We study experimentally the rate of chaotic mixing in viscoplastic fluids by using a rod-stirring protocol with a rotating vessel. Only a limited zone localized around the stirring rods is highly sheared at a given time. Using a dyed spot as the initial condition, we measure the decay of concentration fluctuations of dye as mixing proceeds. The mixing rate is found to be proportional to the volume of highly sheared fluid during a rotation period of the rods and inversely proportional to the number of rotations of the rods over a rotation of the vessel. Due to numerical simulations and experimental measurements, we relate the volume of highly sheared fluid to the parameters of the flow. We propose a quantitative two-zone model for the mixing rate, taking into account the geometry of the highly sheared zone as well as the rate at which fluid is renewed inside this zone. For all experiments, the model predicts correctly the scaling of the exponential mixing rates during a first rapid stage and a second slower one.

While chemically active homogeneous spherical particles do not undergo self-diffusiophoresis in free solution, they may do so when suspended in the vicinity of a solid boundary. We explore this possibility using a first-order kinetic model of solute absorption, where the relative magnitude of reaction to diffusion is characterized by the Damköhler number $\mathrm{Da}$. When the particle is remote from the wall, it is repelled from it with a velocity that scales inversely with the square of distance. The opposite extreme, when the ratio $\delta$ of separation distance to particle size is small, results in the anomalous scaling ${\delta }^{\frac{\sqrt{1+2\mathrm{Da}}-1}{2}}$ of the solute concentration in the narrow gap separating the particle and wall. This irrational power may only be obtained by asymptotic matching with solute transport outside the gap. For $\mathrm{Da}<4$ the self-propulsion speed possesses the same scaling, being set by the large pressures forming in the gap through a lubrication-type mechanism. For $\mathrm{Da}>4$ the particle velocity is $O\left(\delta \right)$, set by the flow in the region outside the gap. Solute advection is subdominant to diffusion in both the remote and near-contact limits and accordingly affects neither the above scaling nor the resulting approximations.

We show that the transition to a $k^{-1}$ spectrum in the enstrophy inertial range of generalized two-dimensional turbulence can be derived analytically using the eddy damped quasinormal Markovianized (EDQNM) closure. The governing equation for the generalized two-dimensional fluid system includes a nonlinear term with a real parameter $\alpha$. This parameter controls the relationship between the stream function and generalized vorticity and the nonlocality of the dynamics. An asymptotic analysis accounting for the overwhelming dominance of nonlocal triads allows the $k^{-1}$ spectrum to be derived based upon a scaling analysis. We thereby provide a detailed analytical explanation for the scaling transition that occurs in the enstrophy inertial range at $\alpha =2$ in terms of the spectral dynamics of the EDQNM closure, which extends and enhances the usual phenomenological explanations.

A cantilevered-capillary force apparatus is used to study the time scale for the coalescence of two droplets compressed together with a constant force. Power-law trends for the coalescence time as a function of droplet radius and compression force are experimentally measured. The measurements are compared against several different scaling theories from the literature. One of the existing theories is found to correctly predict the dependence on the droplet radius, but all of the theories overpredict the dependence on the force. A transition is also observed in the measured drainage time from a small variation around a single deterministic value for droplets with a radius of 125 $\mu \mathrm{m}$ or less to a broad distribution of drainage times for droplets with a radius of 150 $\mu \mathrm{m}$. A qualitative explanation for this transition is provided via scaling arguments.

Plane Couette flow under spanwise, anticyclonic system rotation [rotating plane Couette flow (RPCF)] is studied experimentally using stereoscopic particle image velocimetry for different Reynolds and rotation numbers in the fully turbulent regime. Similar to the laminar regime, the turbulent flow in RPCF is characterized by roll cells, however both instantaneous snapshots of the velocity field and space correlations show that the roll cell structure varies with the rotation number. All three velocity components are measured and both the mean flow and all four nonzero Reynolds stresses are obtained across the central parts of the channel. This also allows us to determine the wall shear stress from the viscous stress and the Reynolds stress in the center of the channel, and for low rotation rates the wall shear stress increases with increasing rotation rate as expected. The results show that zero absolute vorticity is established in the central parts of the channel of turbulent RPCF for high enough rotation rates, but also that the mean velocity profile for certain parameter ranges shows an $\mathsf{S}$ shape giving rise to a negative velocity gradient in the center of the channel. We find that from an analysis of the Reynolds stress transport equation using the present data there is a transport of the Reynolds shear stress towards the center of the channel, which may then result in a negative mean velocity gradient there.

The linear stability of plane Couette flow past a deformable solid is analyzed in the creeping-flow limit with an objective towards elucidating the consequences of employing two widely different formulations for the dissipative stresses in the deformable solid. One of the formulations postulates that the dissipative stress is proportional to the strain-rate tensor based on the left Cauchy-Green tensor, while in the other the dissipative stress in the solid is proportional to the rate-of-deformation tensor. However, it is well known in continuum mechanics that the rate-of-deformation tensor obeys the fundamental principle of material-frame indifference while the strain-rate-tensor formulation does not and hence it is more appropriate to employ the rate-of-deformation tensor in the description of dissipative stresses in deformable solids. In this work we consider the specific context of stability of plane Couette flow past a deformable solid and demonstrate that the results concerning the stability of the system from both models differ drastically. In the rate-of-deformation formulation for the dissipative stress, there is a range of solid-fluid thickness ratios (between 1.21 and 1.46) wherein the system is always stable for nonzero values of solid viscosity, unlike the strain-rate-tensor formulation wherein the system is unstable at all values of solid thickness. Further, for a solid-fluid thickness ratio less than 1, incorporation of dissipative effects in the solid using the rate-of-deformation formulation shows that the flow is more unstable compared to a purely elastic neo-Hookean solid, while for strain-rate-tensor formulation the flow is stabilized with an increase in viscosity of the solid. Using the fundamentally correct dissipative stress formulation, we also address the stability of pressure-driven flow in a deformable channel, wherein previous work carried out for an elastic neo-Hookean solid has shown that only the short-wave instability (driven by the first normal stress difference in the solid) exists while the finite-wave instability is absent in the creeping-flow limit. Using a consistent formulation for viscous stresses in the solid, we show that the short-wave instability is completely stabilized beyond a critical viscosity of the solid. Thus the present study clearly demonstrates the importance of consistent modeling of dissipative effects in the solid in order to accurately predict the stability of fluid flow past deformable solid surfaces.

Expressions for the ratio of the turbulent Schmidt (${\text{Sc}}_t$) to Prandtl (${\text{Pr}}_t$) numbers are derived for an idealized atmospheric surface layer flow over water surfaces. Conditions where the ansatz ${\text{Sc}}_t/{\text{Pr}}_t\approx 1$ are then discussed. It is shown that ${\text{Sc}}_t/{\text{Pr}}_t\approx 1$ is consistent with the active role of temperature in turbulence generation or destruction even when perfect similarity between turbulent transport of heat and water vapor is absent.

Despite the importance of pulsed jets for underwater propulsion, the effect of multiple-jet interactions remains poorly understood. We experimentally investigate how interactions between parallel jets in a pulsed-jet thruster affect the thruster's propulsive performance. Using high-speed fluorescence imaging, we investigate the mutual influence of two pulsed jets under conditions relevant to low-speed maneuvering in a vehicle ($\mathrm{Re}\approx 350, L/D\le 2$). Thrust production and propulsive efficiency are evaluated for different nozzle spacings using a new force estimation technique based on the fluorescence data. This analysis reveals that compared to noninteracting jets, the efficiency and thrust generated by the pair of interacting jets can fall by as much as 10% when the jets are brought into close proximity. Empirically, the thrust $T$ falls off with the nondimensional jet spacing $\tilde{\mathrm{\Delta }}$ as $T=T_{\infty }(1-Co{\tilde{\mathrm{\Delta }}}^{-6})$ for a thrust coupling coefficient $Co=2.04\pm 0.11.$ Finally, we predict this dependence of thrust on spacing using a model that relates the thrust and efficiency drop to streamline curvature and vortex induction at the nozzles.

The relationship between Koopman mode decomposition, resolvent mode decomposition, and exact invariant solutions of the Navier-Stokes equations is clarified. The correspondence rests upon the invariance of the system operators under symmetry operations such as spatial translation. The usual interpretation of the Koopman operator is generalized to permit combinations of such operations, in addition to translation in time. This invariance is related to the spectrum of a spatiotemporal Koopman operator, which has a traveling-wave interpretation. The relationship leads to a generalization of dynamic mode decomposition, in which symmetry operations are applied to restrict the dynamic modes to span a subspace subject to those symmetries. The resolvent is interpreted as the mapping between the Koopman modes of the Reynolds stress divergence and the velocity field. It is shown that the singular vectors of the resolvent (the resolvent modes) are the optimal basis in which to express the velocity field Koopman modes where the latter are not a priori known.

An experimental investigation of nonlinear scale interactions in a forced turbulent boundary layer is presented here. A dynamic wall perturbation mechanism was used to externally force two distinct large-scale synthetic modes with well-defined spatial and temporal wave numbers in a fully turbulent flow. The focus is on characterizing the nonlinear flow response at triadically consistent wave numbers that arises from the direct interactions of the two synthetic modes. These experimental results isolate triadic scale interactions in wall turbulence in a unique fashion, and provide the ability to explore the dynamics of scale coupling in a systematic and detailed manner. The ideas advanced here are intended to contribute towards modeling efforts of high-Reynolds-number wall turbulence.

The impact of a turbulent flow on wind-driven oceanic near-inertial waves is examined using a linearized shallow-water model of the mixed layer. Modeling the flow as a homogeneous and stationary random process with spatial scales comparable to the wavelengths, we derive a transport (or kinetic) equation governing wave-energy transfers in both physical and spectral spaces. This equation describes the scattering of the waves by the flow which results in a redistribution of energy between waves with the same frequency (or, equivalently, with the same wave number) and, for isotropic flows, in the isotropization of the wave field. The time scales for the scattering and isotropization are obtained explicitly and found to be of the order of tens of days for typical oceanic parameters. The predictions inferred from the transport equation are confirmed by a series of numerical simulations. Two situations in which near-inertial waves are strongly influenced by flow scattering are investigated through dedicated nonlinear shallow-water simulations. In the first, a wave packet propagating equatorward as a result from the $\beta$ effect is shown to be slowed down and dispersed both zonally and meridionally by scattering. In the second, waves generated by moving cyclones are shown to be strongly disturbed by scattering, leading again to an increased dispersion.

In a typical Richtmyer-Meshkov experiment a fast moving flat shock strikes a stationary perturbed interface between fluids $A$ and $B$ creating a transmitted and a reflected shock, both of which are perturbed. We propose shock tube experiments in which the reflected shock is stationary in the laboratory. Such a standing perturbed shock undergoes well-known damped oscillations. We present the conditions required for producing such a standing shock wave, which greatly facilitates the measurement of the oscillations and their rate of damping. We define a critical density ratio $R^{\mathrm{critical}}$, in terms of the adiabatic indices of the two fluids, and a critical Mach number $M_s^{\mathrm{critical}}$ of the incident shock wave, which produces a standing reflected wave. If the initial density ratio $R$ of the two fluids is less than $R^{\mathrm{critical}}$ then a standing shock wave is possible at $M_s=M_s^{\mathrm{critical}}$. Otherwise a standing shock is not possible and the reflected wave always moves in the direction opposite the incident shock. Examples are given for present-day operating shock tubes with sinusoidal or inclined interfaces. We consider the effect of viscosity, which affects the damping rate of the oscillations. We point out that nonlinear bubble and spike amplitudes depend relatively weakly on the viscosity of the fluids and that the interface area is a better diagnostic.

In this study we analyze the interaction of a single rising bubble with an inclined wall. We conduct experiments considering different liquids and bubble sizes, to cover a wide range of Reynolds and Weber numbers, with wall angles from nearly horizontal to nearly vertical. For all cases, the bubble initially collides with the wall; after the initial interaction, in accord with previous studies, the bubble either steadily slides on the wall or ascends, colliding repeatedly with it. Considering a force balance for the bubble motion on the wall, we propose a set of conditions for the transition from sliding to bouncing that is validated with the present and previous data.

An object immersed in a fast flow typically experiences fluid forces that increase with the square of speed. Here we explore how this high-Reynolds-number force-speed relationship is affected by unsteady motions of a body. Experiments on disks that are driven to oscillate while progressing through air reveal two distinct regimes: a conventional quadratic relationship for slow oscillations and an anomalous scaling for fast flapping in which the time-averaged drag increases linearly with flow speed. In the linear regime, flow visualization shows that a pair of counterrotating vortices is shed with each oscillation and a model that views a train of such dipoles as a momentum jet reproduces the linearity. We also show that appropriate scaling variables collapse the experimental data from both regimes and for different oscillatory motions into a single drag-speed relationship. These results could provide insight into the aerodynamic resistance incurred by oscillating wings in flight and they suggest that vibrations can be an effective means to actively control the drag on an object.

We perform experiments to investigate the relaxation of a highly deformed elastic filament at a liquid-air interface. The dynamics for filaments of differing length, diameter, and elastic modulus collapse to a single curve when the time dependence is scaled by a time scale $\tau =8\pi \mu L_o^4/B$. Even though the time $\tau$ is obtained by comparing the linear bending and viscous forces, we find that it also controls the relaxation in the highly nonlinear regime of our experiments. The relaxation, however, is completed in a very small fraction of the time $\tau$ due to a prefactor that changes with the tension in the nonlinear regime. Nonlinear numerical simulations show that the force due to tension along the filament is comparable to the bending force, producing a net elastic restoring force that is much smaller than either term. We perform particle image velocimetry at the liquid-air interface to support the results of the numerics. Finally, we find that when the filament is initialized in asymmetric shapes, it rapidly goes to a shape with symmetric stresses. This symmetrization process is entirely nonlinear; we show that the symmetric curvature state minimizes energy at arbitrarily large deformation.

Elastic capsules flowing in small enough tubes, such as red blood cells in capillaries, are well known to line up into regular single-file trains. The stability of such trains in somewhat wider channels, where this organization is not observed, is studied in a two-dimensional model system that includes full coupling between the viscous flow and suspended capsules. A diverse set of linearly amplifying disturbances, both long-time asymptotic (modal) and transient (nonmodal) perturbations, is identified and analyzed. These have a range of amplification rates and their corresponding forms are wavelike, typically dominated by one of five principal perturbation classes: longitudinal and transverse translations, tilts, and symmetric and asymmetric shape distortions. Finite-amplitude transiently amplifying perturbations are shown to provide a mechanism that can bypass slower asymptotic modal linear growth and precipitate the onset of nonlinear effects. Direct numerical simulations are used to verify the linear analysis and track the subsequent transition of the regular capsule trains into an apparently chaotic flow.

Diffusiophoresis, a ubiquitous phenomenon that induces particle transport whenever solute concentration gradients are present, was recently observed in the context of microsystems and shown to strongly impact colloidal transport (patterning and mixing) at such scales. In the present work we show experimentally that this nanoscale mechanism can induce changes in the macroscale mixing of colloids by chaotic advection. Rather than the decay of the standard deviation of concentration, which is a global parameter commonly employed in studies of mixing, we instead use multiscale tools adapted from studies of chaotic flows or intermittent turbulent mixing: concentration spectra and second and fourth moments of the probability density functions of scalar gradients. Not only can these tools be used in open flows, but they also allow for scale-by-scale analysis. Strikingly, diffusiophoresis is shown to affect all scales, although more particularly the small ones, resulting in a change of scalar intermittency and in an unusual scale bridging spanning more than seven orders of magnitude. By quantifying the averaged impact of diffusiophoresis on the macroscale mixing, we explain why the effects observed are consistent with the introduction of an effective Péclet number.

We quantify the dynamics of a prolate leaky dielectric drop upon removal of a uniform dc electric field. Experiments consisting of a castor oil drop suspended in silicone oil are compared against axisymmetric boundary integral computations that account for transient charging, or charge relaxation, of the interface. A temporal asymmetry between the drop deformation and relaxation processes is observed in the experiments and computations: The drop relaxes back to its spherical equilibrium shape faster than the time taken to achieve its steady-state deformation. During the deformation process, the electrical (Maxwell) stress deforms the drop along the direction of the applied field; it is counteracted by the capillary stress. During the relaxation process, i.e., after the field is removed, the electrical stress acts together with the capillary stress to quickly restore the drop back to equilibrium. This change in action of the electrical stress is responsible for the asymmetry between the drop deformation and relaxation. Notably, the electrical stress acts over the charge relaxation time scales of the fluids: Thus, counterintuitively, longer charging time scales yield faster drop relaxation. That is, the longer it takes for the interface to discharge, the faster the drop shape relaxes. We also present computational results for a drop that does not relax back to its initial spherical shape upon removal of the electric field; rather, the drop breaks up via an end-pinching mechanism.

An object moving through a plane interface into a fluid deforms the interface in such a way that fluid from one side of the interface is entrained into the other side, a phenomenon known as Darwin's drift. We investigate this phenomenon experimentally using a disk which is started exactly at the interface of two immiscible fluids, namely, oil and water. First, we observe that due to the density difference between the two fluids the deformation of the interface is influenced by gravity and show that there exists a time window of universal behavior. Second, we show by comparing with boundary integral simulations that, even though the deformation is universal, our results cannot be fully explained by potential flow solutions. We attribute this difference to the starting vortex, which is created in the wake of the disk. Besides contributing significantly to entrainment directly, the vortex also influences the interface deformation due to Darwin's drift. Universal behavior is preserved, however, because the size and strength of the vortex shows the same universality as the potential flow solution.

We demonstrate that porous nematic microfluidics is a potential route for the generation of nematic umbilic defects and regular umbilic defect lattices. By using numerical modeling we show that the mutual (backflow) coupling between the flow velocity and the orientation director field of the nematic liquid crystal leads to the formation of positive umbilic defects at local peaks and to the formation of negative umbilic defects at the local saddles in the flow profile. The number of flow peaks and the index of the flow saddles (i.e., the number of the valleys) are shown to be directly related to the strength of the umbilic defect, effectively relating the two fields at the geometrical level. The regular arrangement of the barriers in the porous channels is demonstrated to lead to the formation of regular lattices of umbilic defects, including square, triangular, and even kagome lattices. Experimental realization of such systems is discussed, with particular focus on microfluidic-tunable birefringent photonic band structures and lattices.