A phenomenological model for the heat transfer coefficient in turbulent pipe flow of power-law (PL) fluids is developed. The analysis employs a two-layer approach, separating the flow into a viscosity-dominated inner region and a turbulence-dominated outer region. By leveraging a scaling framework rooted in Kolmogorov's theory of turbulence, we derive a semi-empirical heat transfer correlation analogous to the Prandtl-Taylor analogy for Newtonian fluids. A key contribution is the introduction of a new dimensionless group, the power-law Prandtl number (${\mathrm{Pr}}_{PL}$), defined to be independent of the Reynolds number, thereby allowing for a more fundamental characterization of heat transfer in these systems. The model incorporates a single calibration function, $c_h\left(n\right)$, which depends on the power-law index $n$ and captures the influence of non-Newtonian rheology on the sublayer-to-core thermal resistance partitioning. This function is calibrated using established literature data for Carbopol solutions. The proposed correlation demonstrates excellent agreement with experimental results across a wide range of Reynolds and Prandtl numbers, achieving a normalized root-mean-square error of 1.14% with the polynomial calibration and exhibiting superior predictive accuracy compared to classical models.

Ionic liquids (ILs), characterized by high electrical conductivity, negligible vapor pressure, and wide physicochemical tunability, hold promise for applications in cooling, coating, and energy systems where droplet impact is critical. We experimentally investigate the effect of an external electric field, $\mathbf{E}$ (0–10 kV/cm), oriented parallel to the impact velocity, on the deformation and impact dynamics of high-conductivity IL droplets across a broad range of Weber $(31\le \mathrm{We}\le 629)$ and Ohnesorge $(0.11\le \mathrm{Oh}\le 1.19)$ numbers. The results show that IL droplets undergo pronounced prolate deformation driven by field-induced Maxwell stresses, with the deformation magnitude increasing with electric field strength and travel time but decreasing with surface tension and viscosity. Despite substantial preimpact deformation $(0\le \mathcal{D}\le 0.62)$, the impact outcomes—spreading and splashing—remain largely unchanged. The maximum spreading factor $\left({\beta }_{\text{max}}\right)$ and splashing threshold exhibit negligible changes compared with spherical droplets. No field-induced jetting or breakup is observed, in contrast to previous findings for water droplets under $\mathbf{E}$. The data for ${\beta }_{\text{max}}$ collapse onto the semiempirical scaling ${\beta }_{\text{max}}=0.61{(23.3+\mathrm{We}/\mathrm{Oh})}^{1/6}$, derived from a simplified energy conservation model in which the initial kinetic energy is dissipated by viscous losses. These findings demonstrate that while electric fields strongly alter droplet morphology in flight, the high viscosity of ionic liquids suppresses electrostatic coupling during impact.

Rare and extreme events in turbulent flows play a critical role in transport, mixing, and transition, yet they are notoriously difficult to capture experimentally. Here we report, to our knowledge, the first direct experimental detection of an upstream-directed jet burst in a backward-facing-step (BFS) flow at ${\text{Re}}_h=2100$, using long-duration live optical flow velocimetry (L-OFV). Continuous monitoring over 1.5 h enabled a data-driven definition of extremes as rare velocity probes excursions deep into the observed distribution's tails; in practice, large negative events ($u:Z<-6$, $v:Z<-5$ at $(x,y)=(2h,h/2)$, where $\left|Z\right|\gg 0$ stands for large deviations from the mean value) triggered the live capture of surrounding velocity fields. The recording is triggered when the probes surpass the defined threshold, using live analysis of the velocity fields. The detected event features a jetlike intrusion into the recirculation region initiated by the collapse of a merged Kelvin–Helmholtz vortex and sustained by counterrotating vortices and is accompanied with heavy-tailed probe statistics and simultaneous amplification of fluctuating kinetic energy and enstrophy. While a single event was recorded, underscoring its rarity, the results establish L-OFV as a viable platform for rare-event detection in separated shear layers and document a previously unreported mechanism of upstream jet bursting in BFS flow.

The rectangular exhaust nozzles of stealth aircraft typically feature a V-shaped trailing edge (VTE) design. However, no study has focused on the influences of VTEs on the flow and acoustic fields of hot supersonic rectangular jets. In light of this, implicit large-eddy simulations of a rectangular nozzle with and without VTEs are performed at a hot overexpanded condition with a nozzle pressure ratio of 3.0 and a temperature ratio of 3.0. Three VTE nozzles with different deflection angles are studied. Results show that the VTE nozzles are able to prompt the formation of Mach disks, shorten the lengths of potential cores, enhance or suppress the shear-layer mixing in the minor- or major-axis plane, and accelerate the axis switching due to counter-rotating vortex pairs induced along the long-lip sides. The VTE nozzles efficiently suppress or eliminate the screech tone propagating upstream in the minor-axis plane, and the levels of turbulent mixing noise and Mach wave radiation propagating downstream are also substantially reduced, although the broadband noise radiation is amplified in the major-axis plane. The azimuthally integrated overall sound pressure levels show a decrease in the upstream and downstream directions and an increase in the sideline regions. These influences become more significant when the VTEs deflect from inward to outward. The VTE nozzles suffer the penalties of thrust losses, and the total thrust losses arise from the reduced momentum and pressure components for the inward- and outward-deflected VTEs, respectively. The shear-layer oscillation transitions from the flapping mode to a helical mode and then to an irregular pattern, as the VTEs deflect from inward to outward. The wave-number spectra analysis reveals that the VTE nozzles are capable of affecting energy redistribution during wave interactions, leading to the suppression of upstream-propagating guided-jet modes in the fluid-acoustic feedback loop.

Accurate estimation of near-wall turbulence from limited surface measurements is critical for various practical and experimental applications; however, it remains challenging due to the complex and multiscale characteristics inherent to turbulent flows. Recent developments in data-driven methods have improved performance over traditional linear approaches but are often limited by loss functions that primarily penalize pointwise errors. In this study, we consider a composite, spectrally informed loss function that augments the conventional mean-squared error with terms that explicitly promote statistical consistency of fluctuation levels and spectral energy distributions. This loss function is evaluated using a baseline convolutional network model applied to direct numerical simulation datasets of turbulent open-channel flow at friction Reynolds numbers, ${\text{Re}}_{\tau }=180$ and 550. We show that the considered loss function substantially reduces reconstruction errors of near-wall fluctuation velocity fields obtained from the network model using wall-shear stress and wall-pressure data as inputs. The proposed method recovers up to a threefold improvement in reconstruction accuracy relative to baseline employing mean-squared loss and retains energy content at small-scale structures. We further show that reconstruction accuracy is only mildly affected when the input wall signals are contaminated with 5–100% Gaussian noise and retains energy content in small flow structures when predicting from coarse wall inputs, indicating that the trained models generalize robustly to noisy, experimentally relevant conditions. Our results demonstrate that properly designed loss functions can improve reconstruction accuracy, highlighting the importance of training strategy in achieving efficient, high-performance neural-network models for practical nonintrusive sensing applications. The present study also provides a foundation for future extensions with advanced architectures, such as generative or diffusion-based models to further improve reconstruction of turbulent flow fields in complex settings.

When two species are initially separated and react according to a second-order reaction scheme, $\mathrm{A}+\mathrm{B}\to \mathrm{C}$, a dynamical reaction zone or $\mathit{reaction}\mathit{front}$ forms. In horizontally oriented layers filled with liquid miscible mixtures, natural convection plays a crucial role. In particular, when the three chemical species (A, B, and C) involved in the reaction affect the surface tension at the air-liquid interface, currents due to Marangoni forces emerge and couple to the preexisting reaction-diffusion dynamics of the front. The system dynamics is then described by coupling the reaction-diffusion-convection equations of the three species to the incompressible Navier-Stokes equations subject to Marangoni stresses at the air-liquid interface. In this work, we analyze the surface tension profiles and the structure of the flows that result from this coupling, such as the number of convective rolls, their rotational direction and relative size, in the general scheme when all the species diffuse at different rates. We show how differential diffusion effects induce and control complex Marangoni-driven convective patterns in solution. We connect the structure of the reactive flows to the surface tension profiles and their monotonic properties. Independently of the model parameters, we find that the number of extrema $\left(m\right)$ and of convective rolls $\left(n\right)$ are related as $n\le m+1$ for all times, with $n$ and $m$ defined as positive integers. The analysis is simplified when the monotonic properties of the surface tension profiles are conserved in time, for which we show that $n=m+1$ for all times. By varying the layer thickness with differential diffusion, more complex convective structures also generically emerge.

A linear flow control strategy designed for weak disturbances may not remain effective in sequences of strong disturbances due to nonlinear interactions, but it is sensible to leverage it for developing a better strategy. In the present study, we propose a transformer-based reinforcement learning framework to learn an effective control strategy for regulating aerodynamic lift in arbitrarily long gust sequences via pitch control. The random gusts produce intermittent, high-variance flows observed only through limited surface pressure sensors, making this control problem inherently challenging compared with stationary flows. The transformer addresses the challenge of partial observability from the limited surface pressures. We demonstrate that the training can be accelerated with two techniques—pretraining with an expert policy (here, linear control) and task-level transfer learning (here, extending a policy trained on isolated gusts to multiple gusts). We show that the learned strategy outperforms the best proportional control, with the performance gap widening as the number of gusts increases. The control strategy learned in an environment with a small number of successive gusts is shown to effectively generalize to an environment with an arbitrarily long sequence of gusts. We investigate the pivot configuration and show that quarter-chord pitching control can achieve superior lift regulation with substantially less control effort than midchord pitching control. Through a decomposition of the lift, we attribute this advantage to the dominant added-mass contribution accessible via quarter-chord pitching.

The process of breaking in surface gravity waves can be characterized by two distinct stages. Breaking onset is defined as the first visible surface manifestation of breaking. This is preceded by breaking inception, which is characterized by the initiation of an irreversible process within the crest that inevitably leads to breaking. Breaking inception diagnostics formulated using the local energetic, kinematic, and geometric properties of the wave crest have recently been proposed that appear to provide generic parametric threshold estimates and may facilitate the effect of wave breaking on larger-scale processes to be parametrized. In a recent numerical study [McAllister et al., J. Fluid Mech. 974, A14 (2023)], a breaking inception diagnostic was proposed based on a threshold value for the maximum local interface angle $\mathrm{\Theta }$. Utlizing an ensemble of breaking wave packet numerical simulations, we extend these findings to include surface tension effects, which are an inescapable feature of ocean surface waves and are known to have a non-negligible effect on wave geometry. The ensemble consists of narrow bandwidth, short wavelength $(\sim 1\mathrm{m})$ gravity wave packets with varying packet size, water depth, and wind speed forcing. For this data, we show that ${\mathrm{\Theta }}_{\text{th}}={60}^{\circ }$ is a robust threshold value for the initiation of breaking inception; twice the magnitude of that for waves in the absence of surface tension. We explore this result in the context of the kinematic inception parameter $B$ [Barthelemy et al., J. Fluid Mech. 841, 463 (2018)] and show that the kinematic and geometric methods are related through the relative flux of energy into the wave crest.

An instability may arise when a hot viscous fluid enters a thin gap and cools through heat transfer to a colder surrounding environment. Fluids whose viscosity increases strongly upon cooling create a positive feedback in which warmer regions flow faster and cool more slowly, leading to the formation of thermoviscous “fingers.” Here we investigate this mechanism in the long-time, small-Biot-number regime, where cooling through the plates is weak but acts over sufficiently long times that the temperature becomes nearly uniform across the gap. This asymptotic limit enables a depth-averaged description that incorporates both thermal diffusion and hydrodynamic (Taylor) dispersion, allowing us to analyze the dependence of the instability on the Péclet number, viscosity contrast, and wall cooling rate. Using numerical simulations of temperature-dependent viscous flow in a Hele-Shaw geometry, we show that fingering instabilities emerge in response to small inlet perturbations within a range of Péclet numbers and viscosity contrasts. From linear stability analysis we find the dispersion relation and quantify how the fastest growth rate ${\gamma }_{max}$ and corresponding wave number $k_{max}$ depend on the global parameters. We further derive analytical expressions for ${\gamma }_{max}$ and $k_{max}$ in the limit of high Péclet number and large viscosity contrast, revealing the scaling behavior that controls pattern selection. These results clarify the physical mechanisms driving thermoviscous fingering in the small-Biot-number regime and have implications for systems in which temperature-dependent viscous fluids are confined within narrow gaps, such as lubrication flows in mechanical components and magma invasion in small-scale fissures.

Turbulence in strongly coupled plasmas (SCP) poses a unique challenge due to long-range interparticle interactions that impart viscoelastic properties to the medium. We report the first observation of intermittent viscoelastic turbulence in such plasmas using large-scale three-dimensional molecular dynamics simulations. In driven dissipative SCP, we observe scale-dependent flow dynamics arising from the interplay between potential (elastic) and viscous stresses at the particle level. Unlike conventional turbulence, which is governed by inertia and viscosity, SCP showcases distinct energy transfer processes due to its inherent viscoelasticity. Both kinetic and elastic energy spectra exhibit power-law scaling $E\left(k\right),\mathrm{\Phi }\left(k\right)\propto k^{-3.5}$. In real space, velocity structure functions demonstrate nontrivial scaling, and the probability distribution functions of velocity increments show significant deviations from Gaussian behavior, particularly in the tails, indicating intermittent behavior. These findings establish SCP as a platform for studying viscoelastic turbulence, with broad relevance to astrophysical plasmas, soft matter, and nonequilibrium statistical physics.

Bubbles bursting at the surface of the ocean produce drops that heavily influence ocean-atmosphere interactions. One of the mechanisms through which drops are formed is called jet drop production, where the collapse of the bubble cavity leads to the formation of a fast upward jet that breaks to form drops. While isolated bubble bursting has been extensively studied, bubbles are often found in rafts (for instance, in the ocean surface or a sparkling wine glass) and the understanding of collective effects remains more limited. We experimentally investigate how jet drop formation is modified by the presence of neighboring bubbles during the collapse. With the help of multiple high-speed views of the collapsing bubble, we show how a change of cavity shape during collapse leads to the selection of smaller, faster, and more numerous drops. The size of the emitted drops is monotonically reduced with increasing number of neighboring bubbles (up to six for hexagonal packing) with the size reduction reaching a factor of 5. The drop size distribution associated with bubbles arranged in rafts of various sizes is therefore much wider than in the case of isolated bubbles and with a peak shifted to smaller sizes.

Wall-bounded turbulence is characterized by coherent, worm-like structures such as hairpin vortices. The attached-eddy model provides a successful statistical framework for the log-law region, yet the complex geometry and multiscale nature of wall-turbulence vortices remain challenging for physics-based modeling. Here, we model wall turbulence as an ensemble of complex vortices, introducing a systematic approach to constructing turbulence fields enriched with hierarchically organized hairpin vortex packets. The geometry and organization of the vortex packets are calibrated to match observations, enabling the model to reproduce both attached and detached motions through a height-dependent core-size variation. Our model successfully reproduces the key statistical and structural features of wall turbulence, matching direct numerical simulations of turbulent channel flow at friction Reynolds numbers from 1000 to 10 000. More importantly, it also reveals new insights into the coherent structures, emphasizing the role of vortex geometry, packet organization, and hierarchy in setting the attached and detached balance, meandering streaks and inclination angles, superstructure alignment, and the overall partition of contributions. Moreover, the constructed channel turbulence rapidly transitions into fully developed turbulence in direct numerical simulation, demonstrating its physical self-consistency and practical utility for initializing high-fidelity simulations. This approach significantly reduces computational costs associated with turbulence development while providing a flexible framework for testing and advancing turbulence models based on vortex structures.

Laser-sustained plasmas (LSP) enable high-resolution optical detection in chip fabrication, yet their spatiotemporal oscillations remain mysterious and strongly impede the development of high-brightness ultrabroadband stabilized light sources. We elucidate the fundamental origin and scaling of periodic oscillations in LSP through experiments and simulations. High-speed shadow imaging and validated multiphysics modeling reveal that the LSP oscillations stem from buoyancy-driven vortex dynamics, where periodic toroidal vortex ring formation and detachment—induced by the core-to-boundary density gradient—directly modulate plasma brightness. Crucially, we find that the Prandtl number (Pr) determines the LSP oscillation threshold (Pr > 0.6), while the Péclet number governs the oscillation patterns and frequency. A generalized scaling law incorporating the LSP core-to-boundary gas density ratio that enhances predictive accuracy is established. Further, we disentangle distinct thermal property roles, that is, a larger specific heat capacity promotes oscillations, whereas an increasing thermal conductivity suppresses instability. This study provides a mechanistic framework for LSP stability, enabling targeted suppression strategies to advance LSP-based light sources for high-resolution optical detection and spectroscopy in the field of the semiconductor industry.

Two-dimensional simulations with detailed chemistry are performed for hydrogen-oxygen-argon mixtures in channels with square obstacles to investigate how obstacle configurations influence detonation attenuation and flame acceleration. The results indicate that at a fixed blockage ratio, increasing obstacle dispersion enhances detonation attenuation by fragmenting the detonation front into discontinuous segments. Two distinct reinitiation modes in postobstacle sections are identified: for concentrated obstacles, reinitiation is directly driven by diffracted wave collisions; for dispersed obstacles, secondary reaction zones form behind the leading shock and develop into localized detonations through wave interactions and viscous dissipation. Subsequently, the stable propagation characteristics of detonation waves after sufficient attenuation within an extended obstacle section are examined. As the blockage ratio increases, propagation transitions from quasidetonation to choking regimes. In the quasidetonation regime, detonation undergoes repeated failures and reinitiations. In the choking regime, the attenuated wave degenerates into a nondetonative but coupled shock–reaction-front complex. The critical blockage ratio (${\xi }_{\mathrm{C}}$) determining the propagation regimes is identified, and the results indicate that ${\xi }_{\mathrm{C}}$ decreases with increasing detonation cell width, indicating that detonations with smaller characteristic scales are more capable of sustaining propagation through obstacle sections. The ratio of obstacle gap to cell width (δ/λ) is employed as a scaling parameter to quantitatively characterize velocity deficits and determine critical conditions for regime transitions. Finally, flame acceleration simulations confirm that obstacles corresponding to quasidetonation and choking regimes lead to detonation formation and nondetonative propagation, respectively. This demonstrates that obstacles exert competing effects, namely, shock attenuation and flame acceleration, and their balance determines the eventual propagation regime.

In this study, we present a deep reinforcement learning (DRL) framework for closed-loop control of canonical flow past a circular cylinder under fully developed turbulent conditions. To enable computationally efficient training for each episode, we employ a graphics processing unit–accelerated flow solver based on the generalized interpolation-supplemented cascaded lattice Boltzmann method. Moreover, we further employ a two-stage exploration process, beginning with a coarse mesh and minimal spanwise flow resolution before progressing to a finer mesh with well-resolved spanwise flow. This approach significantly reduces computational time, making the implementation of intricate and time-consuming DRL-guided active flow control (AFC) feasible in turbulent flows. The resulting DRL-guided AFC achieves an optimal drag reduction of $55\%$ and a lift fluctuation reduction of $26\%$ due to recovered pressure at the rear side, attenuated wake vortical structures, reduced recirculation bubble length, and mitigated downstream turbulent fluctuations. Additional numerical experiments applying second-order smoothing to the DRL-determined control actions confirm that effective turbulent flow control does not necessarily require high-frequency actuation. This work verifies DRL as a powerful machine-learning tool for tackling challenges of AFC in turbulent flows. The revealed phenomena also highlight rich opportunities for further explorations of the underlying physics of flow under control.

The rotor-stator cavity represents a critical component in engine secondary air systems. Due to its large-volume characteristics, the parameter response and flow evolution during engine transient processes exhibit distinctive features that differ from steady-state conditions. Therefore, this study investigates the transient flow evolution in rotor-stator cavities subjected to impulsive changes in rotation rate through theoretical analysis and direct numerical simulation. We examine three distinct conditions: infinite cavities and enclosed cavities under both laminar and turbulent conditions, exploring their transient processes during transient from one stable condition to another. In infinite cavities with initial steady state, the transient process experiences rapid momentum and mass exchange between rotor and stator boundary layers. Compared to start-from-rest scenarios, the transient duration is significantly reduced. For enclosed laminar cavities, the impulsive change generates disturbances at the shroud that excite circular waves propagating radially inward before restabilizing while maintaining laminar conditions. Due to the presence of the shroud and hub in enclosed cavity, the transient process captures circular waves that cannot be observed in infinite cavities, while the presence of the shroud accelerates the transition process. In turbulent enclosed cavities, we observe that following an impulsive increase in rotation rate, the rotor boundary layer adjusts rapidly while the stator side experiences delayed adaptation with small-scale vortex breakdown initiating at high radial positions. After the transient phase ends, the rotor boundary layer maintains laminar characteristics while turbulence intensity increases on the stator side. Although the rotor boundary layer maintains laminar conditions in both laminar and turbulent cavities, these states differ significantly. Under turbulent conditions, disturbances from the turbulent stator penetrate the core region, generating perturbations on the disk. These transient phenomena, distinctly different from steady-state behavior, may significantly impact engine efficiency and operational safety in practical applications. This work provides fundamental insights into impulsive rotation changes in rotor-stator systems, establishing a foundation for future investigations of more complex transient conditions.

Direct numerical simulations of a Couette-Poiseuille flow were performed near the transition to turbulence to investigate the nonlinear relationship between streak waviness and rolls. This relationship is a key step in Waleffe's model for a self-sustaining process. Simulations were conducted for Reynolds numbers ranging from 500 to 940, and a range of initial perturbation amplitudes was used. In these simulations, the streaks, rolls, and streak waviness initially grow. The optimal time for this growth closely matches the linear transient growth period for small perturbations but is much shorter when the initial perturbations are large and highly nonlinear. For higher Reynolds numbers and large initial perturbations, the velocity field reaches a turbulent steady state, while in the remaining cases, the flow relaminarizes. The main result is that the waviness of the streaks is a quadratic function of the rolls, provided that the roll amplitude is sufficiently large.

The stratified inclined duct (SID) provides a canonical setup for sustained, buoyancy-driven exchange flow between two reservoirs of different density, and emerges as a paradigm in geophysical fluid dynamics. Yet, the flow dynamics remain unclear in the highly turbulent regime; laboratory experiments can access this regime but they lack resolution, while direct numerical simulations (DNSs) at realistically high Prandtl number, $\mathrm{Pr}=7$ (for heat in water), have not achieved sufficiently high Reynolds numbers $\mathrm{Re}$. We conduct three-dimensional DNSs up to $\mathrm{Re}=8000$ and observe the transition to the so-called ultimate regime of turbulent convection as evidenced by the Nusselt number scaling $\mathrm{Nu}\sim {\mathrm{Ra}}^{1/2}$, indicating substantially enhanced transport. At the transition, the shear Reynolds number, a key parameter characterizing boundary layer (BL) dynamics, exceeds the threshold range of 420 for turbulent kinetic BLs with the emergence of logarithmic velocity profiles. The nature of the transition toward ultimate SID flow is non-normal-nonlinear, i.e., subcritical and hysteretic, as is typical for the transition to fully turbulent shear flows. Our work connects SID flow with the broader class of wall-bounded turbulent convection flows and gives insight into mixing properties in the vigorously turbulent regime encountered in oceanographic and industrial flows.

Recent investigations have established the physical relevance of spatially localized instability mechanisms in fluid dynamics and their potential for technological innovations in flow control. In this paper, we show that the mathematical problem of identifying spatially localized optimal perturbations that maximize perturbation-energy amplification can be cast as a sparse (cardinality-constrained) optimization problem. Unfortunately, cardinality-constrained optimization problems are nonconvex and combinatorially hard to solve in general. To make the analysis viable within the context of fluid dynamics problems, we propose an efficient iterative method for computing suboptimal spatially localized perturbations. Our approach is based on a generalized Rayleigh quotient iteration algorithm followed by a variational renormalization procedure that reduces the optimality gap in the resulting solution. The approach is demonstrated on a subcritical plane Poiseuille flow at $\text{Re}=4000$, which has been a benchmark problem studied in prior investigations on identifying spatially localized flow structures. Notably, we find that a subset of the perturbations identified by our method yield a comparable degree of energy amplification as their global counterparts. We anticipate our proposed analysis tools will facilitate further investigations into spatially localized flow instabilities, including within the resolvent and input-output analysis frameworks.

We revisit Squire's classical theorem, in which he showed that 2D modes are the more unstable ones, i.e., occur at a smaller Reynolds number, compared to 3D modes. Since this only applies to temporal modes, we extend it to spatiotemporal modes. This results in a new class of subcritical modes, where the 3D modes may occur at a smaller Reynolds number than the corresponding 2D modes. Three special features of the new modes become apparent: (i) they exhibit reflection symmetry breaking in the spanwise ($z$) direction; (ii) the spanwise complex wave number $\beta$ is proportional to the streamwise wave number $\alpha$ ($x$ direction), but with an imaginary prefactor. Hence, real and imaginary parts of $\alpha$ swap their roles for $\beta$, i.e., linking the wave propagation of one direction to the growth in the other direction; and (iii) ${\beta }_r$ constitutes a wave number and therefore refers to an associated wavelength in the $z$ direction. As a consequence, a finite box width in the $z$ direction controls the critical 3D Reynolds number, and, in turn, certain perturbations in slender boxes may not be permitted. We have further analyzed the second part of Squire's theorem, i.e., that the homogeneous Squire equation admits no temporally unstable modes, and this is also lost with the new spatiotemporal modes. Finally, it is emphasized that the use of a single new mode of the above-mentioned type is not meaningful, since its spatial part leads to a diverging perturbation energy. Nevertheless, specific combinations of these modes, such as in Briggs' theory, may lead to new subcritical transitions.