@articleEllero2007, title = Incompressible smoothed particle hydrodynamics,
journal = Journal of Computational Physics, volume = 226, number = 2, pages =
1731-1752, year = 2007, issn = 0021-9991, doi =
https://doi.org/10.1016/j.jcp.2007.06.019, url =
https://www.sciencedirect.com/science/article/pii/S0021999107002598, author =
Marco Ellero and Mar Serrano and Pep Español, keywords = Incompressible flows,
Smoothed particle hydrodynamics, abstract = We present a smoothed particle
hydrodynamic model for incompressible fluids. As opposed to solving a pressure
Poisson equation in order to get a divergence-free velocity field, here
incompressibility is achieved by requiring as a kinematic constraint that the
volume of the fluid particles is constant. We use Lagrangian multipliers to
enforce this restriction. These Lagrange multipliers play the role of non-
thermodynamic pressures whose actual values are fixed through the kinematic
restriction. We use the SHAKE methodology familiar in constrained molecular
dynamics as an efficient method for finding the non-thermodynamic pressure
satisfying the constraints. The model is tested for several flow
configurations.

@articleJFM2019, title=Shear thickening of a non-colloidal suspension with a
viscoelastic matrix, volume=880, DOI=10.1017/jfm.2019.753, journal=Journal of
Fluid Mechanics, publisher=Cambridge University Press, author=Vázquez-Quesada,
Adolfo and Español, Pep and Tanner, Roger I. and Ellero, Marco, year=2019,
pages=1070–1094

@articleMuzio, title = Transition to Turbulence and Mixing in a Viscoelastic
Fluid Flowing Inside a Channel with a Periodic Array of Cylindrical Obstacles,
author = Grilli, Muzio and Vázquez-Quesada, Adolfo and Ellero, Marco, journal
= Phys. Rev. Lett., volume = 110, issue = 17, pages = 174501, numpages = 5,
year = 2013, month = Apr, publisher = American Physical Society, doi =
10.1103/PhysRevLett.110.174501, url =
https://link.aps.org/doi/10.1103/PhysRevLett.110.174501

@articlePAIVA2009306, title = Particle-based viscoplastic fluid/solid
simulation, journal = Computer-Aided Design, volume = 41, number = 4, pages =
306-314, year = 2009, note = Point-based Computational Techniques, issn =
0010-4485, doi = https://doi.org/10.1016/j.cad.2008.10.004, url =
https://www.sciencedirect.com/science/article/pii/S0010448508001887, author =
Afonso Paiva and Fabiano Petronetto and Thomas Lewiner and Geovan Tavares,
keywords = Viscoplastic fluid, Solid deformation, Smoothed particle
hydrodynamics, Radial basis function, Non-Newtonian fluid, Physically-based
animation, abstract = Simulations of viscoplastic materials are traditionally
governed by continuum mechanics. The viscous behavior is typically modeled as
an internal force, defined by diverse quantities. This work introduces a fluid
model to simulate the viscoplastic effect of solid materials, such as plastic,
wax, clay and polymer. Our method consists in modeling a solid object through
a non-Newtonian fluid with high viscosity. This fluid simulation uses the
Smoothed Particle Hydrodynamics method and the viscosity is formulated by
using the General Newtonian Fluid model. This model concentrates the
viscoplasticity in a single parameter. Our results show clear effects of
creep, melting, hardening and flowing.

@ArticleZhu2018, author=Zhu, Chongqiang and Huang, Yu and Zhan, Liang-tong,
title=SPH-based simulation of flow process of a landslide at Hongao landfill
in China, journal=Natural Hazards, year=2018, month=Sep, day=01, volume=93,
number=3, pages=1113-1126, abstract=Landfills are the most popular means of
disposing of construction solid waste owing to the low-cost, large treatment
volume and strong adaptability. However, landfill collapses and landslides are
common, posing a risk to lives and property. A catastrophic landslide, not
triggered by heavy rainfall or earthquakes, occurred on December 20, 2015, at
Hongao landfill in Guangming New district of Shenzhen, China. The collapsed
waste soil flowed 1100 m down a slope with an elevation difference of 100 m.
This catastrophic landslide is typical of long-runout disasters with high
mobility, which result in great loss to life and property. In this paper, we
focused on reproducing the flow processes and simulating the arriving area of
the landslide at Hongao landfill using a full three-dimensional (3D) smoothed
particle hydrodynamics (SPH) method. A series of undrained ring shear tests
were conducted to determine the viscosity coefficient for the numerical
simulation. The calculated geometry of the landslide using the full 3D SPH
method agreed well with the field data. Hence, the combination of ring shear
tests and SPH method can be widely used to predict the dynamic behavior of
potential landslides in landfills, to reduce or prevent similar disasters.,
issn=1573-0840, doi=10.1007/s11069-018-3342-8,
url=https://doi.org/10.1007/s11069-018-3342-8

@articleZHU2010362, title = A numerical study of the flow of Bingham-like
fluids in two-dimensional vane and cylinder rheometers using a smoothed
particle hydrodynamics (SPH) based method, journal = Journal of Non-Newtonian
Fluid Mechanics, volume = 165, number = 7, pages = 362-375, year = 2010, issn
= 0377-0257, doi = https://doi.org/10.1016/j.jnnfm.2010.01.012, url =
https://www.sciencedirect.com/science/article/pii/S0377025710000145, author =
Huaning Zhu and Nicos S. Martys and Chiara Ferraris and Daniel De Kee,
keywords = Vane rheometer, Smoothed particle hydrodynamics (SPH), Bingham
fluid, Non-Newtonian fluid, Papanastasiou model, abstract = In this paper, a
Lagrangian formulation of the Navier–Stokes equations, based on the smoothed
particle hydrodynamics (SPH) approach, was applied to determine how well
rheological parameters such as plastic viscosity can be determined from vane
rheometer measurements. First, to validate this approach, a
Bingham/Papanastasiou constitutive model was implemented into the SPH model
and tests comparing simulation results to well established theoretical
predictions were conducted. Numerical simulations for the flow of fluids in
vane and coaxial cylinder rheometers were then performed. A comparison to
experimental data was also made to verify the application of the SPH method in
realistic flow geometries. Finally, results are presented from a parametric
study of the flow of Bingham fluids with different yield stresses under
various applied angular velocities of the outer cylindrical wall in the vane
and coaxial cylinder rheometers. The stress, strain rate and velocity
profiles, especially in the vicinity of the vane blades, were computed. By
comparing the calculated stress and flow fields between the two rheometers,
the validity of the assumption that the vane could be approximated as a
cylinder for measuring the rheological properties of Bingham fluids at
different shear rates was tested.

@articleKhanpour, title = Mesh-free SPH modeling of sediment scouring and
flushing, journal = Computers & Fluids, volume = 129, pages = 67-78, year =
2016, issn = 0045-7930, doi = https://doi.org/10.1016/j.compfluid.2016.02.005,
url = https://www.sciencedirect.com/science/article/pii/S0045793016300238,
author = M. Khanpour and A.R. Zarrati and M. Kolahdoozan and A. Shakibaeinia
and S.M. Amirshahi, keywords = Smoothed Particle hydrodynamics (SPH),
Continuum-based sediment transport, Sediment scouring, Sediment flushing,
abstract = A two-phase mesh-free Lagrangian model, based on the Smoothed
Particle Hydrodynamics (SPH) method, is developed and evaluated for the
continuum-based study of sediment-water system (a dense submerged granular
flow system). The case studies are (a) sediment scouring downstream of a wall-
jet, and (b) reservoir sediment flushing, both of great interest of engineers
and researchers. The mesh-free Lagrangian nature of the SPH method makes the
model capable of dealing with the large interfacial deformations involved in
these complex water-sediment flow systems. Water and sediment materials are
both treated as continuum and the governing equation of their motion are
solved on a Lagrangian frame using SPH formulation. A pressure-dependent
viscoplastic rheological model is used to describe the behavior of the
sediment phase. A combination of Mohr–Coulomb and Shields yielding criteria
(in place of commonly used Mohr–Coulomb yield criteria) is employed to
determine the critical stress above which the sediment material behaves as a
viscous fluid. Effect of this combination on the model results is
investigated. To validate the developed numerical model, this study also
provides some original experimental measurements for the test cases of this
study. The results show the good compatibility of the numerical and
experimental water and bed profiles. The sediment feature such as scouring and
deposition areas (for wall-jet scouring case) and the sediment failure slope
(for the flushing test case) are reproduced by the developed SPH model, with a
good degree of accuracy. The numerical results as well as experimental
measurements of this study not only provide an insight into better
understanding of the complicated mechanism of sediment transport induced by
rapid flow discharge, but also provide a reliable numerical tool and benchmark
for simulation of such problems.

@articleFourtaka, title = Modelling multi-phase liquid-sediment scour and
resuspension induced by rapid flows using Smoothed Particle Hydrodynamics
(SPH) accelerated with a Graphics Processing Unit (GPU), journal = Advances in
Water Resources, volume = 92, pages = 186-199, year = 2016, issn = 0309-1708,
doi = https://doi.org/10.1016/j.advwatres.2016.04.009, url =
https://www.sciencedirect.com/science/article/pii/S0309170816300926, author =
G. Fourtakas and B.D. Rogers, keywords = SPH, Multi-phase, Scouring, Sediment
transport, Suspension, GPU, abstract = A two-phase numerical model using
Smoothed Particle Hydrodynamics (SPH) is applied to two-phase liquid-sediments
flows. The absence of a mesh in SPH is ideal for interfacial and highly non-
linear flows with changing fragmentation of the interface, mixing and
resuspension. The rheology of sediment induced under rapid flows undergoes
several states which are only partially described by previous research in SPH.
This paper attempts to bridge the gap between the geotechnics, non-Newtonian
and Newtonian flows by proposing a model that combines the yielding, shear and
suspension layer which are needed to predict accurately the global erosion
phenomena, from a hydrodynamics prospective. The numerical SPH scheme is based
on the explicit treatment of both phases using Newtonian and the non-Newtonian
Bingham-type Herschel-Bulkley-Papanastasiou constitutive model. This is
supplemented by the Drucker-Prager yield criterion to predict the onset of
yielding of the sediment surface and a concentration suspension model. The
multi-phase model has been compared with experimental and 2-D reference
numerical models for scour following a dry-bed dam break yielding satisfactory
results and improvements over well-known SPH multi-phase models. With 3-D
simulations requiring a large number of particles, the code is accelerated
with a graphics processing unit (GPU) in the open-source DualSPHysics code.
The implementation and optimisation of the code achieved a speed up of x58
over an optimised single thread serial code. A 3-D dam break over a non-
cohesive erodible bed simulation with over 4 million particles yields close
agreement with experimental scour and water surface profiles.

@ArticleJop2006, author=Jop, Pierre and Forterre, Yoël and Pouliquen, Olivier,
title=A constitutive law for dense granular flows, journal=Nature, year=2006,
month=Jun, day=01, volume=441, number=7094, pages=727-730, abstract=Equations
describing how granular materials move under shear are still a matter of
debate. Jop et al. now propose a new model for dense granular flows in three
dimensions, inspired by the behaviour of visco-plastic fluids such as
toothpaste. The results could serve as a basic tool for modelling complex
flows in geophysical or industrial applications., issn=1476-4687,
doi=10.1038/nature04801, url=https://doi.org/10.1038/nature04801

@articlePouliquen_2006, doi = 10.1088/1742-5468/2006/07/P07020, url =
https://dx.doi.org/10.1088/1742-5468/2006/07/P07020, year = 2006, month = jul,
publisher = , volume = 2006, number = 07, pages = P07020, author = O Pouliquen
and C Cassar and P Jop and Y Forterre and M Nicolas, title = Flow of dense
granular material: towards simple constitutive laws, journal = Journal of
Statistical Mechanics: Theory and Experiment, abstract = Recent experiments
and numerical simulations of dry granular flows suggest that a simple
rheological description in terms of a friction coefficient varying both with
shear rate and pressure through a dimensionless inertial number may be
sufficient to capture the major properties of granular flows. In this paper we
first present the empirical constitutive laws and their interpretation using
dimensional analysis, before analysing the prediction for different flow
configurations. The successes and limits of the approach are discussed based
on comparison with recent studies.

@articleMinattiParis, title = A SPH model for the simulation of free surface
granular flows in a dense regime, journal = Applied Mathematical Modelling,
volume = 39, number = 1, pages = 363-382, year = 2015, issn = 0307-904X, doi =
https://doi.org/10.1016/j.apm.2014.05.034, url =
https://www.sciencedirect.com/science/article/pii/S0307904X14002960, author =
Lorenzo Minatti and Enio Paris, keywords = Smoothed Particles Hydrodynamics,
Granular flows, Viscosity regularisation, abstract = Smoothed Particle
Hydrodynamics (SPH) is a particle-based Lagrangian numerical method,
particularly suitable for modeling free surface flows of fluids with different
constitutive equations. In this work, a SPH model is developed to solve flow
problems of granular matter flowing in a dense regime, featuring a
constitutive equation proposed by Pouliquen et al. (2006) [17] and Jop et al.
(2006) [18]. The problem of resolving the flow close to static conditions has
been solved by introducing a weak compressibility for the grains: this leads
to a third order algebraic equations to be solved for computing granular
material density. A new viscosity regularisation method, suitable for any
visco-plastic fluid with varying yield stress has been developed, in order to
reproduce stopping flow and static conditions. Validation of the model has
been performed by solving the simple uniform granular flow problem, for which
the solution is known analytically and by reproducing a set of experimental
data on granular columns collapses by Lube et al. (2004) [19].

@articleBonetRodriguez-Paz, author = Rodriguez-Paz, M. X. and Bonet, J., title
= A corrected smooth particle hydrodynamics method for the simulation of
debris flows, journal = Numerical Methods for Partial Differential Equations,
volume = 20, number = 1, pages = 140-163, keywords = SPH, debris flows, mesh-
free method, avalanches, doi = https://doi.org/10.1002/num.10083, url =
https://onlinelibrary.wiley.com/doi/abs/10.1002/num.10083, eprint =
https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.10083, abstract = Abstract
This article presents the development and application of a corrected smooth
particle hydrodynamics (CSPH) code to the simulation of debris flow and
avalanches. The advantages of a mesh-free method over other traditional
numerical methods such as the finite element method are discussed. A new
frictional approach for the boundary conditions and modified constitutive
equations are introduced in the SPH method. The resulting technique is then
applied for the simulation of debris flows, comparing the results with those
obtained from experiments reported by other researchers. © 2003 Wiley
Periodicals, Inc. Numer Methods Partial Differential Eq 20: 140–163, 2004.,
year = 2004

@miscmoreno2023morphological, title=Morphological transitions of block
copolymer micelles: implications for isoporous membrane ordering,
author=Nicolas Moreno and Suzana Nunes and Victor Calo, year=2023,
eprint=2304.03900, archivePrefix=arXiv, primaryClass=cond-mat.soft

@miscMorenoGeneral, title=Generalized Lagrangian Heterogenous Multiscale
Modeling of Complex Fluids, author=Nicolas Moreno and Marco Ellero, year=2022,
eprint=2211.05080, archivePrefix=arXiv, primaryClass=physics.flu-dyn

@articleTexier2023, title=Shear-thickening suspensions down inclines: from
Kapitza to Oobleck waves, volume=959, DOI=10.1017/jfm.2023.162,
journal=Journal of Fluid Mechanics, publisher=Cambridge University Press,
author=Darbois Texier, Baptiste and Lhuissier, Henri and Metzger, Bloen and
Forterre, Yoël, year=2023, pages=A27

@articleMaron, title = Application of ree-eyring generalized flow theory to
suspensions of spherical particles, journal = Journal of Colloid Science,
volume = 11, number = 1, pages = 80-95, year = 1956, issn = 0095-8522, doi =
https://doi.org/10.1016/0095-8522(56)90023-X, url =
https://www.sciencedirect.com/science/article/pii/009585225690023X, author =
Samuel H. Maron and Percy E. Pierce, abstract = The generalized flow theory of
Ree and Eyring has been applied successfully to the extensive data of Maron
and Fok (3) on the flow behavior of X-667 latex in both the Newtonian and non-
Newtonian regions. By postulation of the presence of two flow units, a
Newtonian water unit and a latex particle flow unit which may be non-
Newtonian, an equation involving three parameters was obtained which
reproduced very well dependence of the viscosity on rate of shear at all
temperatures and concentrations. Further, the theory predicted the correct
dependence of the parameters on temperature, from which could be obtained the
heats of activation for flow of both the water and latex particles. Both heats
of activation were found to be independent of concentration, with that of the
water being the same as for the pure liquid. The theory does not predict the
concentration dependence of the parameters. However, empirical equations for
such dependence were deduced, from which it was possible to ascertain the
values of the parameters in the Newtonian range below a volume fraction ν =
0.25, and to calculate therefrom Newtonian relative viscosities in good
agreement with the observed ones at all temperatures and concentrations.
Further, on extrapolation to ν = 0, νspνt was found to be 2.60 as against the
expected Einstein value of 2.50 for suspensions of spheres, while
extrapolation to infinite viscosity gave a volume fraction in very good
agreement with that expected for the rhombohedral close packing of nonuniform
spheres. From the combination of the theoretical and empirical equations were
obtained also expressions for the values of the “molecular” parameters of the
theory as functions of temperature and concentration, and the entropy of
activation for the flow of the latex particles. The distance between
equilibrium positions of the water molecules was found to vary from 13 A. to
zero between ν = 0 and the volume fraction at which the viscosity becomes
infinite. On the other hand, the distances between equilibrium positions and
planes of latex particles varied from infinity to zero between the same
concentration limits.

@articleMariSeto, title = Discontinuous Shear Thickening of Frictional Hard-
Sphere Suspensions, author = Seto, Ryohei and Mari, Romain and Morris, Jeffrey
F. and Denn, Morton M., journal = Phys. Rev. Lett., volume = 111, issue = 21,
pages = 218301, numpages = 5, year = 2013, month = Nov, publisher = American
Physical Society, doi = 10.1103/PhysRevLett.111.218301, url =
https://link.aps.org/doi/10.1103/PhysRevLett.111.218301

@articleCAppelaere, title = Rheology, birefringence, and small-angle neutron
scattering in a charged micellar system: Evidence of a shear-induced phase
transition, author = Cappelaere, E. and Berret, J. F. and Decruppe, J. P. and
Cressely, R. and Lindner, P., journal = Phys. Rev. E, volume = 56, issue = 2,
pages = 1869–1878, numpages = 0, year = 1997, month = Aug, publisher =
American Physical Society, doi = 10.1103/PhysRevE.56.1869, url =
https://link.aps.org/doi/10.1103/PhysRevE.56.1869

@ArticleBritton1999, author=Britton, M. M. and Callaghan, P. T., title=Shear
banding instability in wormlike micellar solutions, journal=The European
Physical Journal B - Condensed Matter and Complex Systems, year=1999,
month=Jan, day=01, volume=7, number=2, pages=237-249, abstract=Using NMR
velocimetry and mechanical measurements we study the flow dynamics, within a
cone-and-plate rheometer, of the wormlike micelle system, cetylpyridinium
chloride/sodium salicylate (CPyCl/NaSal) at 100 mM/60 mM concentration in
distilled water. Depending on precise conditions within the system, two
classes of behaviour are observed, one in which the boundary between different
shear rate phases fluctuates rapidly (on the order of tens of milliseconds)
and one in which it migrates slowly with a time constant of many seconds.
These modes of behaviour may depend on minor solution impurities, which
presumably affect the detailed constitutive properties, but also on the
externally applied shear rate within a given system. We argue that the slow
migrations are governed by stress relaxation effects while the rapid
migrations are flow driven and arise from interfacial instability.,
issn=1434-6036, doi=10.1007/s100510050610,
url=https://doi.org/10.1007/s100510050610

@ArticleFielding, author =”Fielding, Suzanne M.”, title =”Complex dynamics of
shear banded flows”, journal =”Soft Matter”, year =”2007”, volume =”3”, issue
=”10”, pages =”1262-1279”, publisher =”The Royal Society of Chemistry”, doi
=”10.1039/B707980J”, url =”http://dx.doi.org/10.1039/B707980J”, abstract
=”Many complex fluids undergo a flow induced transition to a state of
coexisting bands of differing viscosities and internal structuring. This
effect, which is called “shear banding”, is widely observed in wormlike
micellar surfactants, onion surfactants, colloidal suspensions and polymer
solutions. According to a rapidly accumulating body of experimental evidence,
shear bands often exhibit complex dynamics, which can be either oscillatory or
chaotic in nature. This can be seen in the unsteady response of the bulk
rheological signals, and in the motion of the interface between the bands.
After giving a brief overview of this experimental evidence, we review in some
detail recent efforts to address it theoretically.”

@articleOlmsted, author = Olmsted, Peter and Radulescu, Ovidiu and Lu, C-Y,
year = 2000, month = 03, pages = 257-275, title = Johnson-Segalman model with
a diffusion term in cylindrical Couette flow, volume = 44, journal = Journal
of Rheology - J RHEOL, doi = 10.1122/1.551085 @articleHerle, author = Herle,
Vishweshwara and Fischer, Peter and Windhab, Erich J., title = Stress Driven
Shear Bands and the Effect of Confinement on Their StructuresA Rheological,
Flow Visualization, and Rheo-SALS Study, journal = Langmuir, volume = 21,
number = 20, pages = 9051-9057, year = 2005, doi = 10.1021/la0513959, note
=PMID: 16171332,

URL = https://doi.org/10.1021/la0513959

, eprint = https://doi.org/10.1021/la0513959

@articleLIND2011, title = Incompressible smoothed particle hydrodynamics for
free-surface flows: A generalised diffusion-based algorithm for stability and
validations for impulsive flows and propagating waves, journal = Journal of
Computational Physics, volume = 231, number = 4, pages = 1499-1523, year =
2012, issn = 0021-9991, doi = https://doi.org/10.1016/j.jcp.2011.10.027, url =
https://www.sciencedirect.com/science/article/pii/S0021999111006279, author =
S.J. Lind and R. Xu and P.K. Stansby and B.D. Rogers, keywords =
Incompressible SPH, Free-surface flow simulation, Fick’s law, Truncated-kernel
error, abstract = The incompressible smoothed particle hydrodynamics (ISPH)
method with projection-based pressure correction has been shown to be highly
accurate and stable for internal flows and, importantly for many problems, the
pressure field is virtually noise-free in contrast to the weakly compressible
SPH approach (Xu et al., 2009 [31]). However for almost inviscid fluids
instabilities at the free surface occur due to errors associated with the
truncated kernels. A new algorithm is presented which remedies this issue,
giving stable and accurate solutions to both internal and free-surface flows.
Generalising the particle shifting approach of Xu et al. (2009) [31], the
algorithm is based upon Fick’s law of diffusion and shifts particles in a
manner that prevents highly anisotropic distributions and the onset of
numerical instability. The algorithm is validated against analytical solutions
for an internal flow at higher Reynolds numbers than previously, the flow due
to an impulsively started plate and highly accurate solutions for wet bed dam
break problems at zero and small times. The method is then validated for
progressive regular waves with paddle motion defined by linear theory. The
accurate predictions demonstrate the effectiveness of the algorithm in
stabilising solutions and minimising the surface instabilities generated by
the inevitable errors associated with truncated kernels. The test cases are
thought to provide a more thorough quantitative validation than previously
undertaken.

@ARTICLEWagner2017, author = Vázquez-Quesada, A. and Wagner, N. J. and Ellero,
M., title = ”Planar channel flow of a discontinuous shear-thickening model
fluid: Theory and simulation”, journal = Physics of Fluids, year = 2017, month
= oct, volume = 29, number = 10, eid = 103104, pages = 103104, doi =
10.1063/1.4997053, adsurl =
https://ui.adsabs.harvard.edu/abs/2017PhFl…29j3104V, adsnote = Provided by the
SAO/NASA Astrophysics Data System

@articleTanner2005, title = SPH simulations of transient viscoelastic flows at
low Reynolds number, journal = Journal of Non-Newtonian Fluid Mechanics,
volume = 132, number = 1, pages = 61-72, year = 2005, issn = 0377-0257, doi =
https://doi.org/10.1016/j.jnnfm.2005.08.012, url =
https://www.sciencedirect.com/science/article/pii/S0377025705001990, author =
M. Ellero and R.I. Tanner, keywords = SPH, Meshfree methods, Viscoelasticity,
Transient flows, UCM fluid, Oldroyd-B fluid, abstract = A numerical study on
the transient flow of a viscoelastic fluid is presented. The numerical
framework is that of Smoothed Particle Hydrodynamics (SPH) already used by
Ellero et al. in previous simulations of Non-Newtonian flows [J. Non-Newtonian
Fluid. Mech. 105 (2002) 35–51]. In particular, the start-up flow between
parallel plates is simulated for an Oldroyd-B and UCM fluid at low Reynolds
number. Results for a Newtonian fluid are also shown for comparison. The
numerical results are presented and compared with available theoretical
solutions, showing a very good agreement. In particular, the simulations of an
Oldroyd-B fluid have been found to be stable and accurate for a wide range of
the Weissenberg number. In the case of a UCM fluid, the absence of a viscous
term in the momentum equation makes its numerical modelling harder. Namely,
the process is characterised by a travelling damped wave, which, if not
accurately resolved, can lead to the rapid growth of small oscillations in
time eventually causing divergence. On the other hand, if specifically dealing
with transient flow problems, stabilising techniques such as BSD, EVSS or AVSS
can not be used either; whenever used in conjunction with decoupled solution
algorithms, they give an excessive oversmoothing in the results which
deteriorates the final accuracy. In this work, we consider an exact SPH
discretisation of the hyperbolic equation characterising the UCM model. SPH
simulations are finally performed for different Weissenberg numbers showing
very promising results. Finally, a discussion on the SPH treatment of the
boundary conditions for general hydrodynamics problems is also outlined
following the approach of ‘SPH boundary particles’ introduced by Morris for
the simulations of low Reynolds number flows.

@articleKALATEH202051, title = Simulation of cavitating fluid–Structure
interaction using SPH–FE method, journal = Mathematics and Computers in
Simulation, volume = 173, pages = 51-70, year = 2020, issn = 0378-4754, doi =
https://doi.org/10.1016/j.matcom.2020.01.019, url =
https://www.sciencedirect.com/science/article/pii/S0378475420300264, author =
Farhoud Kalateh and Ali Koosheh, keywords = Cavitating flow, Fluid–structure
interaction, Smoothed particle hydrodynamics (SPH), Finite element method
(FEM), Immersed boundary, Multi-phase fluid, Phase change, abstract = In the
present paper, a modified version of SPH–FE is proposed to study the
cavitating fluid interaction with the convergent–divergent nozzle. The
additional terms of mass/momentum transfer, surface tension and pressure based
phase change are added to the standard SPH equations to track the growth,
convection and collapse of the cavitation phenomenon. Due to using the
particle-based method for the fluid, no transitional phase is defined and the
certain particle phase is determined by the absolute pressure value in
comparison with vapor pressure. The comparison of the results of modified SPH
with those of finite volume as a grid-based method shows that the cavitating
region can be accurately modeled in the convergent–divergent nozzle. Then, the
interaction of cavitating fluid flow–nozzle wall is simulated by the improved
SPH–FE algorithm for the different types of steel where the good agreement is
obtained in comparison with the other similar numerical methods. Also, it is
concluded that considering cavitation in the fluid flow can generally change
the nozzle behavior slightly and increase the stress values inside its body.

@ArticleReece2020, author=Reece, Georgina and Rogers, Benedict D. and Lind,
Steven and Fourtakas, Georgios, title=New instability and mixing simulations
using SPH and a novel mixing measure, journal=Journal of Hydrodynamics,
year=2020, month=Aug, day=01, volume=32, number=4, pages=684-698,
abstract=This paper assesses the ability of smoothed particle hydrodynamics
(SPH) to simulate mixing of two-phase flows and their transition to
instabilities under different flow regimes. A new measure for quantification
of the degree of mixing between phases in a Lagrangian framework is also
developed. The method is validated using the lid-driven cavity and two-phase
Poiseuille flow cases. The velocity along the centre of the cavity is compared
with results from the literature, whilst commercial volume-of-fluid code STAR-
CCM+ provides a benchmark for the mixing and different mixing measures are
considered. The velocity of two-phase Poiseuille flow along the channel is
compared to the analytical solution, and the appearance of interfacial
instabilities with perturbation theory. This is the first time SPH has been
used to investigate the onset and development of these instabilities. In
particular, it is able to model the deforming shape of the interface, which is
not given by analytical studies, while also offering improved predictions over
conventional mesh-based computational fluid dynamics simulations.,
issn=1878-0342, doi=10.1007/s42241-020-0045-x,
url=https://doi.org/10.1007/s42241-020-0045-x

@articleShimizu2016, title=Simulation of three-dimensional rapid free-surface
granular flow past different types of obstructions using the SPH method,
volume=62, DOI=10.1017/jog.2016.22, number=232, journal=Journal of Glaciology,
publisher=Cambridge University Press, author=ABDELRAZEK, AHMED M. and KIMURA,
ICHIRO and SHIMIZU, YASUYUKI, year=2016, pages=335–347 @articleJaeger2019,
title = Stress Controlled Rheology of Dense Suspensions Using Transient Flows,
author = Han, Endao and James, Nicole M. and Jaeger, Heinrich M., journal =
Phys. Rev. Lett., volume = 123, issue = 24, pages = 248002, numpages = 6, year
= 2019, month = Dec, publisher = American Physical Society, doi =
10.1103/PhysRevLett.123.248002, url =
https://link.aps.org/doi/10.1103/PhysRevLett.123.248002

@articleOzturk2020, title=Flow-to-fracture transition and pattern formation in
a discontinuous shear thickening fluid, author=Deren Ozturk and Miles L.
Morgan and Bjørnar Sandnes, journal=Communications Physics, year=2020,
volume=3, pages=1-9 @article Peter2016, Title = Direct observation of dynamic
shear jamming in dense suspensions, Author = Peters, Ivo R and Majumdar,
Sayantan and Jaeger, Heinrich M, DOI = 10.1038/nature17167, Number = 7598,
Volume = 532, Month = April, Year = 2016, Journal = Nature, ISSN = 0028-0836,
Pages = 214—217, Abstract = Liquid-like at rest, dense suspensions of hard
particles can undergo striking transformations in behaviour when agitated or
sheared. These phenomena include solidification during rapid impact, as well
as strong shear thickening characterized by discontinuous, orders-of-magnitude
increases in suspension viscosity. Much of this highly non-Newtonian behaviour
has recently been interpreted within the framework of a jamming transition.
However, although jamming indeed induces solid-like rigidity, even a strongly
shear-thickened state still flows and thus cannot be fully jammed.
Furthermore, although suspensions are incompressible, the onset of rigidity in
the standard jamming scenario requires an increase in particle density.
Finally, whereas shear thickening occurs in the steady state, impact-induced
solidification is transient. As a result, it has remained unclear how these
dense suspension phenomena are related and how they are connected to jamming.
Here we resolve this by systematically exploring both the steady-state and
transient regimes with the same experimental system. We demonstrate that a
fully jammed, solid-like state can be reached without compression and instead
purely with shear, as recently proposed for dry granular systems. This state
is created by transient shear-jamming fronts, which we track directly. We also
show that shear stress, rather than shear rate, is the key control parameter.
From these findings we map out a state diagram with particle density and shear
stress as variables. We identify discontinuous shear thickening with a
marginally jammed regime just below the onset of full, solid-like jamming.
This state diagram provides a unifying framework, compatible with prior
experimental and simulation results on dense suspensions, that connects
steady-state and transient behaviour in terms of a dynamic shear-jamming
process., URL = https://doi.org/10.1038/nature17167, @articleHermes2016,
author = Hermes,Michiel and Guy,Ben M. and Poon,Wilson C. K. and Poy,Guilhem
and Cates,Michael E. and Wyart,Matthieu , title = Unsteady flow and particle
migration in dense, non-Brownian suspensions, journal = Journal of Rheology,
volume = 60, number = 5, pages = 905-916, year = 2016, doi =
10.1122/1.4953814,

URL = https://doi.org/10.1122/1.4953814

, eprint = https://doi.org/10.1122/1.4953814

@articleTexier2020, author = Texier, Baptiste and Lhuissier, Henri and
Forterre, Yoël and Metzger, Bloen, year = 2020, month = 12, pages =
1234567890, title = Surface-wave instability without inertia in shear-
thickening suspensions, volume = 3, journal = Communications Physics, doi =
10.1038/s42005-020-00500-4 @articleBalmforth2005, title = Roll waves on
flowing cornstarch suspensions, journal = Physics Letters A, volume = 338,
number = 6, pages = 479-484, year = 2005, issn = 0375-9601, doi =
https://doi.org/10.1016/j.physleta.2005.02.071, url =
https://www.sciencedirect.com/science/article/pii/S0375960105003257, author =
N.J. Balmforth and J.W.M. Bush and R.V. Craster, abstract = We report a
traveling wave instability in low Reynolds number flows of aqueous
concentrated suspensions of corn starch. The experimental observations are
difficult to reconcile with theoretical predictions based on simple
rheological models which indicate that flows are stable at low Reynolds
number. @articleRossi2022, title = SPH simulations of thixo-viscoplastic fluid
flow past a cylinder, journal = Journal of Non-Newtonian Fluid Mechanics,
volume = 308, pages = 104891, year = 2022, issn = 0377-0257, doi =
https://doi.org/10.1016/j.jnnfm.2022.104891, url =
https://www.sciencedirect.com/science/article/pii/S0377025722001161, author =
E. Rossi and I. Garcia de Beristain and A. Vazquez-Quesada and J.E. López-
Aguilar and M. Ellero, keywords = Viscoplasticity, Thixotropy, SPH, abstract =
Thixotropic materials are complex fluids that display time-dependent viscosity
and/or yield-stress response upon the application of a fixed deformation,
while recovering their original structured-state when the deformation is
discontinued. Thixotropic effects are presents in many different systems and
applications, ranging from food products, such as ketchup, to metals, such as
molten aluminum. In this work we present a first attempt to simulate the
rheological properties of thixo-viscoplastic flows using a Smoothed Particle
Hydrodynamic (SPH) method. The study set up is a 2D flow around a circular
cylinder as well as a simple shear flow between parallel plates to validate
our numerical results. SPH solutions are compared with simulations performed
using the open-source Finite Volume Method solver RheoTool, based on OpenFOAM.
The viscoplastic model used in this work is the Papanastasiou model combined
with a recently developed microstructural one, in order to include thixotropy.
In this thixo-viscoplastic framework, we analyze the flow properties in terms
of yield-fronts, streamlines and structure-parameter fields at different
Bingham and Thixotropy numbers, through microstructural thixotropic and yield-
stress parameters variation. Obtained results show an important novelty: an
asymmetry in the thixo-viscoplastic flow around the cylinder. @articleXu2016,
title = An improved weakly compressible SPH method for simulating free surface
flows of viscous and viscoelastic fluids, journal = Computer Physics
Communications, volume = 201, pages = 43-62, year = 2016, issn = 0010-4655,
doi = https://doi.org/10.1016/j.cpc.2015.12.016, url =
https://www.sciencedirect.com/science/article/pii/S0010465515004543, author =
Xiaoyang Xu and Xiao-Long Deng, keywords = Smoothed particle hydrodynamics,
Free surface flow, Viscous fluid, Viscoelastic fluid, abstract = In this
paper, an improved weakly compressible smoothed particle hydrodynamics (SPH)
method is proposed to simulate transient free surface flows of viscous and
viscoelastic fluids. The improved SPH algorithm includes the implementation of
(i) the mixed symmetric correction of kernel gradient to improve the accuracy
and stability of traditional SPH method and (ii) the Rusanov flux in the
continuity equation for improving the computation of pressure distributions in
the dynamics of liquids. To assess the effectiveness of the improved SPH
algorithm, a number of numerical examples including the stretching of an
initially circular water drop, dam breaking flow against a vertical wall, the
impact of viscous and viscoelastic fluid drop with a rigid wall, and the
extrudate swell of viscoelastic fluid have been presented and compared with
available numerical and experimental data in literature. The convergent
behavior of the improved SPH algorithm has also been studied by using
different number of particles. All numerical results demonstrate that the
improved SPH algorithm proposed here is capable of modeling free surface flows
of viscous and viscoelastic fluids accurately and stably, and even more
important, also computing an accurate and little oscillatory pressure field.
@articleEllero2012, title = SPH simulations of a viscoelastic flow around a
periodic array of cylinders confined in a channel, journal = Journal of Non-
Newtonian Fluid Mechanics, volume = 167-168, pages = 1-8, year = 2012, issn =
0377-0257, doi = https://doi.org/10.1016/j.jnnfm.2011.09.002, url =
https://www.sciencedirect.com/science/article/pii/S0377025711002199, author =
A. Vázquez-Quesada and M. Ellero, keywords = Smoothed Particle Hydrodynamics,
Oldroyd-B model, Elastic instabilities, abstract = In this paper a numerical
study on the performance of the Smoothed Particle Hydrodynamics method (SPH)
in the case of a flow of a viscoelastic liquid around a linear array of
cylinders confined in a channel is presented. Numerical convergence in the
case of a low Reynolds number Newtonian flow was demonstrated by Ellero et al.
in [Int. J. Numer. Methods Eng. 86 (2011) 1027]. Here viscoelastic effects are
incorporated in the SPH scheme according to the Oldroyd-B model presented in
[Phys. Rev. E 79 (2009) 056707]. Good agreement of the dimensionless drag
force acting on the cylinder with literature data is observed for a wide range
of Weissenberg numbers We. The case of closely spaced cylinders is also
investigated and the impact of We on the solution discussed. It turns out that
in this case the Newtonian solution exhibits a stable secondary flow
represented by two counter-rotating vortices. When elasticity is considered,
above a critical Weissenberg number Wec these vortices become unstable,
breaking the plane symmetry and producing a quasi-periodic flow of mass in and
out of the gap region between cylinders. Correspondingly, the flow becomes
increasingly unsteady and a dramatic increase in the cylinder’s drag is
observed. The results are in qualitative agreement with experimental
observations made on Boger liquids under similar flow conditions.
@articleLiu2012, title=SPH simulations of three-dimensional non-Newtonian free
surface flows, author=Xiaoyang Xu and Jie Ouyang and Bin-Xin Yang and Zhijun
Liu, journal=Computer Methods in Applied Mechanics and Engineering, year=2013,
volume=256, pages=101-116 @articleHosseini2007, title = An incompressible SPH
method for simulation of unsteady viscoelastic free-surface flows, journal =
International Journal of Non-Linear Mechanics, volume = 42, number = 10, pages
= 1210-1223, year = 2007, issn = 0020-7462, doi =
https://doi.org/10.1016/j.ijnonlinmec.2007.09.006, url =
https://www.sciencedirect.com/science/article/pii/S0020746207001862, author =
A. Rafiee and M.T. Manzari and M. Hosseini, keywords = Smoothed particle
hydrodynamics, Viscoelastic, Artificial stress, Free surface, abstract = In
this paper, an incompressible smoothed particle hydrodynamics (SPH) method is
presented to solve unsteady free-surface flows. Both Newtonian and
viscoelastic fluids are considered. In the case of viscoelastic fluids, both
the Maxwell and Oldroyd-B models are investigated. The proposed SPH method
uses a Poisson pressure equation to satisfy the incompressibility constraints.
The solution algorithm is an explicit predictor-corrector scheme and employs
an adaptive smoothing length based on density variations. To alleviate the
numerical difficulties encountered when fluid is highly stretched, an
artificial stress term is incorporated into the momentum equation which
reduces the risk of unrealistic fractures in the material. Two challenging
test cases, the impacting drop and the jet buckling problems, are solved to
demonstrate the capability of the proposed scheme in handling viscoelastic
flows with complex free surfaces. The jet buckling test case was solved for a
wide range of Weissenberg numbers. It was shown that in all cases the method
is stable and fairly accurate and agrees well with the available data.
@articleFang2009, title = Improved SPH methods for simulating free surface
flows of viscous fluids, journal = Applied Numerical Mathematics, volume = 59,
number = 2, pages = 251-271, year = 2009, issn = 0168-9274, doi =
https://doi.org/10.1016/j.apnum.2008.02.003, url =
https://www.sciencedirect.com/science/article/pii/S0168927408000305, author =
Jiannong Fang and Aurèle Parriaux and Martin Rentschler and Christophe Ancey,
keywords = Meshless/meshfree method, Smoothed Particle Hydrodynamics, Finite
particle method, Viscous fluids, Free surface flows, abstract = In this paper
we present two enhanced variants of the smoothed particle hydrodynamics (SPH)
method for the numerical simulation of free surface flows of viscous fluids.
Improvements are achieved by deriving a new set of general discrete SPH-like
equations under an energy-based framework and applying a corrected (high-
order) or coupled particle approximation scheme for function derivatives. By
doing so, we ensure that the enhanced variants retain the conservative nature
of SPH which is important for the stability of long-term simulations. Among
various corrected approximations, we here implement the one obtained by the
so-called finite particle method (FPM) within the framework to produce a
higher-order SPH method which conserves both linear and angular momentums. In
order to improve the efficiency of the higher-order variant, a coupled
approach with the idea of using the SPH approximation for the interior
particles and the FPM approximation for the exterior particles is also
proposed and tested in this paper. Three prototype tests concerning free
deformation of a viscous fluid patch with free surface are presented with
comparisons between different methods to demonstrate the performance of the
two proposed methods. Numerical results show that both the higher-order
version using FPM and the coupled version using FPM/SPH outperform the
original version of SPH in respect of accuracy and stability.
@articleFang2006, title=A numerical study of the SPH method for simulating
transient viscoelastic free surface flows, author=Jiannong Fang and Robert G.
Owens and Laurent Tacher and Aurèle Parriaux, journal=Journal of Non-newtonian
Fluid Mechanics, year=2006, volume=139, pages=68-84 @articleEllero2002, title
= Viscoelastic flows studied by smoothed particle dynamics, journal = Journal
of Non-Newtonian Fluid Mechanics, volume = 105, number = 1, pages = 35-51,
year = 2002, issn = 0377-0257, doi =
https://doi.org/10.1016/S0377-0257(02)00059-9, url =
https://www.sciencedirect.com/science/article/pii/S0377025702000599, author =
Marco Ellero and Martin Kröger and Siegfried Hess, keywords = Smoothed
particle hydrodynamics, Viscoelasticity, Isothermal flow, Fluid-pseudo-
particles, Jaumann–Maxwell model, abstract = A viscoelastic numerical scheme
based on smoothed particle dynamics is presented. The concept goes a step
beyond smoothed particle hydrodynamics (SPH) which is a grid-free Lagrangian
method describing the flow by fluid-pseudo-particles. The relevant properties
are interpolated directly on the resulting movable grid. In this work, the
effect of viscoelasticity is incorporated into the ordinary conservation laws
by a differential constitutive equation supplied for the stress tensor. In
order to give confidence in the methodology we explicitly consider the non-
stationary simple corotational Maxwell model in a channel geometry. Without
further developments the scheme is applicable to ‘realistic’ models relevant
for three-dimensional (3D) viscoelastic flows in complex geometries.
@articleMonaghan2005, doi = 10.1088/0034-4885/68/8/R01, url =
https://dx.doi.org/10.1088/0034-4885/68/8/R01, year = 2005, month = jul,
publisher = , volume = 68, number = 8, pages = 1703, author = J J Monaghan,
title = Smoothed particle hydrodynamics, journal = Reports on Progress in
Physics, abstract = In this review the theory and application of Smoothed
particle hydrodynamics (SPH) since its inception in 1977 are discussed.
Emphasis is placed on the strengths and weaknesses, the analogy with particle
dynamics and the numerous areas where SPH has been successfully applied.
@articleMonaghan1977, author = Gingold, R. A. and Monaghan, J. J., title =
”Smoothed particle hydrodynamics: theory and application to non-spherical
stars”, journal = Monthly Notices of the Royal Astronomical Society, volume =
181, number = 3, pages = 375-389, year = 1977, month = 12, abstract = ”A new
hydrodynamic code applicable to a space of an arbitrary number of dimensions
is discussed and applied to a variety of polytropic stellar models. The
principal feature of the method is the use of statistical techniques to
recover analytical expressions for the physical variables from a known
distribution of fluid elements. The equations of motion take the form of
Newtonian equations for particles. Starting with a non-axisymmetric
distribution of approximately 80 particles in three dimensions, the method is
found to reproduce the structure of uniformly rotating and magnetic polytropes
to within a few per cent. The method may be easily extended to deal with more
complicated physical models.”, issn = 0035-8711, doi =
10.1093/mnras/181.3.375, url = https://doi.org/10.1093/mnras/181.3.375, eprint
= https://academic.oup.com/mnras/article-
pdf/181/3/375/3104055/mnras181-0375.pdf,

@ARTICLELucy1977, author = Lucy, L. B., title = ”A numerical approach to the
testing of the fission hypothesis.”, journal = Astronomical Journal, keywords
= Binary Stars, Hypotheses, Nuclear Fission, Numerical Analysis, Protostars,
Stellar Evolution, Astronomical Models, Difference Equations, Gas Dynamics,
Monte Carlo Method, Numerical Stability, Particle Motion, Stellar Rotation,
Astrophysics, year = 1977, Month = Dec, volume = 82, pages = 1013–1024, doi =
10.1086/112164, adsurl = https://ui.adsabs.harvard.edu/abs/1977AJ…..82.1013L,
adsnote = Provided by the SAO/NASA Astrophysics Data System

@articleLind2020, author = Lind, Steven J. and Rogers, Benedict D. and
Stansby, Peter K. , title = Review of smoothed particle hydrodynamics: towards
converged Lagrangian flow modelling, journal = Proceedings of the Royal
Society A: Mathematical, Physical and Engineering Sciences, volume = 476,
number = 2241, pages = 20190801, year = 2020, doi = 10.1098/rspa.2019.0801,
URL = https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2019.0801,
eprint = https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2019.0801 ,
abstract = This paper presents a review of the progress of smoothed particle
hydrodynamics (SPH) towards high-order converged simulations. As a mesh-free
Lagrangian method suitable for complex flows with interfaces and multiple
phases, SPH has developed considerably in the past decade. While original
applications were in astrophysics, early engineering applications showed the
versatility and robustness of the method without emphasis on accuracy and
convergence. The early method was of weakly compressible form resulting in
noisy pressures due to spurious pressure waves. This was effectively removed
in the incompressible (divergence-free) form which followed; since then the
weakly compressible form has been advanced, reducing pressure noise. Now
numerical convergence studies are standard. While the method is
computationally demanding on conventional processors, it is well suited to
parallel processing on massively parallel computing and graphics processing
units. Applications are diverse and encompass wave–structure interaction,
geophysical flows due to landslides, nuclear sludge flows, welding, gearbox
flows and many others. In the state of the art, convergence is typically
between the first- and second-order theoretical limits. Recent advances are
improving convergence to fourth order (and higher) and these will also be
outlined. This can be necessary to resolve multi-scale aspects of turbulent
flow. @articleWilsonCates2019, title = Constitutive Model for Time-Dependent
Flows of Shear-Thickening Suspensions, author = Gillissen, J. J. J. and Ness,
C. and Peterson, J. D. and Wilson, H. J. and Cates, M. E., journal = Phys.
Rev. Lett., volume = 123, issue = 21, pages = 214504, numpages = 6, year =
2019, month = Nov, publisher = American Physical Society, doi =
10.1103/PhysRevLett.123.214504, url =
https://link.aps.org/doi/10.1103/PhysRevLett.123.214504

@articleWilsonCates2020, author = Gillissen,J. J. J. and Ness,C. and
Peterson,J. D. and Wilson,H. J. and Cates,M. E. , title = Constitutive model
for shear-thickening suspensions: Predictions for steady shear with superposed
transverse oscillations, journal = Journal of Rheology, volume = 64, number =
2, pages = 353-365, year = 2020, doi = 10.1122/1.5129657,

URL = https://doi.org/10.1122/1.5129657

, eprint = https://doi.org/10.1122/1.5129657

@articleKamrin2019, author = Aaron S. Baumgarten and Ken Kamrin , title = A
general constitutive model for dense, fine-particle suspensions validated in
many geometries, journal = Proceedings of the National Academy of Sciences,
volume = 116, number = 42, pages = 20828-20836, year = 2019, doi =
10.1073/pnas.1908065116, URL =
https://www.pnas.org/doi/abs/10.1073/pnas.1908065116, eprint =
https://www.pnas.org/doi/pdf/10.1073/pnas.1908065116, abstract = Fine-particle
suspensions (such as cornstarch mixed with water) exhibit dramatic changes in
viscosity when sheared, producing fascinating behaviors that captivate
children and rheologists alike. Examination of these mixtures in simple flow
geometries suggests intergranular repulsion and its influence on the
frictional nature of granular contacts is central to this effect—for mixtures
at rest or shearing slowly, repulsion prevents frictional contacts from
forming between particles, whereas when sheared more forcefully, granular
stresses overcome the repulsion allowing particles to interact frictionally
and form microscopic structures that resist flow. Previous constitutive
studies of these mixtures have focused on particular cases, typically limited
to 2D, steady, simple shearing flows. In this work, we introduce a predictive
and general, 3D continuum model for this material, using mixture theory to
couple the fluid and particle phases. Playing a central role in the model, we
introduce a microstructural state variable, whose evolution is deduced from
small-scale physical arguments and checked with existing data. Our space- and
time-dependent model is implemented numerically in a variety of unsteady,
nonuniform flow configurations where it is shown to accurately capture a
variety of key behaviors: 1) the continuous shear-thickening (CST) and
discontinuous shear-thickening (DST) behavior observed in steady flows, 2) the
time-dependent propagation of “shear jamming fronts,” 3) the time-dependent
propagation of “impact-activated jamming fronts,” and 4) the non-Newtonian,
“running on oobleck” effect, wherein fast locomotors stay afloat while slow
ones sink.

@articleWC2014, title = Discontinuous Shear Thickening without Inertia in
Dense Non-Brownian Suspensions, author = Wyart, M. and Cates, M. E., journal =
Phys. Rev. Lett., volume = 112, issue = 9, pages = 098302, numpages = 5, year
= 2014, month = Mar, publisher = American Physical Society, doi =
10.1103/PhysRevLett.112.098302, url =
https://link.aps.org/doi/10.1103/PhysRevLett.112.098302

@articleBrown_2014, doi = 10.1088/0034-4885/77/4/046602, url =
https://dx.doi.org/10.1088/0034-4885/77/4/046602, year = 2014, month = apr,
publisher = IOP Publishing, volume = 77, number = 4, pages = 046602, author =
Eric Brown and Heinrich M Jaeger, title = Shear thickening in concentrated
suspensions: phenomenology, mechanisms and relations to jamming, journal =
Reports on Progress in Physics, abstract = Shear thickening is a type of non-
Newtonian behavior in which the stress required to shear a fluid increases
faster than linearly with shear rate. Many concentrated suspensions of
particles exhibit an especially dramatic version, known as Discontinuous Shear
Thickening (DST), in which the stress suddenly jumps with increasing shear
rate and produces solid-like behavior. The best known example of such counter-
intuitive response to applied stresses occurs in mixtures of cornstarch in
water. Over the last several years, this shear-induced solid-like behavior
together with a variety of other unusual fluid phenomena has generated
considerable interest in the physics of densely packed suspensions. In this
review, we discuss the common physical properties of systems exhibiting shear
thickening, and different mechanisms and models proposed to describe it. We
then suggest how these mechanisms may be related and generalized, and propose
a general phase diagram for shear thickening systems. We also discuss how
recent work has related the physics of shear thickening to that of granular
materials and jammed systems. Since DST is described by models that require
only simple generic interactions between particles, we outline the broader
context of other concentrated many-particle systems such as foams and
emulsions, and explain why DST is restricted to the parameter regime of hard-
particle suspensions. Finally, we discuss some of the outstanding problems and
emerging opportunities.

@articlelind2021, title = Incompressible smoothed particle hydrodynamics for
free-surface flows: A generalised diffusion-based algorithm for stability and
validations for impulsive flows and propagating waves, journal = Journal of
Computational Physics, volume = 231, number = 4, pages = 1499-1523, year =
2012, issn = 0021-9991, doi = https://doi.org/10.1016/j.jcp.2011.10.027, url =
https://www.sciencedirect.com/science/article/pii/S0021999111006279, author =
S.J. Lind and R. Xu and P.K. Stansby and B.D. Rogers, keywords =
Incompressible SPH, Free-surface flow simulation, Fick’s law, Truncated-kernel
error, abstract = The incompressible smoothed particle hydrodynamics (ISPH)
method with projection-based pressure correction has been shown to be highly
accurate and stable for internal flows and, importantly for many problems, the
pressure field is virtually noise-free in contrast to the weakly compressible
SPH approach (Xu et al., 2009 [31]). However for almost inviscid fluids
instabilities at the free surface occur due to errors associated with the
truncated kernels. A new algorithm is presented which remedies this issue,
giving stable and accurate solutions to both internal and free-surface flows.
Generalising the particle shifting approach of Xu et al. (2009) [31], the
algorithm is based upon Fick’s law of diffusion and shifts particles in a
manner that prevents highly anisotropic distributions and the onset of
numerical instability. The algorithm is validated against analytical solutions
for an internal flow at higher Reynolds numbers than previously, the flow due
to an impulsively started plate and highly accurate solutions for wet bed dam
break problems at zero and small times. The method is then validated for
progressive regular waves with paddle motion defined by linear theory. The
accurate predictions demonstrate the effectiveness of the algorithm in
stabilising solutions and minimising the surface instabilities generated by
the inevitable errors associated with truncated kernels. The test cases are
thought to provide a more thorough quantitative validation than previously
undertaken.

@articleAdami2012, title = A generalized wall boundary condition for smoothed
particle hydrodynamics, journal = Journal of Computational Physics, volume =
231, number = 21, pages = 7057-7075, year = 2012, issn = 0021-9991, doi =
https://doi.org/10.1016/j.jcp.2012.05.005, url =
https://www.sciencedirect.com/science/article/pii/S002199911200229X, author =
S. Adami and X.Y. Hu and N.A. Adams, keywords = Boundary condition, SPH, Solid
walls, abstract = In this paper we present a new formulation of the boundary
condition at static and moving solid walls in SPH simulations. Our general
approach is both applicable to two and three dimensions and is very simple
compared to previous wall boundary formulations. Based on a local force
balance between wall and fluid particles we apply a pressure boundary
condition on the solid particles to prevent wall penetration. This method can
handle sharp corners and complex geometries as is demonstrated with several
examples. A validation shows that we recover hydrostatic equilibrium
conditions in a static tank, and a comparison of the classical dam break
simulation with state-of-the-art results in literature shows good agreement.
We simulate various problems such as the flow around a cylinder and the
backward facing step at Re=100 to demonstrate the general applicability of
this new method.

@articlewagner_dst, author = Cwalina,Colin D. and Wagner,Norman J. , title =
Rheology of non-Brownian particles suspended in concentrated colloidal
dispersions at low particle Reynolds number, journal = Journal of Rheology,
volume = 60, number = 1, pages = 47-59, year = 2016, doi = 10.1122/1.4935445,
URL = https://doi.org/10.1122/1.4935445, eprint =
https://doi.org/10.1122/1.4935445

@articleSeto2013, title = Discontinuous Shear Thickening of Frictional Hard-
Sphere Suspensions, author = Seto, Ryohei and Mari, Romain and Morris, Jeffrey
F. and Denn, Morton M., journal = Phys. Rev. Lett., volume = 111, issue = 21,
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