| Volume 8, Issue 1 January 2023 | |
Advertisement | Witness groundbreaking physics research, network with potential employers, and prepare for future success at March Meeting 2023. Regular registration rates for one of the largest and most exciting conferences in physics is open through February 28. Register today » | | | | | | Not an APS member? Join today to start connecting with a community of more than 50,000 physicists. | | | | EDITORIALS AND ANNOUNCEMENTS | Eric Lauga and Beverley McKeon Phys. Rev. Fluids 8, 010001 (2023) – Published 24 January 2023 | | | Featured in Physics Editors' Suggestion Gen Li, Dmitry Kolomenskiy, Hao Liu, Ramiro Godoy-Diana, and Benjamin Thiria Phys. Rev. Fluids 8, 013101 (2023) – Published 13 January 2023  | Intermittent swimming has been recognized as a strategy for fish to enhance their energetical efficiency. In this study, a hybrid computational fluid dynamic model is used to assess the swimming performance in intermittent swimming parametrically and quantitatively. The results show that the energetical performance of intermittent swimming can be better than that of continuous swimming, but also that an unoptimized intermittent gait may become very energetically expensive. | | | | | | Featured in Physics Editors' Suggestion Andrew P. Grace, Marek Stastna, Kevin G. Lamb, and K. Andrea Scott Phys. Rev. Fluids 8, 014502 (2023) – Published 5 January 2023  | Freshwater has been shown to have a maximum density at about four degrees Celsius, and this leads to a phenomenon known as cabbeling. Cabbeling occurs when masses of water on different sides of the temperature of maximum density mix and create a denser mass. What happens when intruding and ambient temperatures in a gravity current are on opposite sides of the temperature of maximum density? How does cabbeling affect the evolution characteristics of gravity currents, and what sort of long term behavior arises? | | | | | | Editors' Suggestion Christopher J. Howland, Roberto Verzicco, and Detlef Lohse Phys. Rev. Fluids 8, 013501 (2023) – Published 4 January 2023  | At steep ice faces submerged in the ocean, convection is driven by differences in salt concentration, yet ablation of the ice is controlled by a mixture of heat and salt fluxes. The differing molecular diffusivities of heat and salt play a key role in setting these fluxes due to thin, diffusive boundary layers at the ice-ocean interface. We analyze such boundary layers by simulating convection between two vertical plates of fixed temperature and salinity, and varying their relative diffusivities, finding that the ratio of fluxes transitions between scaling regimes at a critical Prandtl number. These results provide physical insight for future parameterization of steep ice-ocean interfaces. | | | | | | Editors' Suggestion Lorenzo Betti, Céline Cohen, and Xavier Noblin Phys. Rev. Fluids 8, 013601 (2023) – Published 9 January 2023  | We performed a new experiment of mechanical impact on a plate supporting a breath figure on its bottom surface. Although breath figures mature usually slowly, we show that the droplets pattern evolves here dramatically in a few milliseconds leading to a strong droplets number reduction above a threshold in acceleration. We interpret this by the droplets radii oscillations produced which then induce droplets contacting and coalescing with each other in a fascinating manner. Introducing an effective Bond number, coupling the droplets' average initial radii and the acceleration amplitude, leads to a data collapse on a single master curve. | | | | | | Letter Wenbo Li and Dali Kong Phys. Rev. Fluids 8, L011501 (2023) – Published 30 January 2023  | Rotating convection in spheres or spherical shells has been modeled in order to understand dynamics in planetary or stellar interiors. Existing theories have predicted various different types of symmetry for the dynamics at the onset of thermal convection. The only form of global convection not found so far is axisymmetric and equatorially symmetric. In this Letter, we for the first time, report the discovery of a solution bearing such symmetry. It is the geometric oblateness due to rotational flattening that is the key factor, which had not previously been considered. | | | | | | Laminar and Viscous Flows | Letter Ehud Yariv Phys. Rev. Fluids 8, L012101 (2023) – Published 6 January 2023  | This paper addresses shear flows over a superhydrophobic grating made up of a periodic array of grooves separated by solid slats, with air bubbles trapped in the grooves. At large pressures, the liquid partially invades the grooves, and the menisci are no longer pinned at the edges of the slats. Asymptotic methods and conformal maps are used to derive approximations for the slip length in the limit of small solid fractions. | | | | | | Letter David Greenblatt, Hanns Müller-Vahl, and Christoph Strangfeld Phys. Rev. Fluids 8, L012102 (2023) – Published 19 January 2023  | We identified a new dynamic stall mechanism in surging flows on thick airfoils at low angles-of-attack. Counterintuitively, the laminar separation bubble bursts during early imposition of the favorable temporal pressure gradient, which rapidly drives the bubble aft, rendering it unable to reattach. We also introduced a new generalized pressure coefficient, without which the experimental data are impossible to interpret. | | | | | | Multiphase, Granular, and Particle-Laden Flows | Letter Ashkan Irannezhad, Bauyrzhan K. Primkulov, Ruben Juanes, and Benzhong Zhao Phys. Rev. Fluids 8, L012301 (2023) – Published 11 January 2023  | Wettability exerts fundamental control over multiphase flow in porous media, which has been extensively studied in uniform-wet porous media. In contrast, multiphase flow in mixed-wet porous media is less well-understood. We combine microfluidic experiments and pore-scale simulations to study the displacement of oil by water in a mostly oil-wet porous media patterned with discrete water-wet clusters. Our work demonstrates the complex nature of wettability control in mixed-wet porous media, and it presents experimental and numerical platforms upon which further insights can be drawn. | | | | | | Biological and Biomedical Flows | Featured in Physics Editors' Suggestion Gen Li, Dmitry Kolomenskiy, Hao Liu, Ramiro Godoy-Diana, and Benjamin Thiria Phys. Rev. Fluids 8, 013101 (2023) – Published 13 January 2023 | Intermittent swimming has been recognized as a strategy for fish to enhance their energetical efficiency. In this study, a hybrid computational fluid dynamic model is used to assess the swimming performance in intermittent swimming parametrically and quantitatively. The results show that the energetical performance of intermittent swimming can be better than that of continuous swimming, but also that an unoptimized intermittent gait may become very energetically expensive. | | | | | | Compressible and Rarefied Flows, Kinetic Theory | Wei Su, Livio Gibelli, Jun Li, Matthew K. Borg, and Yonghao Zhang Phys. Rev. Fluids 8, 013401 (2023) – Published 31 January 2023  | In nonequilibrium flow of dense gases composed of hard-sphere molecules, a kinetic model is needed to simplify the Enskog equation and reduce computational cost, while also preserving the most important physical properties of high-density gas systems, including the correct transport coefficients. Moreover, density inhomogeneity in the vicinity of a solid boundary, which may influence transport processes, has been overlooked. We have developed such a model using a method based on density functional theory. The inhomogeneous density near the solid wall is found to enhance the oscillation amplitude, while the bulk viscosity causes stronger sound wave attenuation. | | | | | | Editors' Suggestion Christopher J. Howland, Roberto Verzicco, and Detlef Lohse Phys. Rev. Fluids 8, 013501 (2023) – Published 4 January 2023 | At steep ice faces submerged in the ocean, convection is driven by differences in salt concentration, yet ablation of the ice is controlled by a mixture of heat and salt fluxes. The differing molecular diffusivities of heat and salt play a key role in setting these fluxes due to thin, diffusive boundary layers at the ice-ocean interface. We analyze such boundary layers by simulating convection between two vertical plates of fixed temperature and salinity, and varying their relative diffusivities, finding that the ratio of fluxes transitions between scaling regimes at a critical Prandtl number. These results provide physical insight for future parameterization of steep ice-ocean interfaces. | | | | | | Drops, Bubbles, Capsules, and Vesicles | Editors' Suggestion Lorenzo Betti, Céline Cohen, and Xavier Noblin Phys. Rev. Fluids 8, 013601 (2023) – Published 9 January 2023 | We performed a new experiment of mechanical impact on a plate supporting a breath figure on its bottom surface. Although breath figures mature usually slowly, we show that the droplets pattern evolves here dramatically in a few milliseconds leading to a strong droplets number reduction above a threshold in acceleration. We interpret this by the droplets radii oscillations produced which then induce droplets contacting and coalescing with each other in a fascinating manner. Introducing an effective Bond number, coupling the droplets' average initial radii and the acceleration amplitude, leads to a data collapse on a single master curve. | | | | | | Azmaine Iqtidar, Joseph J. Kilbride, Fouzia F. Ouali, David J. Fairhurst, Howard A. Stone, and Hassan Masoud Phys. Rev. Fluids 8, 013602 (2023) – Published 10 January 2023  | Theoretical calculations and experimental measurements reveal that, in any array of identical sessile droplets, the appropriately scaled drying dynamics of the 1st, 2nd, and n-th fastest evaporating droplets are (almost) universal across different arrangements. | | | | | | Diego Taglienti, Fabio Guglietta, and Mauro Sbragaglia Phys. Rev. Fluids 8, 013603 (2023) – Published 24 January 2023  | Droplet deformation at finite capillary numbers is accurately studied in shear flows via Immersed Boundary - Lattice Boltzmann (IB-LB) simulations. A reduced model is introduced to capture nonlinear effects in droplet deformation and orientation. | | | | | | Huiyong Feng, Jianguo Zheng, Bei Wei, Jian Hou, and Haibo Huang Phys. Rev. Fluids 8, 013604 (2023) – Published 31 January 2023  | The distribution of capsules and rheological properties of suspensions in curved tubes are investigated numerically. At limited inertia, the effective viscosity decreases with increasing Reynolds number (Re), which is different from the variation trend in straight tubes. Dean's vortices play an important role. They greatly promote the capsules' circumferential transportation by trapping the capsules into their centers and making the location of maximum azimuthal velocity close to them. | | | | | | Instability, Transition, and Control | Min Chan Kim and Satyajit Pramanik Phys. Rev. Fluids 8, 013901 (2023) – Published 17 January 2023  | Linear and nonlinear analyses on miscible viscous fingering (VF) in a cylindrical packed column is conducted. The effects of boundary conditions on the onset conditions of instability are suggested as a function of log-viscosity ratio and Péclet number. Through two- and three-dimensional (2D and 3D) simulations based on the finite element method, we clearly visualize the effect of the lateral boundary on the onset and the growth of VF. Our 3D simulations explain the experimental results available in the literature by suitably choosing the parameters. | | | | | | Moritz Stieneker, Leon Topp, Svetlana V. Gurevich, and Andreas Heuer Phys. Rev. Fluids 8, 013902 (2023) – Published 18 January 2023  | We present a general mapping approach between the microscopic molecular dynamics model and a mesoscopic model based on the thin-film equation for the description of the dynamics of droplets on partially-wettable, switchable substrates. We apply the mapping approach to the adaption of a droplet to a new wettability upon switching and the coalescence of two droplets making the first step toward quantitative comparisons of dynamics between models acting on mesoscopic and microscopic scales. | | | | | | Interfacial Phenomena and Flows | Liam C. Morrow, Nicolas De Cock, and Scott W. McCue Phys. Rev. Fluids 8, 014001 (2023) – Published 10 January 2023  | Viscous fingering patterns that occur at the interface between two immiscible fluids of differing viscosities in a Hele-Shaw cell are normally studied mathematically via a nonlinear moving boundary problem with a single interface. Here we study a more realistic model which involves a doubly connected region of the more viscous fluid bounded by two interfaces. We simulate this model numerically using a level set method, supported by linear stability analysis and some experiments. Various results are presented for configurations in which the flow is driven by either a pressure difference or by rotating the entire Hele-Shaw cell. | | | | | | Manas Ranjan Behera, Anvith Rao, Anirvan Dasgupta, and Sudipto Chakraborty Phys. Rev. Fluids 8, 014002 (2023) – Published 31 January 2023  | We study the impingement of a liquid drop on a liquid pool and find four new flow structures: (i) blob, (ii) blob and secondary ring, (iii) primary ring, secondary ring and jet, and (iv) primary ring and jet. We have delineated comprehensive pictures of coalescence and splashing regimes and their associated flow structures as a function of the controlling dimensionless numbers in a series of phase diagrams. These findings may not only provide physical insight into phenomena associated with the drop impact dynamics but also assist researchers to identify the specific impact conditions under which each phenomenon occurs, so that it can be avoided or allowed depending on the application. | | | | | | Laminar and Viscous Flows | Christina Kurzthaler, Rodolfo Brandão, Ory Schnitzer, and Howard A. Stone Phys. Rev. Fluids 8, 014101 (2023) – Published 3 January 2023  | Starting from a slender-body formulation, we numerically and asymptotically characterize the steady-state deformation of a tethered, elastic filament in viscous flows with uniform, shear, and parabolic profiles. | | | | | | Ahmad Ababaei and Bogdan Rosa Phys. Rev. Fluids 8, 014102 (2023) – Published 17 January 2023  | The gravitational collision efficiency of a pair of cloud droplets settling in quiescent air is computed using various models for the aerodynamic interaction forces. The employed models consider (i) the effect of spherical fluid drops with mobile interfaces, and (ii) the noncontinuum molecular effect which changes short-range lubrication forces when the gap size between a pair is comparable to the air mean free path. These cases are compared with the widely used case that considers the droplets to be spherical rigid particles. We find that assuming rigid particles is accurate for water droplets interacting in air, but noncontinuum lubrication has to be taken into account. | | | | | | Arkava Ganguly and Ankur Gupta Phys. Rev. Fluids 8, 014103 (2023) – Published 20 January 2023  | We investigate the self-propelling motion of a bent rod. Our analysis reveals that the planar motion of the rod is always circular, and the radius of the circle and the speed of the particle can be tuned by changing the relative length of the two arms and the angle between them. | | | | | | E. Kirkinis Phys. Rev. Fluids 8, 014104 (2023) – Published 26 January 2023  | Is the full odd viscous (anomalous) stress tensor responsible for the normal pressure experienced by a cylinder rotating in a viscous liquid? The answer is negative, in general. When liquid velocities (and not stresses) are prescribed on the boundaries, it is the null divergence part of the odd viscous stress that generates this pressure. | | | | | | Multiphase, Granular, and Particle-Laden Flows | Y. Zhang, A. D. Bragg, and G. Wang Phys. Rev. Fluids 8, 014301 (2023) – Published 13 January 2023  | Statistical equations for inertial particle transport in turbulent boundary layers are usually closed using the quasi-Normal approximation (QNA), but this leads to large errors when the particle inertia is significant. Here we develop a new closure based on an asymptotic solution to the exact transport equations, referred to as the ACA. Results show that the ACA is in far better agreement with direct numerical simulation (DNS) data than the QNA. | | | | | | Matthew R. Turner and Richard P. Sear Phys. Rev. Fluids 8, 014302 (2023) – Published 17 January 2023  | Suspended particles in the atmosphere, such as water droplets or pollen, flowing over obstacles such as an aircraft wing or a tree branch may not follow the same path as the air, due to the particle's inertia. The particles may then collide with and deposit onto the obstacle. This is particularly significant in the case of water droplets that freeze to ice on an aircraft wing, as an ice deposit on the wing can cause a crash. In this paper we identify a scaling law for the amount of deposition, above a critical particle Stokes number, for particles in inviscid, irrotational flow past a cylinder. | | | | | | B. Siddani and S. Balachandar Phys. Rev. Fluids 8, 014303 (2023) – Published 17 January 2023  | Point-particle closure models that are utilized in Euler-Lagrange simulations play an important role in replicating true dynamics of particle-laden flows. The accuracy of these point-particle models depends on how well they incorporate the local microstructural information of neighboring particles. The current work presents a physics-based hierarchical machine learning approach for developing robust N-body closures. The inclusion of ternary interactions, in addition to binary interactions, enabled by the hierarchical approach leads to improved predictions. | | | | | | James A. Robinson, Daniel J. Holland, and Luke Fullard Phys. Rev. Fluids 8, 014304 (2023) – Published 19 January 2023  | Various continuum models have been developed to describe granular flows, but most of these were developed based on discrete simulations of isochoric flows, such as shear cells, where flow is parallel and there is no variation in velocity or packing fraction along streamlines. Many real granular flows are nonisochoric and may exhibit significant packing variations along streamlines. In our work we use discrete element modeling to simulate granular flows in nonisochoric flow geometries. Using this discrete data, we compare the performance of some existing compressible continuum models. | | | | | | Benjamin Musci, Britton Olson, Samuel Petter, Gokul Pathikonda, and Devesh Ranjan Phys. Rev. Fluids 8, 014501 (2023) – Published 3 January 2023  | High-speed experiments studying the Blast Driven Instability are used to validate Reynolds-Averaged Navier-Stokes (RANS) and Large eddy simulation (LES) turbulent mixing models. The work helps to elucidate the three mixing regimes of the instability and shows that the LES can successfully capture, at lease to the first order, the complex physics therein. The work also highlights the limitations of RANS models for transitioning instabilities and how LES results remain highly sensitive to the characterization of initial conditions. | | | | | | Featured in Physics Editors' Suggestion Andrew P. Grace, Marek Stastna, Kevin G. Lamb, and K. Andrea Scott Phys. Rev. Fluids 8, 014502 (2023) – Published 5 January 2023 | Freshwater has been shown to have a maximum density at about four degrees Celsius, and this leads to a phenomenon known as cabbeling. Cabbeling occurs when masses of water on different sides of the temperature of maximum density mix and create a denser mass. What happens when intruding and ambient temperatures in a gravity current are on opposite sides of the temperature of maximum density? How does cabbeling affect the evolution characteristics of gravity currents, and what sort of long term behavior arises? | | | | | | Jiahao Kong, Luke G. Bennetts, Bagus Nugroho, and R. C. Chin Phys. Rev. Fluids 8, 014601 (2023) – Published 10 January 2023  | Two-dimensional square-bars roughness significantly influences the drag coefficient and coherent structures of turbulent boundary layer flow. A pitch ratio of 8 can induce the maximum drag effect for the turbulent boundary layer compared to other pitch ratios at a moderate friction Reynolds number. Hot-wire experiment results show that a wide range of pitch ratios has similar energy modulation by transferring energy from the largest scale structures to the smaller ones. | | | | | | Jiaxing Song, Fenghui Lin, Yabiao Zhu, Zhen-Hua Wan, Nansheng Liu, Xi-Yun Lu, and Bamin Khomami Phys. Rev. Fluids 8, 014602 (2023) – Published 13 January 2023  | More than two decades ago it was discovered that very viscous fluids can become turbulent even when the flow speed is vanishingly small. We perform challenging three-dimensional direct numerical simulation of this purely elastic turbulence in the prototypical Taylor-Couette flow with the aim to unravel the underlying mechanism. We demonstrate that in this unique inertialess turbulent state, large-scale solitary vortices and anisotropic elastic traveling waves as well as random velocity perturbations delicately sustain the turbulent dynamic cycle. | | | | | | Hao Su (苏豪), Yue Yang (杨越), and Stephen B. Pope Phys. Rev. Fluids 8, 014603 (2023) – Published 27 January 2023  | We propose a simple model for the bottleneck effect in isotropic turbulence based on Kolmogorov's hypotheses. The model of the longitudinal structure function consists of two quadratic functions representing large- and small-scale motions, with parameters derived from asymptotic behaviors of the structure function. The model shows a decaying power law for the bump height in the compensated spectrum, which agrees with the result of direct numerical simulations at a range of Reynolds numbers. | | | | | | Arvind T. Mohan, Nicholas Lubbers, Misha Chertkov, and Daniel Livescu Phys. Rev. Fluids 8, 014604 (2023) – Published 31 January 2023  | We demonstrate an approach to enforce mass conservation constraints for three-dimensional incompressible turbulence inside the convolutional neural network architecture. Our method shows increased interpretability and adheres to periodic boundary conditions, while showing high accuracy. This approach is generic for differential constraints L of the form L(V) = G, and can be extended to different applications and neural network architectures. | | | | | | Esmaeel Masoudi, David Sims-Williams, and Lian Gan Phys. Rev. Fluids 8, 014701 (2023) – Published 17 January 2023  | Flow around bluff bodies has been studied for more than 200 years, especially since the discovery of Kármán vortex streets, however less attention has been paid to polygonal cylinders. In this study we used Large Eddy Simulations (LES) to investigate flow behavior, and especially shear layer separation, of polygonal cylinders with side number of N=5-8 in various incidence angles. The flow separation mechanism is then explained with a focus on the separation pattern as well as the flapping motion of the separated shear layer and its signature in time mean fields. | | | | | | Luca Galantucci, Giorgio Krstulovic, and Carlo F. Barenghi Phys. Rev. Fluids 8, 014702 (2023) – Published 19 January 2023  | Numerical simulations of superfluid helium turbulent flows show the emergence of hydrodynamic cooperation between quantum vortices leading to dynamics almost without dissipation. This energy saving mechanism accounts for the enhanced vortex lifetime observed in experiments and is similar to the collective dynamics observed in active fluids, such as bacteria in aqueous suspensions, fungal spores in the atmosphere, and cyclists in pelotons. This active fluid characteristic of superfluid helium potentially determines properties of turbulence in both the inviscid and the viscous fluid components. | | | | | | Jacob M. Neal and Michael Amitay Phys. Rev. Fluids 8, 014703 (2023) – Published 19 January 2023  | We conduct experiments on the structure of the 3D flow fields for unswept cantilevered wings at high angle of attack. Stall cell counter-rotating vortices on the suction surface form for a range of aspect ratios (AR), though AR changes the angle of attack at which the flow separates. Analysis of mean flow volume over the suction surface and near wake shows that the arch vortex forms in the wake and connects to the surface at the stall cell foci. Reynolds stress peaks indicate a strong reliance on the arch vortex location in the mean flow. Spectral analysis of the velocity field at select spanwise planes shows that shedding is coherent but intermittent, and varies strongly along the wingspan. | | | | | | Juan Carlos Bilbao-Ludena and George Papadakis Phys. Rev. Fluids 8, 014704 (2023) – Published 20 January 2023  | This study investigates the flow physics in some of the most inhomogeneous and complex regions of a separated flow around a wing tip and the near wake. The region is highly convoluted, strongly three-dimensional, multiscale, and far from being self-similar. We elucidate the early formation mechanisms of vortices close to the leading edge and investigate the intensification/suppression mechanisms of the different types of vortical structures as they evolve in space. The production of turbulent kinetic energy and Reynolds stresses is also investigated and discussed in conjunction with the identified vortex patterns. | | | | | | Wave Dynamics, Free Surface Flows, Stratified, and Rotating Flows | Zibo Zheng, Yan Li, and Simen Å. Ellingsen Phys. Rev. Fluids 8, 014801 (2023) – Published 13 January 2023  | We investigate the effect of a vertically sheared current on wave statistics, including the probability of rogue waves, and apply it to a real-world case using measured spectral and shear current data from the mouth of the Columbia River. A theory for weakly nonlinear waves valid to second order in wave steepness is derived and used to analyze statistical properties of surface waves. With the wave spectrum and velocity profile measured in the Columbia River estuary, our theory predicts that the probability of rogue waves is significantly reduced and enhanced during ebb and flood, respectively, supporting the need for shear currents to be accounted for in wave modeling and prediction. | | | | | | H. M. Yin, Q. Pan, and K. W. Chow Phys. Rev. Fluids 8, 014802 (2023) – Published 17 January 2023  | Crossing sea states, with two or more wave trains propagating at an oblique angle to each other, constitute a plausible generation mechanism of rogue waves in the oceans. This underlying principle of enhanced modulation instability is extended to layered and stratified fluids. Long wave-short wave resonance in a two-layer flow and three-wave interaction in a fluid with constant buoyancy frequency are studied. Maxima in the growth rates imply preferred configurations of crossing sea states. A cascading mechanism is employed to demonstrate the emergence of breathers beyond linear modulation instability. | | | | | | S. Zheng, H. Liang, S. Michele, and D. Greaves Phys. Rev. Fluids 8, 014803 (2023) – Published 23 January 2023  | A Hankel transform based model is developed to study water wave interaction with an array of thin submerged horizontal perforated and/or elastic circular plates. The integral equations are formulated in terms of unknown functions related to the jump in velocity potential across each plate. A Galerkin method is adopted to the solution of these integral equations and the velocity potential jump across the plate is expressed in terms of Fourier-Gegenbauer series. The present model is found to be valid for multiple plates distributed arbitrarily, including the staggered arrangement, for which the traditional eigenfunction matching method would not work. | | | | | | Guillaume Ricard and Eric Falcon Phys. Rev. Fluids 8, 014804 (2023) – Published 31 January 2023  | An experiment is performed to examine the previously unobserved transition from a dispersive wave-turbulence regime to a nondispersive shock-wave regime on the surface of a fluid. Using a magnetic liquid subjected to a high enough external magnetic field, a set of random shock waves is indeed observed on the free surface. Their spectrum is then found to be well modeled by a spectrum of singularities (Kuznetsov-like spectrum), and their statistics are close to the predictions from the one-dimensional random-forced Burgers' equation. | | | | | | | |
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