32 results found
Westwood TA, Delmotte B, Keaveny EE, 2022, A generalised drift-correcting time integration scheme for Brownian suspensions of rigid particles with arbitrary shape, Journal of Computational Physics, Vol: 467, ISSN: 0021-9991
The efficient computation of the overdamped, random motion of micron and nanometre scale particles in a viscous fluid requires novel methods to obtain the hydrodynamic interactions, random displacements and Brownian drift at minimal cost. Capturing Brownian drift is done most efficiently through a judiciously constructed time-integration scheme that automatically accounts for its contribution to particle motion. In this paper, we present a generalised drift-correcting (gDC) scheme that accounts for Brownian drift for suspensions of rigid particles with arbitrary shape. The scheme seamlessly integrates with fast methods for computing the hydrodynamic interactions and random increments and requires a single full mobility solve per time-step. As a result, the gDC provides increased computational efficiency when used in conjunction with grid-based methods that employ fluctuating hydrodynamics to obtain the random increments. Further, for these methods the additional computations that the scheme requires occur at the level of individual particles, and hence lend themselves naturally to parallel computation. We perform a series of simulations that demonstrate the gDC obtains similar levels of accuracy as compared with the existing state-of-the-art. In addition, these simulations illustrate the gDC's applicability to a wide array of relevant problems involving Brownian suspensions of non-spherical particles, such as the structure of liquid crystals and the rheology of complex fluids.
Maretvadakethope S, Hwang Y, Keaveny E, 2022, Synchronized states of hydrodynamically coupled filaments andtheir stability, Physical Review Fluids, ISSN: 2469-990X
Cilia and flagella are organelles that play central roles in unicellular locomotion, embryonicdevelopment, and fluid transport around tissues. In these examples, multiple cilia are often foundin close proximity and exhibit coordinated motion. Inspired by the flagellar motion of biflagellatecells, we examine the synchrony exhibited by a filament pair surrounded by a viscous fluid andtethered to a rigid planar surface. A geometrically-switching base moment drives filament motion,and we characterize how the stability of synchonized states depends of the base torque magnitude.In particular, we study the emergence of bistability that occurs when the anti-phase, breast-strokebranch becomes unstable. Using a bisection algorithm, we find the unstable edge-state that existsbetween the two basins of attraction when the system exhibits bistability. We establish a bifurcationdiagram, study the nature of the bifurcation points, and find that the observed dynamical systemcan be captured by a modified version of Adler’s equation. The bifurcation diagram and presenceof bistability reveal a simple mechanism by which the anti-phase breast stroke can be modulated, orswitched entirely to in-phase undulations through the variation of a single bifurcation parameter.
Aceves Sanchez P, Degond P, Keaveny E, et al., 2020, Large-scale dynamics of self-propelled particles moving through obstacles: model derivation and pattern formation, Bulletin of Mathematical Biology, Vol: 82, Pages: 1-39, ISSN: 0092-8240
We model and study the patterns created through the interactionof collectively moving self-propelled particles (SPPs) and elastically tetheredobstacles. Simulations of an individual-based model reveal at least three distinct large-scale patterns: travelling bands, trails and moving clusters. Thismotivates the derivation of a macroscopic partial differential equations modelfor the interactions between the self-propelled particles and the obstacles, forwhich we assume large tether stiffness. The result is a coupled system of non linear, non-local partial differential equations. Linear stability analysis showsthat patterning is expected if the interactions are strong enough and allowsfor the predictions of pattern size from model parameters. The macroscopicequations reveal that the obstacle interactions induce short-ranged SPP aggregation, irrespective of whether obstacles and SPPs are attractive or repulsive.
Schoeller SF, Holt W, Keaveny EE, 2020, Collective dynamics of sperm cells, PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY B-BIOLOGICAL SCIENCES, Vol: 375, ISSN: 0962-8436
- Author Web Link
- Citations: 13
Keaveny E, Townsend A, Westwood T, et al., 2020, Methods for suspensions of passive and active filaments, Publisher: arXiv
Flexible filaments and fibres are essential components of important complex fluids that appear in many biological and industrial settings. Direct simulations of these systems that capture the motion and deformation of many immersed filaments in suspension remain a formidable computational challenge due to the complex, coupled fluid--structure interactions of all filaments, the numerical stiffness associated with filament bending, and the various constraints that must be maintained as the filaments deform. In this paper, we address these challenges by first describing filament kinematics using quaternions to resolve both bending and twisting, applying implicit time-integration to alleviate numerical stiffness, and using quasi-Newton methods to obtain solutions to the resulting system of nonlinear equations. In particular, we employ geometric time integration to ensure that the quaternions remain unit as the filaments move. We also show that our framework can be used with a variety of models and methods, including matrix-free fast methods, that resolve low Reynolds number hydrodynamic interactions. We provide a series of tests and example simulations to demonstrate the performance and possible applications of our method. Finally, we provide a link to a MATLAB/Octave implementation of our framework that can be used to learn more about our approach and as a tool for filament simulation.
Michelin S, Game S, Lauga E, et al., 2020, Spontaneous onset of convection in a uniform phoretic channel., Soft Matter, Vol: 16, Pages: 1259-1269, ISSN: 1744-683X
Phoretic mechanisms, whereby gradients of chemical solutes induce surface-driven flows, have recently been used to generate directed propulsion of patterned colloidal particles. When the chemical solutes diffuse slowly, an instability further provides active isotropic particles with a route to self-propulsion by spontaneously breaking the symmetry of the solute distribution. Here we show theoretically that, in a mechanism analogous to Bénard-Marangoni convection, phoretic phenomena can create spontaneous and self-sustained wall-driven mixing flows within a straight, chemically-uniform active channel. Such spontaneous flows do not result in any net pumping for a uniform channel but greatly modify the distribution and transport of the chemical solute. The instability is predicted to occur for a solute Péclet number above a critical value and for a band of finite perturbation wavenumbers. We solve the perturbation problem analytically to characterize the instability, and use both steady and unsteady numerical computations of the full nonlinear transport problem to capture the long-time coupled dynamics of the solute and flow within the channel.
Maretvadakethope S, Keaveny E, Hwang Y, 2019, The instability of gyrotactically-trapped cell layers, Journal of Fluid Mechanics, Vol: 868, ISSN: 0022-1120
Several metres below the coastal ocean surface there are areas of high ecological activity that contain thin layers of concentrated motile phytoplankton. Gyrotactic trapping has been proposed as a potential mechanism for layer formation of bottom-heavy swimming algae cells, especially in flows where the vorticity varies linearly with depth (Durham et al., Science, vol. 323(5917), 2009, pp. 1067–1070). Using a continuum model for dilute microswimmer suspensions, we report that an instability of a gyrotactically trapped cell layer can arise in a pressure-driven plane channel flow. The linear stability analysis reveals that the equilibrium cell-layer solution is hydrodynamically unstable due to negative microswimmer buoyancy (i.e. a gravitational instability) over a range of biologically relevant parameter values. The critical cell concentration for this instability is found to be Nc≃104 cells cm−3 , a value comparable to the typical maximum cell concentration observed in thin layers. This result indicates that the instability may be a potential mechanism for limiting the layer’s maximum cell concentration, especially in regions where turbulence is weak, and motivates the study of its nonlinear evolution, perhaps, in the presence of turbulence.
Maretvadakethope S, Keaveny EE, Hwang Y, 2019, The instability of gyrotactically trapped cell layers
Several metres below the coastal ocean surface there are areas of high ecological activity that contain thin layers of concentrated motile phytoplankton. Gyrotactic trapping has been proposed as a potential mechanism for layer formation of bottom-heavy swimming algae cells, especially in flows where the vorticity varies linearly with depth (Durham et al.Science, vol. 323(5917), 2009, pp. 1067-1070). Using a continuum model for dilute microswimmer suspensions, we report that an instability of a gyrotactically trapped cell layer can arise in a pressure-driven plane channel flow. The linear stability analysis reveals that the equilibrium cell-layer solution is hydrodynamically unstable due to negative microswimmer buoyancy (i.e. a gravitational instability) over a range of biologically relevant parameter values. The critical cell concentration for this instability is found to be, a value comparable to the typical maximum cell concentration observed in thin layers. This result indicates that the instability may be a potential mechanism for limiting the layer's maximum cell concentration, especially in regions where turbulence is weak, and motivates the study of its nonlinear evolution, perhaps, in the presence of turbulence.
Bao Y, Rachh M, Keaveny E, et al., 2018, A fluctuating boundary integral method for Brownian suspensions, Journal of Computational Physics, Vol: 374, Pages: 1094-1119, ISSN: 0021-9991
We present a fluctuating boundary integral method (FBIM) for overdamped Brownian Dynamics (BD) of two-dimensional periodic suspensions of rigid particles of complex shape immersed in a Stokes fluid. We develop a novel approach for generating Brownian displacements that arise in response to the thermal fluctuations in the fluid. Our approach relies on a first-kind boundary integral formulation of a mobility problem in which a random surface velocity is prescribed on the particle surface, with zero mean and covariance proportional to the Green's function for Stokes flow (Stokeslet). This approach yields an algorithm that scales linearly in the number of particles for both deterministic and stochastic dynamics, handles particles of complex shape, achieves high order of accuracy, and can be generalized to three dimensions and other boundary conditions. We show that Brownian displacements generated by our method obey the discrete fluctuation–dissipation balance relation (DFDB). Based on a recently-developed Positively Split Ewald method Fiore et al. (2017) , near-field contributions to the Brownian displacements are efficiently approximated by iterative methods in real space, while far-field contributions are rapidly generated by fast Fourier-space methods based on fluctuating hydrodynamics. FBIM provides the key ingredient for time integration of the overdamped Langevin equations for Brownian suspensions of rigid particles. We demonstrate that FBIM obeys DFDB by performing equilibrium BD simulations of suspensions of starfish-shaped bodies using a random finite difference temporal integrator.
Kamal A, Keaveny EE, 2018, Enhanced locomotion, effective diffusion and trapping of undulatory micro-swimmers in heterogeneous environments, JOURNAL OF THE ROYAL SOCIETY INTERFACE, Vol: 15, ISSN: 1742-5689
- Author Web Link
- Open Access Link
- Citations: 6
Mingarelli L, Keaveny EE, Barnett R, 2018, Vortex lattices in binary mixtures of repulsive superfluids, Physical Review A, Vol: 97, ISSN: 1050-2947
We present an extension of the framework introduced in previous work [L. Mingarelli, E. E. Keaveny, and R. Barnett, J. Phys.: Condens. Matter 28, 285201 (2016)JCOMEL0953-898410.1088/0953-8984/28/28/285201] to treat multicomponent systems, showing that new degrees of freedom are necessary in order to obtain the desired boundary conditions. We then apply this extended framework to the coupled Gross-Pitaevskii equations to investigate the ground states of two-component systems with equal masses, thereby extending previous work in the lowest Landau limit [E. J. Mueller and T.-L. Ho, Phys. Rev. Lett. 88, 180403 (2002)PRLTAO0031-900710.1103/PhysRevLett.88.180403] to arbitrary interactions within Gross-Pitaevskii theory. We show that away from the lowest Landau level limit, the predominant vortex lattice consists of two interlaced triangular lattices. Finally, we derive a linear relation which accurately describes the phase boundaries in the strong interacting regimes.
Schoeller S, Keaveny EE, 2018, From flagellar undulations to collective motion: predicting the dynamics of sperm suspensions, Journal of the Royal Society Interface, Vol: 15, Pages: 1-10, ISSN: 1742-5662
Swimming cells and microorganisms are as diverse in their collective dynamics as they are in their indi-vidualshapesandpropulsionmechanisms. Evenforspermcells, whichhaveastereotypedshapeconsistingof a cell body connected to a flexible flagellum, a wide range of collective dynamics is observed spanningfrom the formation of tightly packed groups to the display of larger-scale, turbulence-like motion. Using adetailed mathematical model that resolves flagellum dynamics, we perform simulations of sperm suspen-sions containing up to 1000 cells and explore the connection between individual and collective dynamics.We find that depending on the level of variation in individual dynamics from one swimmer to another,the sperm exhibit either a strong tendency to aggregate, or the suspension exhibits large-scale swirling.Hydrodynamic interactions govern the formation and evolution of both states. In addition, a quantitativeanalysis of the states reveals that the flows generated at the time-scale of flagellum undulations contributesignificantly to the overall energy in the surrounding fluid, highlighting the importance of resolving theseflows.
Delmotte B, Keaveny EE, Climent E, et al., 2018, Simulations of Brownian tracer transport in squirmer suspensions, IMA Journal of Applied Mathematics, ISSN: 0272-4960
In addition to enabling movement towards environments with favourable living conditions, swim-ming by microorganisms has also been linked to enhanced mixing and improved nutrient uptake by theirpopulations. Experimental studies have shown that Brownian tracer particles exhibit enhanced diffusiondue to the swimmers, while theoretical models have linked this increase in diffusion to the flows generated bythe swimming microorganisms, as well as collisions with the swimmers. In this study, we perform detailedsimulations based on the force-coupling method and its recent extensions to the swimming and Brownianparticles to examine tracer displacements and effective tracer diffusivity in squirmer suspensions. By iso-lating effects such as hydrodynamic or steric interactions, we provide physical insight into experimentalmeasurements of the tracer displacement distribution. In addition, we extend results to the semi-diluteregime where the swimmer-swimmer interactions affect tracer transport and the effective tracer diffusiv-ity no longer scales linearly with the swimmer volume fraction.Tracer dispersion - Squirmers - Activesuspensions - Simulations
Li K, Javer A, Keaveny E, et al., 2018, Recurrent Neural Networks with Interpretable Cells Predict and Classify Worm Behaviour, Twenty-ninth Annual Conference on Neural Information Processing Systems (NIPS)
An important goal in behaviour analytics is to connect disease state or genomevariation with observable differences in behaviour. Despite advances in sensortechnology and imaging, informative behaviour quantification remains challenging.The nematode worm C. elegans provides a unique opportunity to test analysisapproaches because of its small size, compact nervous system, and the availabilityof large databases of videos of freely behaving animals with known genetic differences.Despite its relative simplicity, there are still no reports of generative modelsthat can capture essential differences between even well-described mutant strains.Here we show that a multilayer recurrent neural network (RNN) can produce diversebehaviours that are difficult to distinguish from real worms’ behaviour andthat some of the artificial neurons in the RNN are interpretable and correlate withobservable features such as body curvature, speed, and reversals. Although theRNN is not trained to perform classification, we find that artificial neuron responsesprovide features that perform well in worm strain classification.
Game SE, Hodes M, Keaveny EE, et al., 2017, Physical mechanisms relevant to flow resistance in textured microchannels, Physical Review Fluids, Vol: 2, ISSN: 2469-990X
Flow resistance of liquids flowing through microchannels can be reduced by replacing flat, no-slip boundaries with boundaries adjacent to longitudinal grooves containing an inert gas, resulting in apparent slip. With applications of such textured microchannels in areas such as microfluidic systems and direct liquid cooling of microelectronics, there is a need for predictive mathematical models that can be used for design and optimization. In this work, we describe a model that incorporates the physical effects of gas viscosity (interfacial shear), meniscus protrusion (into the grooves), and channel aspect ratio and show how to generate accurate solutions for the laminar flow field using Chebyshev collocation and domain decomposition numerical methods. While the coupling of these effects are often omitted from other models, we show that it plays a significant role in the behavior of such flows. We find that, for example, the presence of gas viscosity may cause meniscus protrusion to have a more negative impact on the flow rate than previously appreciated. Indeed, we show that there are channel geometries for which meniscus protrusion increases the flow rate in the absence of gas viscosity and decreases it in the presence of gas viscosity. In this work, we choose a particular definition of channel height: the distance from the base of one groove to the base of the opposite groove. Practically, such channels are used in constrained geometries and therefore are of prescribed heights consistent with this definition. This choice allows us to easily make meaningful comparisons between textured channels and no-slip channels occupying the same space.
Keaveny E, Brown AE, 2017, Predicting path from undulations for C. elegans using linear and nonlinear resistive force theory, Physical Biology, Vol: 14, ISSN: 1478-3975
A basic issuein the physics of behaviouris the mechanical relationship between an animal and its surroundings. The nematode and model organism C. elegans provides an excellent platform to explore this relationship due to its anatomical simplicity. Nonetheless,the physics of nematode crawling, in which the worm undulates its body to move on a wet surface, is not completely understoodand the mathematical models often used to describe this phenomenon are empirical. We confirm that linear resistive force theory, one such empirical model,is effective at predicting a worm’s path from its sequence of body postures for forward crawling, reversing, and turning and for a broad range of different behavioural phenotypes observedin mutant worms. However, agreement between the predicted and observed path is lost when using this model with recently measured valuesof the drag anisotropy. A recently proposed nonlinear extensionof the resistive force theory model also provides accurate predictions, but does not resolve the discrepancy between the parameters required to achieve good path prediction and the experimentally measured parameters. This meansthat while we have good effective models of worm crawling that can be used in applications such as whole-animal simulations and advanced tracking algorithms, there are still unanswered questions about the precise nature of the physical interaction between worms and their most commonly studied laboratory substrate.
Mingarelli L, Keaveny EE, Barnett R, 2016, Simulating infinite vortex lattices in superfluids, Journal of Physics: Condensed Matter, Vol: 28, ISSN: 0953-8984
We present an efficient framework to numerically treat infinite periodic vortex lattices in rotating superfluids described by the Gross–Pitaevskii theory. The commonly used split-step Fourier (SSF) spectral methods are inapplicable to such systems as the standard Fourier transform does not respect the boundary conditions mandated by the magnetic translation group. We present a generalisation of the SSF method which incorporates the correct boundary conditions by employing the so-called magnetic Fourier transform. We test the method and show that it reduces to known results in the lowest-Landau-level regime. While we focus on rotating scalar superfluids for simplicity, the framework can be naturally extended to treat multicomponent systems and systems under more general 'synthetic' gauge fields.
Delmotte B, Keaveny EE, 2015, Simulating Brownian suspensions with fluctuating hydrodynamics, Journal of Chemical Physics, Vol: 143, ISSN: 0021-9606
Fluctuating hydrodynamics has been successfully combined with several computational methods torapidly compute the correlated random velocities of Brownian particles. In the overdamped limitwhere both particle and fluid inertia are ignored, one must also account for a Brownian drift term inorder to successfully update the particle positions. In this paper, we present an efficient computationalmethod for the dynamic simulation of Brownian suspensions with fluctuating hydrodynamics thathandles both computations and provides a similar approximation as Stokesian Dynamics for diluteand semidilute suspensions. This advancement relies on combining the fluctuating force-couplingmethod (FCM) with a new midpoint time-integration scheme we refer to as the drifter-corrector(DC). The DC resolves the drift term for fluctuating hydrodynamics-based methods at a minimalcomputational cost when constraints are imposed on the fluid flow to obtain the stresslet correctionsto the particle hydrodynamic interactions. With the DC, this constraint needs only to be imposed onceper time step, reducing the simulation cost to nearly that of a completely deterministic simulation.By performing a series of simulations, we show that the DC with fluctuating FCM is an effective andversatile approach as it reproduces both the equilibrium distribution and the evolution of particulatesuspensions in periodic as well as bounded domains. In addition, we demonstrate that fluctuatingFCM coupled with the DC provides an efficient and accurate method for large-scale dynamicsimulation of colloidal dispersions and the study of processes such as colloidal gelation.
Delmotte B, Keaveny E, Plouraboue F, et al., 2015, Large-scale simulation of steady and time-dependent active suspensions with the force-coupling method, Journal of Computational Physics, Vol: 302, Pages: 524-547, ISSN: 1090-2716
We present a new development of the force-coupling method (FCM) to addressthe accurate simulation of a large number of interacting micro-swimmers. Ourapproach is based on the squirmer model, which we adapt to the FCM framework,resulting in a method that is suitable for simulating semi-dilute squirmersuspensions. Other effects, such as steric interactions, can be readilyconsidered with our model. We test our method by comparing the velocity fieldaround a single squirmer and the pairwise interactions between two squirmerswith exact solutions to the Stokes equations and results given by othernumerical methods. We also illustrate our method's ability to describespheroidal swimmer shapes and biologically-relevant time-dependent swimminggaits. We detail the numerical algorithm used to compute the hydrodynamiccoupling between a large collection ($10^4-10 ^5$) of micro-swimmers. Usingthis methodology, we investigate the emergence of polar order in a suspensionof squirmers and show that for large domains, both the steady-state polar orderparameter and the growth rate of instability are independent of system size.These results demonstrate the effectiveness of our approach to achieve nearcontinuum-level results, allowing for better comparison with experimentalmeasurements while complementing and informing continuum models.
Kalogirou A, Keaveny EE, Papageorgiou DT, 2015, An in-depth numerical study of the two-dimensional Kuramoto-Sivashinsky equation, Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences, Vol: 471, ISSN: 1364-5021
The Kuramoto–Sivashinsky equation in one spatial dimension (1D KSE) is one of the most well-known and well-studied partial differential equations. It exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. In this study, we perform an extensive numerical study of the Kuramoto–Sivashinsky equation (2D KSE) to complement this analytical work. We explore in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling.
Keaveny EE, 2014, Fluctuating force-coupling method for simulations of colloidal suspensions, Journal of Computational Physics, Vol: 269, Pages: 61-79, ISSN: 0021-9991
Keaveny EE, Walker SW, Shelley MJ, 2013, Optimization of Chiral Structures for Microscale Propulsion, NANO LETTERS, Vol: 13, Pages: 531-537, ISSN: 1530-6984
- Author Web Link
- Citations: 73
Walker SW, Keaveny EE, 2013, Analysis of Shape Optimization for Magnetic Microswimmers, SIAM J. Control Optim., Vol: 51, Pages: 3093-3126
Majmudar T, Keaveny EE, Zhang J, et al., 2012, Experiments and theory of undulatory locomotion in a simple structured medium., J R Soc Interface, Vol: 9, Pages: 1809-1823
Undulatory locomotion of micro-organisms through geometrically complex, fluidic environments is ubiquitous in nature and requires the organism to negotiate both hydrodynamic effects and geometrical constraints. To understand locomotion through such media, we experimentally investigate swimming of the nematode Caenorhabditis elegans through fluid-filled arrays of micro-pillars and conduct numerical simulations based on a mechanical model of the worm that incorporates hydrodynamic and contact interactions with the lattice. We show that the nematode's path, speed and gait are significantly altered by the presence of the obstacles and depend strongly on lattice spacing. These changes and their dependence on lattice spacing are captured, both qualitatively and quantitatively, by our purely mechanical model. Using the model, we demonstrate that purely mechanical interactions between the swimmer and obstacles can produce complex trajectories, gait changes and velocity fluctuations, yielding some of the life-like dynamics exhibited by the real nematode. Our results show that mechanics, rather than biological sensing and behaviour, can explain some of the observed changes in the worm's locomotory dynamics.
Keaveny EE, Shelley MJ, 2011, Applying a second-kind boundary integral equation for surface tractions in Stokes flow, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 230, Pages: 2141-2159, ISSN: 0021-9991
- Author Web Link
- Citations: 40
Liu D, Keaveny EE, Maxey MR, et al., 2009, Force-coupling method for flows with ellipsoidal particles, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 228, Pages: 3559-3581, ISSN: 0021-9991
- Author Web Link
- Citations: 32
Keaveny EE, Shelley MJ, 2009, Hydrodynamic mobility of chiral colloidal aggregates, PHYSICAL REVIEW E, Vol: 79, ISSN: 1539-3755
- Author Web Link
- Citations: 8
Keaveny EE, Maxey MR, 2008, Modeling the magnetic interactions between paramagnetic beads in magnetorheological fluids, JOURNAL OF COMPUTATIONAL PHYSICS, Vol: 227, Pages: 9554-9571, ISSN: 0021-9991
- Author Web Link
- Citations: 81
Keaveny EE, Maxey MR, 2008, Interactions between comoving magnetic microswimmers, PHYSICAL REVIEW E, Vol: 77, ISSN: 1539-3755
- Author Web Link
- Citations: 18
Keaveny EE, Maxey MR, 2008, Spiral swimming of an artificial micro-swimmer, JOURNAL OF FLUID MECHANICS, Vol: 598, Pages: 293-319, ISSN: 0022-1120
- Author Web Link
- Citations: 48
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