12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022

EXPERIMENTAL INVESTIGATION OF SHEAR-INDUCED MIGRATION IN
PARTICLE-LADEN PIPE FLOW USING MRIWillian Hogendoorn
David FrankMultiphase Systems (3ME-P&E)
Lehrstuhl Strömingsmechanik
Delft University of Technology
Universität Rostock
Leeghwaterstraat 39, 2628 CB Delft, The Netherlands Albert-Einstein-Str. 2, 18059 Rostock, Germany
w.j.hogendoorn@tudelft.nl
david.frank@uni-rostock.deMartin Bruschewski
Christian Poelma
Lehrstuhl Strömingsmechanik
Multiphase Systems (3ME-P&E)
Universität Rostock
Delft University of Technology
Albert-Einstein-Str. 2, 18059 Rostock, Germany Leeghwaterstraat 39, 2628 CB Delft, The Netherlands
martin.bruschewski@uni-rostock.dec.poelma@tudelft.nl

ABSTRACT

direction of the velocity gradient, where the particles are transported in the direction of decreasing shear rate. These concentration gradients are subsequently responsible for (strong)
viscosity gradients. Therefore, the effective viscosity needs
to be modelled as a local viscosity instead of a constant viscosity for particle-laden shear flows (Gadala-Maria & Acrivos,1980). The phenomena of shear-induced migration is observedin a variety of flow configurations, including Couette systems(Gadala-Maria & Acrivos, 1980; Abbott et al., 1991; Blanc
et al., 2013), channel flows (Lyon & Leal, 1998; Zade et al.,2018) and pipe flows (Altobelli et al., 1991; Sinton & Chow,1991). In addition to experimental work, theoretical models
were introduced by Leighton & Acrivos (1987) and Phillips
et al. (1992) in order to explain the experimental observations.
Furthermore, a suspension balance model (SBM) was introduced by Nott & Brady (1994). In a revision of this model, awell-defined particle phase stress is added, instead of the identification of the particle phase stress with the particle contribution to the suspension stress (Nott et al., 2011). By taking this particle stress contribution into account, this model
is distinct from other models, which only take into account aforce acting on the particle phase. However, experimental andtheoretical studies are predominantly limited to the Stokesian
regime in order to avoid inertial effects, as these effects willfurther complicate the modeling efforts. In this study experiments are performed for a range of Reynolds numbers, spanning the laminar, transitional and turbulent region, as for many
practical applications inertial effects cannot be neglected. Furthermore, in order to resolve discrepancies between numericaland experimental studies, a reliable experimental data set will
significantly aid the the validation and development of numerical models.

Using magnetic resonance imaging we are able to obtain average velocity and volume fraction profiles in a pipe
flow with a neutrally buoyant suspension. In this experimental work, the effect of increasing Reynolds number and particle volume fraction on shear-induced migration is studied.
For increasing bulk volume fraction, the initially nearly homogeneous suspension gradually changes to a strongly nonhomogeneous suspension. This is observed for all studied
Reynolds numbers. In contrast to the majority of previous(MRI) studies, experiments are also performed for suspension
Reynolds numbers of approximately 5000 in order to study inertial effects on shear-induced migration.

INTRODUCTION
Particle-laden flows are gaining a growing and continuous interest over the last decades (Guazzelli & Pouliquen,2018; Morris, 2020). This interest is motivated by the relevance of these suspensions in industrial, geophysical and societal contexts. A variety of practical applications include thefood processing industry, 3D bioprinting of organs, and sediment transport. The modeling of suspension flow dynamics
in moderate and concentrated regimes is considered to be achallenge, in particular when inertial effects cannot longer beignored. Despite significant theoretical, experimental and numerical progress, open questions still remain. In this experimental work a systematic study is conducted to investigate
the effect of the Reynolds number (Re) and particle volume
fraction (φ ) on shear-induced migration. To this end, magnetic resonance imaging (MRI) experiments are performed ona neutrally-buoyant particle-laden system. In contrast to otherexperimental techniques, MRI allows for accurate measurements in dense suspensions, where velocity and volume fractions can be simultaneously measured.
Shear-induced migration will be observed when studying suspension flow dynamics in shear flows. Finite-sizedparticles experience a force resulting from the velocity gradient: an initially homogeneous suspension rearranges to a nonhomogeneous mixture. Particle clusters are formed along theEXPERIMENTAL DETAILSExperiments are performed in a 30.35±0.12 mm diameter (D) pipe flow setup using unexpanded polystyrene particles(Synthos EPS) with a diameter, d = 1.75±0.12 mm, resulting
in a particle-to-pipe diameter ratio, d/D = 0.058. For the continuous phase a glycerol-water mixture (13.6/86.4%) is used1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
Table 1: Used MRI parameters

Parameter

Value

Matrix size

640 × 640 × 1

Non-isotropic resolution

0.3 × 0.3 × 50 mm3

Repetition time (TR)22 ms

Echo time (TE)

9 ms

RF flip angle

5◦

Receiver bandwidth

280 Hz/pixel

Velocity Encoding

0.1 –1.7 m/s

Number of averages

32

profiles of less than 1% is found. Furthermore, the error in
the volume fraction is determined using error propagation (Bruschewski et al., 2016). For all 19 cases studied a maximum
error of 2.3% is found. The order of magnitude of this error
is confirmed by comparing the integrated volume fraction profiles with the target bulk volume fraction set by preparing thesuspension. For the highest volume fraction case (φ = 0.5), a
discrepancy of 2.7% is found, as the integrated volume fraction
was determined to be 0.473. Note that this error comprises all
experimental and measurement uncertainties, i.e., errors when
preparing the suspension, inhomogeneities in the system, calibration errors of the MRI system, reconstruction errors, etc.
The actual bulk volume fraction in the pipe is particularly important for the determination of the corresponding suspension
based Reynolds number (Res ). This Reynolds number is basedon the effective suspension dynamic viscosity, µs , which is determined using Eilers’s viscosity correction (Eilers, 1941):2µsφ.
= 1 + 1.25
µ01 − φ /φmin order to match the density of the fluid with the particles,ρ = 1032±1.17 kg/m3 . Furthermore, coppersulfate (CuSO4 ;1g/L) is added in order to enhance the signal-to-noise ratio ofthe MRI measurements. A progressive cavity pump (AxFlow
B.V., Lelystad, the Netherlands) is used to transport the suspension. The flowloop is constructed as follows: first, a settling chamber is used. An orifice is placed 10D after this settling chamber in order to ensure a transition at a fixed critical
Reynolds number, Rec ≈ 2000 for single-phase flows. Notethat this onset of turbulence for particle-laden flows is a function of d/D and the volume fraction, φ (Hogendoorn et al.,2022). MRI measurements are obtained at a downstream distance of 132D from this orifice. The flowrate is monitored using an inline Coriolis mass flow meter (KROHNE OPTIMASS7050c), which is located in the return section. This return loop
is connected to a reservoir, closing the flow loop, as the pumpis fed from this reservoir. In order to be able to achieve low
Reynolds numbers in the system, a bypass is installed fromthe pump exit to the reservoir. Additionally, a valve is placedin the main flow loop to be able to reduce the Reynolds number even further. A cooling system (JULABO FT 402) is usedto minimise the temperature increase of the suspension fromheat addition by the pump. Particles are added in steps of
10% until a volume fraction of 50%. In addition, experiments
are performed for a volume fraction of 45%. Experiments areperformed for three different target Reynolds numbers of 800,
2000 and 5000.
A MAGNETOM Trio 3T Whole-Body MRI scanner(Siemens, Erlangen, Germany) is used to obtain velocity andconcentration data. This system has a maximum gradient amplitude of 40 mT/m and a maximum gradient slew rate of 200
T/m/s. Two standard receive-only body coils were used to receive the signal, with a receiver bandwidth of 280 Hz/pixel.
Repetition (TR) and echo (TE) times of respectively 22 and9 ms are used, in combination with a RF flip angle of 5◦ . A
non-isotropic resolution of 0.3×0.3×50 mm3 is used in thex, y, and z direction, respectively. The z component is in thestreamwise direction. The x − y plane is perpendicular to thestreamwise direction with x the horizontal and y the verticalcomponent. The used parameters are summarised in Table 1.
In order to validate the complete experimental setup andthe MRI system, experiments are compared with reference
data from literature. To this end single-phase experiments are
performed for Re = 5300, 10000, and 25000, for which average velocity profiles are available. A total average error (averaged along the radial direction) between the mean velocity(1)Here µ0 is the viscosity of the single-phase fluid, in our casea glycerol-water mixture. For the maximum packing fraction,
φm , a value of 0.64 is used. In previous studies this value isshown to be an appropriate choice for our batch of particles(Hogendoorn & Poelma, 2018; Hogendoorn et al., 2021). Furthermore, the volume fraction used to determine the apparent
viscosity is based on the integrated volume fraction profile,as this is considered to result in a more accurate determination of Res . This in contrast of using the mass fraction of theparticles and fluid when preparing the suspension, as due tothe radial concentration gradients the apparent volume fraction in the pipe differs from the bulk volume fraction in themain reservoir. Also, the average volume fractions reported
in this study are based on the same integrated volume fraction
profiles.

RESULTSA schematic overview of all conducted experiments, including the single-phase validation experiments, are shownin Fig. 1. The marker color indicate the corresponding par1 1ticle perturbation amplitude, ε = Dd 2 φ 6 , as introduced by
Hogendoorn et al. (2022). This parameter was shown to distinguish between different transition scenarios and allows aprediction of the onset of turbulence for a range of suspensions. The three different target Reynolds numbers (i.e., Res= 800, 2000, and 5000) are indicated by the horizontal dashedlines. The deviation of the actual measurements from the target Reynolds numbers is due to the fact that the actual volumefraction in the measurement section is determined a posteriori.
This is done by circumferential integration of the particle volume fraction profile, obtained from the MRI measurements.
This method is considered to result in a more accurate determination of the bulk volume fraction, rather than using themass fractions of the fluid and solid phase when preparing thesuspension (see also the discussion in the validation section).
Note that the suspension Reynolds number is also affected by
the determined bulk volume fraction.
The average velocity and concentration profiles for an approximate suspension Reynolds number of 2000 are shown inFig. 2a and Fig. 2b, respectively. For this selected Reynolds
number several observations can be made. First, for increasing volume fraction, the velocity profiles gradually changes

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
30000

0.25

migration is observed for a range of Reynolds numbers. For
increasing volume fraction, the initially nearly homogeneous
suspension gradually changes to a strongly non-homogeneoussuspension. This shows that particle-particle interactions play
an important role in the underlying dynamics. In particular forφb ≥ 0.27 a dense particle core is observed, explaining the observed blunted velocity profiles. Currently, direct numericalsimulations are pursued in order to explain the obtained velocity and volume fraction profiles and to distinguish between thedifferent contributions involved.

0.2

10000
0.15

5000

0.1

2000
0.05

800000.10.2

0.3

0.4

0.5

REFERENCES
Abbott, JR, Tetlow, N, Graham, AL, Altobelli, SA,Fukushima, Eiichi, Mondy, LA & Stephens, TS 1991
Experimental observations of particle migration in concentrated suspensions: Couette flow. Journal of rheology
35 (5), 773–795.
Altobelli, SA, Givler, RC & Fukushima, Eiichi 1991 Velocity
and concentration measurements of suspensions by nuclear
magnetic resonance imaging. Journal of Rheology 35 (5),721–734.
Blanc, Frédéric, Lemaire, Elisabeth, Meunier, Alain & Peters,François 2013 Microstructure in sheared non-Brownian
concentrated suspensions. Journal of rheology 57 (1), 273–292.
Bruschewski, Martin, Freudenhammer, Daniel, Buchenberg,
Waltraud B, Schiffer, Heinz-Peter & Grundmann, Sven2016 Estimation of the measurement uncertainty in magnetic resonance velocimetry based on statistical models. Experiments in Fluids 57 (5), 1–13.
Eilers, von H 1941 Die viskosität von emulsionen
hochviskoser stoffe als funktion der konzentration.
Kolloid-Zeitschrift 97 (3), 313–321.
Gadala-Maria, F & Acrivos, Andreas 1980 Shear-inducedstructure in a concentrated suspension of solid spheres.
Journal of Rheology 24 (6), 799–814.
Guazzelli, Elisabeth & Pouliquen, Olivier 2018 Rheology ofdense granular suspensions. Journal of Fluid Mechanics
852.
Hampton, RE, Mammoli, AA, Graham, AL, Tetlow, N & Altobelli, SA 1997 Migration of particles undergoing pressuredriven flow in a circular conduit. Journal of Rheology
41 (3), 621–640.
Hogendoorn, Willian, Chandra, Bidhan & Poelma, Christian2021 Suspension dynamics in transitional pipe flow. Physical Review Fluids 6 (6), 064301.
Hogendoorn, Willian, Chandra, Bidhan & Poelma, Christian2022 Onset of turbulence in particle-laden pipe flows. Phys.
Rev. Fluids 7, L042301.
Hogendoorn, Willian & Poelma, Christian 2018 Particle-ladenpipe flows at high volume fractions show transition withoutpuffs. Physical Review Letters 121 (19), 194501.
Leighton, David & Acrivos, Andreas 1987 The shear-induced
migration of particles in concentrated suspensions. Journalof Fluid Mechanics 181, 415–439.
Lyon, MK & Leal, LG 1998 An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 1. Monodisperse systems. Journal of fluid mechanics 363, 25–56.
Morris, Jeffrey F. 2020 Toward a fluid mechanics of suspensions. Phys. Rev. Fluids 5, 110519.
Nott, Prabhu R & Brady, John F 1994 Pressure-driven flow ofsuspensions: simulation and theory. Journal of Fluid Mechanics 275, 157–199.

Figure 1: Schematic overview of performed experiments,

as function of Res and φ . The marker color indicate theparticle perturbation amplitude, ε.

from a parabolic profile to a blunted profile, eventually resulting in a uniform velocity in the pipe center for the case with
φb = 0.47. Note that the velocity profile corresponding to Res= 2307, φb = 0.08 deviates from this trend. This is most likely
explained by the change in transition mechanism which occurs
for these specific conditions (Hogendoorn & Poelma, 2018;
Hogendoorn et al., 2021). Also, from the corresponding volume fractions profiles in Fig. 2b, it can be seen that the particle
distribution for this specific case is nearly uniform. For increasing volume fraction this nearly homogeneous suspension
gradually changes to a strongly non-homogeneous suspension(see, e.g., φb ≥ 0.27). This suggest that particle-particle interactions are playing an important role in the suspension dynamics (see, e.g., Guazzelli & Pouliquen, 2018). Additionally,
the local shear-rate will be an important parameter for the volume fraction distribution. The volume fraction profiles also
explains the (degree of) blunting of the velocity profiles for increasing φb . For φb > 0.27 the particle distribution in the pipe
center is found to be approaching the random close packing,which is inherently ordered and restricts shear. In the directionof the pipe wall, the volume fraction decreases. This results ina (more) dis-ordered system, allowing more shear; this is also
visible in the velocity profiles. Eventually, in the near-wall region (r/D ≈ 0.47) a concentration ‘peak’ can be observed forthe dense cases. This ordered particle layer was observed before by e.g., Hampton et al. (1997). The peak becomes more
pronounced for increasing φb . Moreover, additional particle
rings can be observed, as can be interpreted from the additional ‘wiggles’ in the profiles of the dense cases.
These particle rings are even better visible for the highest
volume fraction experiment at a lower Res . The average velocity and volume fraction profile are shown in Fig. 3a. The velocity profile is normalised using the bulk velocity and corresponds to the left y-axis, whereas the average volume fraction
profile corresponds to the right y-axis. A pipe cross-section ofthe particle distribution is shown in the 2D grayscale map of
the volume fraction in Fig. 3b. This contains similar information as the average volume fraction profile.

CONCLUSION AND OUTLOOKThe effect of increasing volume fraction and Reynolds
numbers is studied on shear-induced migration. Average velocity and concentration profiles are obtained for neutrally
buoyant particle-laden pipe flow using MRI. Shear-induced

312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
2

0.8

1.5

0.6

1

0.4

0.5

0.2

000

0.1

0.2

0.3

0.4

0.5

(a)

00.1

0.2

0.3

0.4

0.5

(b)

Figure 2: (a) Normalised average velocity profiles as function of the pipe radius for approximately constant suspension

Reynolds number, Res ≈ 2000, and increasing volume fraction. (b) Average concentration profiles as function of the piperadius. The markers are corresponding to the conditions indicated in the legend in Fig. 2a.

1
1.2
1

0.8

0.8

0.6

0.6
0.4
0.4
0.2

0.2
00

0.1

0.2

0.3

0.4

0
0.5

(a)

(b)

Figure 3: (a) Normalised average velocity profile (blue squares, left y-axis) and average volume fraction profile (red circles,right y-axis) as function of the pipe radius for a suspension Reynolds number, Res = 859, and bulk volume fraction, φb =0.473. (b) 2D map of the volume fraction corresponding to the profile of Fig. 3a.

Nott, Prabhu R, Guazzelli, Elisabeth & Pouliquen, Olivier
2011 The suspension balance model revisited. Physics ofFluids 23 (4), 043304.
Phillips, Ronald J, Armstrong, Robert C, Brown, Robert A,
Graham, Alan L & Abbott, James R 1992 A constitutive equation for concentrated suspensions that accounts forshear-induced particle migration. Physics of Fluids A: FluidDynamics 4 (1), 30–40.

Sinton, Steven W & Chow, Andrea W 1991 NMR flow imaging of fluids and solid suspensions in Poiseuille flow. Journal of Rheology 35 (5), 735–772.
Zade, Sagar, Costa, Pedro, Fornari, Walter, Lundell, Fredrik &Brandt, Luca 2018 Experimental investigation of turbulent
suspensions of spherical particles in a square duct. Journalof Fluid Mechanics 857, 748–783.

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