonlinear propagation of unidirectional wave trains. It has been recently demonstrated that the PS can emerge locally, as an asymptotic structure arising from the propagation of an arbitrary large decaying pulse, independently of its solitonic content. This mathematical discovery has changed the widely accepted paradigm of the solitonic nature of rogue waves by enabling the PS to emerge from partially radiative or even completely solitonless initial data. In this work, we realize this scenario in a water tank experiment with a particular aim to control the point of the PS occurrence in space-time by imposing an appropriately chosen initial chirp. By employing the inverse scattering transform for the synthesis of the initial data, we are able to engineer a localized wave packet with a prescribed solitonic and radiative content. This enables us to control the position of the emergence of the rogue wave by adjusting the inverse scattering spectrum. The proposed method of nonlinear spectral engineering is found to be robust to higher-order nonlinear effects, preceding the wave breaking dynamics, that are inevitable in realistic wave propagation conditions.

Smooth muscle-lined organs like the gut, the ureter, and the fallopian tubes transport matter by generating traveling contractile waves. Intestinal peristalsis is characterized by rhythmic trains of shallow, low-amplitude myogenic waves and high-amplitude, lumen-obliterating neurogenic waves. In this paper, we develop a simple analytical Poiseuille-flow model to predict the flow rates induced by these different contractions as a function of all relevant wave parameters, and compare them to a numerical fluid-solid finite element model. We rationalize experimentally observed bolus to-and-fro motion induced by shallow myogenic waves. We show that occluding waves induce considerable bolus mixing due to an upstream vortex. We then investigate the hydrodynamics induced by two waves propagating either in the same direction (colinear) or in opposite directions, as happens in the digestive tract. For colinear waves, we find that the bolus reflux is maximal at a distance between successive myogenic waves close to the one observed physiologically. Colliding waves create a high pressure region that gives rise to rapid fluid flow, high shear stress, and radial mixing upon annihilation. Our paper provides fundamental insight on the fluid dynamics (reflux, propulsion, and mixing) generated by different contraction patterns of the intestine.

While landing, birds often perform a perching maneuver, which involves pitching their wings upward while decelerating to a complete stop. This study is motivated by the observation that some birds fold their wings to create a wing sweep during such perching. The objective of this study is to find out whether such a wing sweep helps during a perching maneuver. We use two flat plates, one with a sweep and another without any sweep, and consider a deceleration maneuver where both of them decelerate to stop from Reynolds number $\text{Re}=13000$. We consider two cases: one, where the wings undergo only heaving, and two, where the wings perform both heaving and pitching. The latter maneuver was designed to mimic perching. By performing experiments and simulations, we compare the temporal evolution of the instantaneous forces and the vortex dynamics of both these plates. We show that during a major part of the deceleration, the instantaneous lift forces are higher in the case of the plate with sweep compared to the plate with no sweep during both kinematics. Our results indicate that the higher lift in the swept plate case was contributed by a stable leading-edge-vortex (LEV) which remains attached to the plate. This increase in stability was contributed by the spanwise vorticity convection caused by a distinct spanwise flow on the swept plate, as revealed by the numerical simulation. We also show that combined pitching and heaving resulted in higher force peaks, and the forces also decayed at a faster rate in this case compared to the heave-only case. Finally, by using an analytical model for unsteady flows, we prove that the higher lift characteristics of the swept plate were entirely due to higher circulatory forces.

The Childress-Soward model, consisting of a an electrically conducting Rayleigh-Bénard layer rotating about a vertical axis, is the conceptually simplest model configuration that captures all essentials of convectively driven dynamos. Here we study the effects of different boundary conditions on this type of dynamo. This closes a gap in the existing literature, which has primarily focused on simple, but geophysically not very realistic, cases. Furthermore, the extensive literature on boundary layers in nonrotating and rotating Rayleigh-Bénard convection makes the Childress-Soward model an ideal test bed for illuminating the fundamental effects that the boundary conditions have on convection-driven dynamos. In this study we systematically vary the thermal, mechanical, and electrical boundary conditions in direct numerical simulations, with a focus on flow regimes characterized by geostrophic turbulence. One key result applies to both dynamos and nonmagnetic, rotating convection. We show that for no-slip boundaries, the Nusselt number increases significantly when a fixed heat flux is imposed instead of a given temperature difference. This effect can be explained by an interplay of Ekman pumping and the internal structure of the thermal boundary layer, which is very sensitive to thermal boundary conditions. Dynamical changes are shown to exist only within the boundary layers, such that the bulk dynamics remains largely unaffected. In the dynamo case, we argue that care needs to be taken in order to ensure comparable energetics when changing from fixed-temperature to fixed-flux conditions. If done properly, no significant differences in the leading order features of the resulting dynamos are observed. In particular, the thermal boundary conditions have no noticeable influence on the horizontal flow scales, as might have been expected from studies in spherical geometry. In contrast, the mechanical boundary conditions largely control whether or not large-scale flows and fields can be generated. For no-slip boundaries, Ekman friction strongly suppresses upscale transport and eliminates the large-scale vortex dynamos that exist in the stress-free case. Finally, the magnetic boundary conditions are shown to strongly affect the topology of the generated magnetic fields. For an insulating exterior, the magnetic field is most intense in the central part of the fluid layer, whereas we observe a strong magnetic field buildup close to perfectly conducting boundaries. For large magnetic Reynolds numbers, we further observe thin layers of intense horizontal field attached to perfect conductors. The horizontal flow scales, however, remain largely unaffected in the investigated parameter regime.

The economics, environmental impact, and mechanical properties of paper-based storage containers have been widely studied. However, knowledge of the physical processes relevant to the end-user experience is unavailable. This paper outlines the main effects associated with the closing and opening of telescoping boxes, which are used, for instance, to store and transport board games, footwear, mobile phones, and tablet computers. The sliding motion of the lid is controlled by the flow in a thin film of air in the gap separating the lid and the base of the box. Based on a broad comparison between theory and experiments on real and synthetic boxes, we find that the process is primarily controlled by the shape of the gap between the base and the lid. We derive a master equation for the lid motion and identify the origin of three distinct experimental regimes. Finally, an optimal design for a rapidly closing box is identified.