Over the last decade, the physics of discontinuous shear thickening (DST) in dense suspensions has undergone a paradigm shift. Growing evidence indicates that it stems from a frictional transition at the particle scale induced by repulsive forces between the suspended grains. However, in dry granular materials, interparticle friction can also exhibit a transition when the grains are coated, typically with polymers or lubricants widely used in industry. In this work, we use discrete-element method simulations to demonstrate that DST occurs in dry granular materials when considering non-Coulombian behavior, where the interparticle friction varies with the contact normal force to mimic the influence of a coating. By performing pressure- and volume-imposed measurements, we reveal all regimes of dry shear thickening and present a mean-field model that predicts the observed scenarios by relating the bulk rheology to the microscopic friction law.

The plastron on submerged microgrooved surfaces enables substantial slip velocity, showing great potential in hydrodynamic drag deduction. Previous studies have indicated that hydrophobic surfaces exhibit considerable drag reduction when the plastron fully covers the microgrooves. However, few studies have examined drag reduction effects in relation to longitudinal plastron deformation. This study experimentally investigates longitudinal plastron deformation and drag reduction characteristics of microgrooved hydrophobic surfaces in a recirculating water tunnel. An interfacial visualization system, skin friction measurement system, and high-resolution particle image velocimetry are employed to evaluate longitudinal plastron deformation, drag reduction rates, and turbulent boundary layer flow simultaneously. Our experiments reveal a decrease in drag reduction from 40% to 20% as the Reynolds number $\left({\text{Re}}_x\right)$ increases from $0.67\times {10}^6$ to $2.15\times {10}^6$. Longitudinal plastron deformation, induced by shear stress, enhances with increasing ${\text{Re}}_x$, which leads to additional form drag and reduces drag reduction. We find that longitudinal plastron deformation over hydrophobic surfaces with shorter grooves is effectively mitigated, resulting in enhanced drag reduction. This work provides valuable insights into designing microgrooved hydrophobic surfaces to improve drag reduction in practical applications.

In the field of acoustic suspension or levitation of droplets against gravity, the application of Gor'kov's acoustic radiation force for small particles (within the Rayleigh limit, i.e., $d\ll \lambda$) or its extensions to larger ones (beyond the Rayleigh limit, i.e., $d\gtrsim \lambda$) is limited to predicting the suspension position of the droplet. Since this approach treats the droplet as a rigid particle, it fails to capture the fluid dynamics of the droplet and is also unsuitable for studying interfacial phenomena such as droplet deformation, splitting, or coalescence. In this work, we employ our recently developed acoustic body force in Eulerian form [J. Thirisangu et al., Phys. Fluids 37, 052012 (2025)], which models the droplet as a fluid, to theoretically investigate the suspension dynamics of a droplet subjected to standing waves through the interaction between acoustic, interfacial, and gravitational forces in a liquid-liquid system. Our theory predicts that when interfacial forces are dominant, the presence of positive and negative acoustic force regions within droplets beyond the Rayleigh limit reduces the net acoustic force counteracting gravity. As a result, the suspension dynamics become highly dependent on droplet size, leading to phenomena such as size-dependent switching of droplet suspending position between nodes and antinodes, alternating patterns in the critical acoustic energy density ($E_{\mathrm{cr}}$) required for suspension, and enabling droplet sorting based on critical energy method, which agrees with the recent experimental work [J. Thirisangu et al., Phys. Fluids 35, 122012 (2023)]. Unlike the case where interfacial force dominates, when this force weakens, acoustic body force can deform the droplet. If the droplet contains regions with opposing acoustic forces, these forces compete with each other, potentially causing the droplet to split. If gravitational force is not sufficiently weak, droplet splitting becomes unequal/asymmetric.

In an infinitely long cylinder horizontally rotating about its axis partially filled with liquid, convection cells whose axes are perpendicular to the rotation axis of the cylinder appear, even when the liquid-gas interface is hardly deformed. In the present study, we conduct direct numerical simulations to investigate the effects of axial flow on the convection cells. Assuming that the Froude number is sufficiently small and the Bond number is large enough, we treat the liquid-gas interface as an undeformable slip boundary. Even if axial flow exists, non-trivial three-dimensional flow structures emerge through a supercritical bifurcation as the rotational Reynolds number ${\mathrm{Re}}^{*}$ increases. For lower axial-flow Reynolds number ${\mathrm{Re}}_a^{*}$, the bifurcated three-dimensional flow is composed of convection cells drifting in the axial direction, which are similar to those in the absence of axial flow. In contrast, for higher ${\mathrm{Re}}_a^{*}$, vortices, whose rotation axis is approximately parallel to the axial direction, first appear at a critical ${\mathrm{Re}}^{*}$, and convection cells can coexist with these vortices at even higher ${\mathrm{Re}}^{*}$.

Many-body hydrodynamic interactions (HIs) play an important role in the dynamics of fluid suspensions. While many-body HIs have been studied extensively using particle simulations, there is a dearth of experimental frameworks with which to quantify fluid-mediated multibody interactions. To address this, we design an experimental method that utilizes optical laser tweezers for quantifying fluid-mediated colloidal interactions with exquisite precision and control. By inducing translation-rotation hydrodynamic coupling between trapped fluorescently labeled colloids, we obtain a direct report of few- to many-body HIs experimentally. We leverage the torque-free nature of laser tweezers to enable sensitive measurements of signals between trapped colloids. First, we measure the hydrodynamic interaction between a stationary tracer probe and a translating particle as a function of their separation distance, agreeing with microhydrodynamic theory. We discover that our technique can precisely quantify distant fluid disturbances that are generated by $\sim 2\mathrm{pN}$ of hydrodynamic force at 12 particle radii of separation. Many-body HIs are studied by measuring the rotational mobility of a tracer probe in a three-particle setup and in a model material, a two-dimensional hexagonal colloidal cluster, that undergoes oscillatory strain. To corroborate our measurements, we modify the Stokesian dynamics formalism to compute the induced particle rotations from prescribed linear velocities. Respectively, we discover that the probe and its nearest neighbor can spin in opposite directions in certain three-body configurations, and we find that rotational mobility in the hexagonal structure is strongly attenuated by particle rigidity. The excellent agreement between our measurements and calculations highlights our ability to accurately measure many-body HIs. Further, the robustness of our experimental methodology enables us to extend our theoretical framework to manipulate colloidal-scale fluid flows. With experimental validation, we compute the required trajectory of a moving particle to induce a desired angular velocity of a probe.

This study investigates the spatial evolution of a zero pressure gradient turbulent boundary layer (TBL) imposed by a square-wave (SqW) of steady spanwise wall-forcing, which varies along the streamwise direction ($x$). The SqW wall-forcing is imposed experimentally via a series of streamwise periodic belts running in opposite spanwise directions, following the methodology of Knoop et al. [Exp. Fluids 65, 65 (2024)], with the streamwise extent increased to beyond $\sim 11$ times the boundary layer thickness (${\delta }_o$) in the present study. This unique setup is leveraged to investigate the influence of viscous-scaled wavelength of SqW wall-forcing on the turbulent drag reduction efficacy for ${\lambda }_x^{+}=471$ (suboptimal), 942 (near-optimal), and 1884 (postoptimal conditions), at fixed viscous-scaled wall-forcing amplitude, $A^{+}=12$, and friction Reynolds number, ${\text{Re}}_{\tau }=960$. The TBL's response to this wall-forcing is elucidated by drawing inspiration from established knowledge on traditionally studied sinusoidal forcing, based on analysis of the streamwise-phase variation of the Stokes strain rate (SSR). The analysis reveals the SqW forcing to be characterized by a combination of two markedly different SSR regimes whose influence on the overlying turbulence is found to depend on the forcing waveform: subphase I of local and strong impulses of SSR downstream of the half- (${\lambda }_x/2$) and full-phase (${\lambda }_x$) locations, associated with a reversal in spanwise forcing directions, leading to significant turbulence attenuation, and subphase II of near-zero SSR over the remainder of forcing phase that enables turbulence recovery (when wall-forcing magnitudes and direction remain constant). Upon the initial imposition of the SqW forcing, the Reynolds stresses are strongly attenuated over the short streamwise extent of $x/{\delta }_0<0.5$ for all wavelengths, whereas the skin-friction transient is more gradual. Thereafter, once the forcing is ultimately established, the suboptimum and optimum wavelength regimes display no distinctive responses to the individual SSR subphases; rather, the drag-reduced TBL response is quasi-streamwise homogeneous. In contrast, an SSR-related phenomenology establishes itself clearly for the postoptimal case, in which a local attenuation of near-wall turbulence characterizes subphase I, while the turbulent energy recovers in subphase II owing to the extended region of near-zero SSR.

We experimentally investigate the forced synchronization of a turbulent, lean-premixed, bluff-body-stabilized flame undergoing self-excited oscillations due to global hydrodynamic instability. We acoustically force the flame at different amplitudes ($\alpha$) and frequencies ($f_f$) around its natural global frequency ($f_n$), while measuring its heat release rate (HRR) response via time-resolved CH* chemiluminescence imaging. As $\alpha$ increases at a fixed $f_f$, the flame initially oscillates quasiperiodically at both $f_n$ and $f_f$, but synchronizes with the forcing above a critical $\alpha$, consistent with the behavior of a canonical self-excited oscillator. The minimum $\alpha$ required for synchronization increases with the detuning ($|f_f-f_n|$), but not symmetrically around $f_f/f_n=1$, resulting in a skewed Arnold tongue. The HRR amplitude grows at small detuning due to resonant amplification but decays at large detuning due to asynchronous quenching. Comparing the globally unstable flame with an equivalent globally stable flame, we find that both exhibit a range of coupled states, including desynchronization, phase synchronization, and generalized synchronization. Crucially, the phase locking value varies spatially throughout the flame body, with the shear layers synchronizing more readily than the wake and recirculation zones. At higher detuning levels (i.e., when the forcing frequency is below 0.9 or above 1.1 times the natural frequency, $f_f/f_n<0.9$ or $f_f/f_n>1.1$), the HRR amplitude of the globally unstable flame is observed to be lower than that of the globally stable flame. This finding suggests that global hydrodynamic instability in a flame may serve as a passive mechanism to weaken self-excited thermoacoustic oscillations in combustion systems such as gas turbines and rocket engines.

The unsteady separation motion of a Mach 2 sharp-fin induced shock-wave/boundary-layer interaction was studied using a variant of the detached eddy simulation technique. Unsteadiness throughout the interaction region was observed to be driven by synthetically generated inflow turbulence. Unsteadiness in the separation shock foot (located via a wall pressure criterion) was characterized by a lower frequency range compared to the unsteadiness in the separation position (located via a skin friction criterion). Significant correlation was found between streamwise velocity perturbations within the incoming boundarylayer and unsteadiness in the separation motion. To investigate the effects of controlled upstream disturbances on the separation unsteadiness, a streamwise time-periodic body force was simulated through momentum and energy source terms. The results indicated that unsteadiness in the separation motion was influenced by upstream forcing, where the strength of the response depended on the frequency and spanwise form of forcing. Forcing at a lower frequency, representative of the baseline separation shock unsteadiness, produced significant sinusoidal phase-averaged motions of both the separation shock and separation position. When forced at a higher frequency, representative of the baseline separation position unsteadiness, a strong phase-averaged response was obtained only in the separation position data; the separation shock motion appeared to be uncorrelated with the imposed high-frequency forcing. With a spanwise form that synchronized the incoming velocity perturbations to leverage on the spanwise flow effects in the baseline flow, a constructive effect was observed in the perturbations of the separation line, enhancing the strength of the phase-averaged response. These findings demonstrate that upstream forcing of a particular spatial-temporal form can modulate the separation unsteadiness of the sharp-fin flow. It highlights important design factors for flow control of similar three-dimensional swept shock-wave/boundary-layer interactions.

We report experiments on the dynamics and characteristics of surface waves generated by the low Weber number impact of immiscible silicone oil droplets on a water pool. The study employed high-speed videography and background-oriented schlieren (BOS), in tandem, to capture the droplet impact and spatiotemporally resolved interfacial topography and surface waves, respectively. In addition to the applications of fundamental research interests, the importance of this study lies in the generation of whole-field experimental data, which can be used to develop and validate models for surface waves. The investigation explored effects of droplet viscosity, Weber number, and pool height on the characteristics of generated waves. The impact parameters led to various modes of droplet-pool interactions, including bouncing, coalescence, secondary jet formation, droplet pinning, and lens formation. The study revealed that the wave amplitude increases with the Weber number, while the wave velocity is independent of the wave amplitude and droplet Weber number. The wave velocity was found to be lower for thin films and higher for shallow and deep pools, and it also increases with the droplet viscosity for shallow and deep pools. The study identified three distinct types of waves: primary, leading, and trailing waves, each with different velocities and amplitudes. In comparing the experimental wave velocities with the theoretical model, it was observed that the experimentally measured wave velocity corresponded to the different critical velocities observed at corresponding pool heights. The velocity of the primary wave aligns with the mean phase velocity, while the velocities of the leading and trailing waves correspond to the group velocity predicted by the wave dispersion equations.

We conduct experimental and numerical investigations to understand how subperiodic variability in oscillatory flows drives fluid stirring and facilitates the transition to chaos in slow viscous flows. Our work introduces pulsatile fluid flow through a bifurcating network called the T-section model, which incorporates piston assemblies driven by stepping motors. In this configuration, the velocity disparity between the pistons—defined by the phase angle $\varphi$—is the basis for subperiodic variability. The interplay between $\varphi$ and the normalized oscillation frequency $\mathfrak{f}$ governs the transition from regular to chaotic trajectories. Our analyses identify that this subperiodic variability within the bifurcating T-section setup results in the coexistence of open and closed streamlines, with periodic switching between them effectively driving stretch-and-fold mechanisms that are crucial for stirring. We have created a regime map in the $\mathfrak{f}\text{−}\varphi$ parameter space to identify regions of chaotic and nonchaotic stirring while also delineating the boundaries of the chaotic zone. These findings highlight the significant role of subperiodic variability in enhancing stirring within viscous flows, with broad implications for transportation and mixing processes in physical, biological, and industrial systems.

In an experiment on a turbulent jet, we detect interfacial turbulent layers in a frame that moves, on average, along with the turbulent-nonturbulent interface. This significantly prolongs the observation time of scalar and velocity structures and enables the measurement of two types of Lagrangian coherent structures. One structure, the finite-time Lyapunov field (FTLE), quantifies advective transport barriers of fluid parcels while the other structure highlights barriers of diffusive momentum transport. These two complementary structures depend on large-and small-scale motion and are therefore associated with the growth of the turbulent region through engulfment or nibbling, respectively. We detect the turbulent-nonturbulent interface from cluster analysis, where we divide the measured scalar field into four clusters. Not only the turbulent-nonturbulent interface can be found this way, but also the next, internal, turbulent-turbulent interface. Conditional averages show that these interfaces are correlated with barriers of advective and diffusive transport when the Lagrangian integration time is smaller than the integral timescale. Diffusive structures decorrelate faster since they have a smaller timescale. Conditional averages of these structures at internal turbulent-turbulent interfaces show the same pattern with a more pronounced jump at the interface indicative of a shear layer. This is quite an unexpected outcome, as the internal interface is now defined not by the presence or absence of vorticity, but by conditional vorticity corresponding to two uniform concentration zones. The long-time diffusive momentum flux along Lagrangian paths represents the growth of the turbulent flow into the irrotational domain, a direct demonstration of nibbling. The diffusive flux parallel to the turbulent-nonturbulent interface appears to be concentrated in a diffusive superlayer whose width is comparable with the Taylor microscale, which is relatively invariant in time.

Predicting drag caused by turbulent flow over rough surfaces remains one of the most challenging problems in engineering due to the intricate interactions between turbulent flow and surface roughness. This complexity is further heightened by the wide variability in rough-wall topographies and the roughness statistics required to accurately represent rough surfaces. In this study, we propose an approach to rough-wall modeling for drag prediction using deep convolutional autoencoders. We first compress complex data of 93 distinct rough surfaces into a low-dimensional space of just three variables using an autoencoder. These rough-wall topographies are obtained from various experiments and direct numerical simulation studies. We then utilize this reduced-order representation and relate these to the equivalent sandgrain roughness height $\left(k_s\right)$ with the help of a feedforward neural network. The predictive accuracy of the resulting rough-wall model is further assessed against unseen rough surfaces generated from reconstructions obtained from the latent space in unexplored areas. We observe that the formulated rough-wall model predicts $k_s$ values for these previously unseen surfaces to a reasonable accuracy. The present findings suggest the potential for ultra-low-dimensional data-driven representations of complex surface roughness and demonstrate their relevance in constructing generalizable predictive models for rough-wall bounded turbulence.

One of the most intriguing characteristics of red blood cells is their ability to form asymmetric, rotating slipper states even in symmetric channel flows. Here we argue that the common association of these slipper states with the well-understood tank-treading in shear flow is incomplete. We use boundary-integral simulations for a systematic decomposition of the channel flow field into linear shear and quadratic curvature components. Our findings show that a fore-aft asymmetry induced by flow curvature is a plausible mechanism behind the formation of slipper shapes.

When a droplet is laden with particles, evaporation dynamics depends not only on vapour diffusion in the atmosphere, substrate stiffness, and thermocapillarity but also on local particle concentration. Here, based on lubrication theory and the finite-element method, we derive evolution equations for the droplet profile, the displacement of the elastic solid, the particle concentration in the bulk and the profile of deposited particles. These evolution equations are coupled with the Laplace equation for the vapor concentration in the gas phase solved in two dimensions. A precursor film models the contact line motion. Our simulations indicate that on softer substrates, a secondary wetting ridge forms, with its height increasing in response to greater substrate deformability, higher particle loads, and stronger thermocapillary stresses. The substrate's deformability can also lead to a spontaneous symmetry break, which becomes more pronounced as initial particle loads decrease. This symmetry breaking may be accompanied by an oscillatory droplet motion driven by an instability in the contact line region, an effect observed only in particle-laden droplets. We demonstrate that this oscillatory instability arises from a complex interplay among capillary forces at the liquid-gas interface, the substrate's elastic forces, and osmotic pressure effects.

We conduct a study on linear thermocapillary instability for a two-dimensional gravity-driven viscous film flowing over a uniformly heated impermeable plane, even though the film is contaminated with a soluble surfactant. Specifically, we revisit the recent study by S. D'Alessio et al. [J. Fluid Mech. 887, A20 (2020)] to decipher the role that the soluble surfactant plays in different unstable temporal modes accountable for thermocapillary instabilities. In the present study, we have identified a total of four dominant temporal modes, namely, the H-mode (surface mode), shear mode, and two additional thermocapillary S- and P-modes. From the long-wave stability analysis, we can report that the surfactant Marangoni number stabilizes, whereas the thermal Marangoni number destabilizes the H-mode instability. As a result, we establish a mathematical connection between the surfactant and thermal Marangoni numbers, demonstrating that the onset of H-mode instability is independent of the effects of heat transfer and soluble surfactant. Moreover, we see that the nondimensional base surface surfactant concentration ${\mathrm{\Gamma }}_E$ has a dual role in the H-mode instability in its permissible range. However, the numerical results based on the Chebyshev spectral collocation method in the arbitrary wave-number regime reveal that the measure of surfactant solubility parameter ${\xi }_s$ stabilizes the H-mode, S-mode, P-mode, and shear-mode instabilities by diminishing their unstable zones as long as it increases. However, the parameter ${\mathrm{\Gamma }}_E$ exhibits both stabilizing and destabilizing roles in the H-mode, S-mode, P-mode, and shear-mode instabilities. In particular, increasing ${\mathrm{\Gamma }}_E$ has a stabilizing impact, but eventually, it becomes destabilizing as it approaches very close to ${\mathrm{\Gamma }}_E\approx 1$. Finally, a modal coalescence happens between the H-mode and the S-mode for a specific set of parameter values.

Direct numerical simulations have been conducted in order to quantify effects caused by upstream uncertainties on flow past a circular cylinder confined within a duct. The Reynolds number considered is ${\text{Re}}_b=630$ based on the bulk velocity, cylinder diameter, and kinematic viscosity. Two geometric parameters also govern the flow, being the blockage ratio, ratio of cylinder diameter to duct height, and aspect ratio, ratio of duct width to cylinder diameter, which have been fixed to 0.6 and 4.44, respectively. Two sources of uncertainties were assumed to affect measurement of flow statistics: spanwise asymmetries in upstream flow profiles and location of measurement plane. We find that a combination of both uncertainties is responsible for variations at the end of the recirculation region in time-averaged velocity and crossflow velocity fluctuations measured along the center line of the midspan plane. In the near wake, spanwise recirculating structures were found, resulting in variations of time-averaged streamwise velocity with measurement plane uncertainties. However, these were found to only be partially responsible for discrepancies with the prior literature. It is argued that limitations of 2D2C PIV for such a flow configuration also contribute to discrepancies observed.

Motivated by previous work on kinetic energy cascades in the ocean, atmosphere, plasmas, and other fluids, we develop a spatiotemporal spectral transfer tool that can be used to study scales of variability in generalized dynamical systems. In particular, we use generalized time-frequency methods from signal analysis to broaden the applicability of frequency transfers from theoretical to practical fluids applications such as the study of observational data or simulation output. We also show that triad interactions in wavenumber used to study kinetic energy and enstrophy cascades can be generalized to study triad interactions in frequency or wavenumber frequency. We study the effects of sweeping on the locality of frequency transfers and frequency triad interactions to better understand the locality of spatiotemporal frequency transfers. As an illustrative example, we use the spatiotemporal spectral transfer tool to study the results of a simulation of two-dimensional homogeneous isotropic turbulence. This simulated fluid is forced at a well-defined wavenumber and frequency with dissipation occurring at both large and small scales, making this one of the first studies of “modulated turbulence” in two dimensions. Our results show that the spatiotemporal transfers we develop in this paper are robust to potential practical problems such as low sampling rates or nonstationarity in time series of interest. We anticipate that this method will be a useful tool in studying scales of spatiotemporal variability in a wide range of fluids applications as higher resolution observations and simulations of fluids become more widely available.

To investigate the effects of finite arithmetic precision on large-scale direct numerical simulations (DNS) of three-dimensional turbulence using a spectral method, we performed a comparative analysis between two sets of DNS: one using single arithmetic precision (AP) and the other using double AP. Each set simulated the turbulent flows of an incompressible fluid under periodic boundary conditions at Reynolds numbers ($R_{\lambda }$) of 170 and 268, with a resolution defined by $k_{\max }\eta =4$, where $k_{\max }$ denotes the maximum wave number in the DNS and $\eta$ represents the Kolmogorov length scale. Special focus was given to the temporal evolution of the maximum values of local enstrophy ($\mathrm{\Omega }$) and local energy dissipation rate ($\epsilon$), as well as the moments $\langle {\mathrm{\Omega }}^p\rangle$ and $\langle {\epsilon }^p\rangle$. The comparison shows that at $R_{\lambda }=268$, the differences in the time-dependent maximum values between double AP and single AP DNSs are negligible for short duration ($t<1.6T$), where $T$ denotes the eddy turnover time. However, these differences become significant for longer duration ($t>1.6T$). For moments with $p<4$, no notable differences were observed, but significant discrepancies emerged for $p>4$ and $t>1.6T$. In DNSs at $R_{\lambda }=170$, the differences in both maximum values and moments were insignificant throughout the entire simulated time range. The results in the present paper suggest that the difference becomes more significant in DNSs of turbulence at higher Reynolds numbers.

The velocity gradient tensor (VGT) has been extensively employed to describe many fundamental and intrinsic spatial characteristics of turbulence. However, the temporal evolution characteristics following fluid particles are crucial for understanding the dynamics of turbulence. Therefore, we utilize the acceleration gradient tensor (AGT) including both temporal evolution and spatial characteristics of flow as a tool to probe turbulence. Our focus is directed towards the flow characteristics described by three principal invariants of the AGT. It is confirmed that the AGT principal invariants include not only the spatial characteristics described by VGT principal invariants, but also the evolution features such as the temporal evolutions of dilatation, strain rate, and rotation rate. Moreover, the statistical analysis of AGT principal invariants is conducted in compressible channel flows to discern the flow characteristics affected by compressibility. The distributions of normalized AGT principal invariants with different bulk Mach numbers, ${\text{Ma}}_b$, collapse together far from the viscous sublayer but exhibit significant differences in the viscous sublayer. These differences are attributed to the intense temporal evolutions of dilatation, strain rate, and rotation rate at high ${\text{Ma}}_b$, induced by the generation of alternating positive and negative structures (APNSs) near the wall in high-speed flows with strong compressibility. By examining the principal invariants of filtered AGT, it is confirmed that the compressibility effect on flow characteristics near the wall becomes negligible as the flow scale increases. We further prove that AGT is determined by the spatial nonuniformities of the density and the force acting on the fluid particles. Hence, the compelling spatial nonuniformity of the force in the viscous sublayer has been identified as the primary dynamics driving the intense temporal evolutions and the generation of APNSs at high ${\text{Ma}}_b$.

Our experiments on a sphere falling from rest under gravity in Stokes flow show the development of unsteady flow structures and significant history effects. Unlike in steady Stokes flow, we observe a vortex ring forming around the sphere and drifting away, which is captured by our unsteady Stokes theory. Consistently, we observe an algebraic, not exponential, relaxation rate of the sphere to its terminal velocity, validating the Basset-Boussinesq-Oseen equation, which is based on unsteady Stokes theory. As the Reynolds number nears unity, the vortex ring lags behind the sphere, departing from Stokesian theory, though the sphere's algebraic response persists. These findings are critical for particle interactions in the Stokes limit.