3.1.1 Sphere in square duct

We start with the simulation of a single spherical rigid particle in a square duct. Most of the simulations are performed for a size ratio, the ratio between the duct height (width) to the sphere diameter, $H/D_{s}=5$ . We define the average particle Reynolds number by $\overline{Re_{p}}=\overline{\dot{\unicode[STIX]{x1D6FE}}}D_{s}^{2}/\unicode[STIX]{x1D708}$ where $\overline{\dot{\unicode[STIX]{x1D6FE}}}=2U_{b}/D_{h}$ is the average shear rate in the duct and $D_{h}=2WH/(W+H)$ is the hydrodynamic diameter of a duct with height, $H$ , and width, $W$ . Note that $D_{h}=H$ for a square duct. With these definitions, the bulk Reynolds number, $Re_{b}=100$ , corresponds to the average particle Reynolds number, $\overline{Re_{p}}=2Re_{b}(D_{s}/H)^{2}=8$ when $H/D_{s}=5$ . Due to the geometrical symmetry, we vary the initial position of the particle only over one-eighth of the duct cross-section. In figure 2(a), top part, we show the sphere lateral trajectory for different simulations where open circles and triangles represent the start and end positions of the particle centre, respectively. We show as an example the initial and final position of the particle projection onto the cross-section using dashed and solid light blue lines. We clearly observe that all the trajectories end close to the centre of a duct wall, a face-centred equilibrium position, independent of the initial position of the particles. This position is one of the four equilibrium positions observed in the square duct already in previous works, among others Chun & Ladd (Reference Chun and Ladd2006) and Di Carlo et al. (Reference Di Carlo, Irimia, Tompkins and Toner2007). For $Re_{b}=100$ and size ratio $H/D_{s}=5$ we report the equilibrium position to be approximately $0.2H$ away from the wall of the duct, i.e. $0.3H$ from the centre.

Owing to the absence of radial symmetry in ducts, the lateral motion of the particles cannot be simply explained by the balance between shear-induced and wall-induced lift forces. In duct flows, the particle lateral motion, from the initial to the final equilibrium position, can be seen as a two-step process. (i) The fast migration towards the equilibrium wall, the wall adjacent to the equilibrium position, resulting from the interplay between the strong shear-induced and wall-induced lift forces. (ii) The slow migration towards the equilibrium position induced by the weak rotation-induced lift force (see also Choi, Seo & Lee Reference Choi, Seo and Lee2011; Zhou & Papautsky Reference Zhou and Papautsky2013). Interestingly, the trajectories collapse onto a single curve, denoted the equilibrium manifold, during the second slow lateral motion towards the equilibrium position. The equilibrium manifold is a unique lateral path leading to the equilibrium position regardless of their initial positions. Even though the existence of this manifold has been reported in the recent works by Prohm & Stark (Reference Prohm and Stark2014) and Liu et al. (Reference Liu, Hu, Jiang and Sun2015), the underlying physical mechanism is still not explained. We will attempt an explanation in connection with the results of the particle migration in a rectangular duct.

We display the effect of the size ratio, $H/D_{s}$ , on the equilibrium position of a sphere in a square duct in the lower part of figure 2(a). We simulate the motion of the spheres of three different sizes at $Re_{b}=100$ starting from the same lateral position in the duct cross-section and observe a similar behaviour: the particles focus at closest face-centred equilibrium position. At equilibrium, the larger particles are closer to the centre. This is mainly attributed to the steric effect as discussed by Amini et al. (Reference Amini, Lee and Di Carlo2014). The final location of spheres with size ratio $H/D_{s}=[3.5,5,10]$ is $[0.27H,0.3H,0.34H]$ from the duct centre. The distribution of different sizes at the equilibrium position resembles the schematic diagram of Hood et al. (Reference Hood, Lee and Rope2015) based on the theoretical prediction of the lateral force on the particles.

Figure 2.

(a) Upper side of duct cross-section: lateral trajectory of a spherical rigid particle in one-eighth of a square duct at $Re_{b}=100$ . The ratio between the duct height to particle diameter is $H/D_{s}=5$ . Open circles and triangles show the starting and ending points of the lateral trajectories. We show as an example the initial and final position of a sphere with light blue dashed and solid lines. Lower side of the cross-section: particle size effect on the lateral trajectory and the equilibrium position of a single sphere in square duct at $Re_{b}=100$ . The dashed and solid lines display the initial and final position of the spheres. Right side of the cross-section: lift coefficient versus downstream traveling length for some of the particle trajectories. (b) Focusing length for different particle trajectories (symbols are displayed at a fixed time interval). The focusing length of larger and smaller spheres is also shown by solid lines. (c) Reynolds number dependence of the equilibrium position of a spherical particle with size ratio $H/D_{s}=5$ .

To better understand the origins of these lateral motions, we examine the particle lift coefficient in the duct. Following the formulation introduced by Asmolov (Reference Asmolov1999), we express the lift coefficient of a spherical particle as

where $F_{L}$ is the lateral force on the particle in the opposite direction of its cross-flow velocity, $\unicode[STIX]{x1D70C}$ is the density of the fluid and particle and $a$ denotes the particle radius. Since the particle lateral velocity, $U_{L}$ , is low the force on the particle can be estimated by Stokes law, $F_{L}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}aU_{L}$ , where $\unicode[STIX]{x1D708}$ is the fluid dynamic viscosity (Zhou & Papautsky Reference Zhou and Papautsky2013). Combing with (3.1), the lift coefficient can be written as

This is estimated employing the particle lateral velocity from the simulation data for a particle moving through the duct. The lift coefficients, corresponding to some particle trajectories, are shown in figure 2(a) versus the downstream distance, right-hand side of the cross-section. The profiles reflect the initial acceleration and final deceleration of the particle toward the equilibrium position due to the shear-induced and wall-induced lift forces. In particular, the particle experiences the highest lift coefficient at the beginning of its lateral motion, especially if it is started close to a diagonal symmetry line; i.e. red trajectory.

In figure 2(b) we show the downstream length needed for a sphere to focus, the focusing length, as a function of the distance from the vertical duct symmetry line. For each particle trajectory the same colour as in panel (a) is used. As regards spheres with size ratio $H/D_{s}=5$ , we observe that if the particle is initially located close to the duct centre and in particular close to the diagonal symmetry line, the focusing length is longer. For the particular particle trajectory indicated by the red solid line, the focusing length is approximately 250 times the channel height. Considering the particle motion on the longest possible trajectory towards the equilibrium position, we can predict the minimum streamwise length required for the particle focusing. In addition, we note that if the particle starts close to the duct centre, its lateral motion shows the classical slow and fast motion (the symbols represent the same time intervals). The opposite is true if the particle starts close to the wall; in this case, the lateral motion is more uniform. Considering spheres of different sizes, we report that small spheres require longer ducts, approximately 600 times the channel height, to focus for the same bulk Reynolds number. This behaviour is exploited in the design of a microfluidic system to efficiently separate small and large particles (Zhou et al. Reference Zhou, Giridhar, Kasperb and Papautsky2013).

Next we examine the effect of Reynolds number on the equilibrium position of a sphere in a square duct. It is known that the particle in the centre of the channel always tends to move toward the wall regardless of Reynolds number. The final equilibrium position however gets closer to the wall with increasing the Reynolds number. This effect is first observed by Segre & Silberberg (Reference Segre and Silberberg1962) and then studied theoretically by Schonberg & Hinch (Reference Schonberg and Hinch1989) for $Re_{b}<75$ and by Asmolov (Reference Asmolov1999) for larger $Re_{b}$ . This effect is attributed to the narrowing of the wake behind the particle so that the wake no longer feels the wall as the particle Reynolds number increases. As a consequence, the interaction between the wall and the particle decreases and the particle migrates towards the wall (see also Liu et al. Reference Liu, Hu, Jiang and Sun2015). In order to examine the Reynolds number effects, we continue a simulation where the particle is close to the equilibrium position and vary the bulk Reynolds number. We show the final part of the particle trajectories ( $z$ -component versus $x$ -component) as a function of the Reynolds number in figure 2(c). Interestingly, we observe a non-monotonic behaviour of the final equilibrium position of the particle. As we increase the bulk Reynolds number up to 200, the particle ultimately focuses closer to the wall however at $Re_{b}\geqslant 300$ the trend reverses and the particle focuses closer to the centre. Note that we have also considered two additional cases at $Re_{b}=200$ and $Re_{b}=300$ varying the initial position of the sphere and obtain the same result. The non-monotonic positioning of the particle as a function of Reynolds number is also observed in the experiment by Ciftlik, Ettori & Gijs (Reference Ciftlik, Ettori and Gijs2013) in a rectangular duct. These authors also report the threshold value of $Re_{b}\approx 300$ for the reversing of the behaviour. This observation is however opposite to the results of the numerical study by Chun & Ladd (Reference Chun and Ladd2006) and Liu et al. (Reference Liu, Hu, Jiang and Sun2015) where the particle equilibrium position moves monotonically toward the wall as the Reynolds number increases. The origin of this discrepancy is still not clear and requires further investigations. The final location of sphere at $Re_{b}=[50,100,200,300,400]$ are $[0.287H,0.3H,0.307H,0.3H,0.281H]$ from the channel centre. Finally we note that the lateral force on the particle increases with the Reynolds number, i.e. a shorter focusing length is needed. As example the particle focusing length, when staring from the initial position shown by dark blue line in figure 2(a), is [150, 87, 55] in units of $H$ at $Re_{b}=[100,200,300]$ .

3.1.2 Sphere in rectangular duct

We next study the lateral motion of a single spherical particle in a duct with aspect ratio 2. The objective is to examine the effect of the shear and lift force asymmetry on the inertial migration of the particles. The results are summarised in figure 3(a) where we show the initial position, trajectory and final position of particles from different realisations exploiting the symmetry over one-quarter of the cross-section. The size ratio between the duct height and particle diameter is $H/D_{s}=5$ , indicated by the blue circles in the figure, and the bulk Reynolds number $Re_{b}=100$ as for the results in the previous section. Since the hydrodynamic diameter of the rectangular duct $D_{h}=4H/3$ , the average particle Reynolds number $\overline{Re_{p}}=\overline{\dot{\unicode[STIX]{x1D6FE}}}D_{s}^{2}/\unicode[STIX]{x1D708}=6$ for these simulations. In the rectangular duct, the shear-induced lift force is stronger towards the longer wall. Therefore, the general trend is that the particle first tends to move laterally toward the long wall and then slowly along the long wall towards the equilibrium position at the face centre (a similar behaviour is observed among others in Ciftlik et al. (Reference Ciftlik, Ettori and Gijs2013), Zhou et al. (Reference Zhou, Giridhar, Kasperb and Papautsky2013) and Liu et al. (Reference Liu, Hu, Jiang and Sun2015)). The particle may also focus along the symmetry line at the face centre of the short wall if it is initially located close enough to that point (Bhagat et al. Reference Bhagat, Kuntaegowdanahalli and Papautsky2008). The final distance between the sphere centre and the centre of the rectangular duct is approximately $0.275H$ and $0.82H$ at the equilibrium position close to the longer and shorter wall respectively. The boundary delimiting the initial positions leading to the two different equilibrium positions, depicted by the dashed black line in the figure, is extrapolated from the data by fitting a parabola.

Figure 3.

(a) Lateral trajectories of a spherical rigid particle over one-quarter of a rectangular duct at $Re_{b}=100$ . The particle diameter is one-fifth of the duct height whereas open circles and triangles show the initial and final equilibrium position of different realisations. The light blue dashed line indicates the equilibrium manifold over the entire cross-section. The black dashed line defines the initial position leading to two different equilibrium positions, at the face centre of the long or short wall as indicated by the individual trajectories (only shown on one side).The dotted red and green lines indicate isolevels of total shear rate equal to 0.75 and 0.81 and are tangential to the equilibrium manifold (only shown on one side). (b) Focusing length for different particle trajectories. The dashed lines represent the horizontal ( $z=2.5$ ) and vertical ( $x=5$ ) duct symmetry line, as the particle focuses on either of them.

Similar to the behaviour in the square duct, we observe that all the trajectories collapse onto an equilibrium manifold before they end at the equilibrium position. The equilibrium manifold is obtained by connecting the different trajectories over the entire cross-section and is shown by a dashed light blue line in the figure. Here, we also display isocontours of constant total shear rate (dotted lines). Interestingly, these are tangential to the manifold in the vicinity of the equilibrium positions. This indicates that once the particle starts the secondary slow motion towards the equilibrium, they move along lines of constant shear rate. The isolines of total shear rate that are tangential to the manifold close to the longer and shorter wall have values of 0.75 and 0.81 in units of  $U_{b}/H$ .

Next, we compare the features of the particle inertial migration in square and rectangular ducts. First, we note that if the particle is released close to the duct centre and focuses at the equilibrium position at the centre of a longer wall, it does not experience the initial lateral motion towards a shorter wall. This is directly connected to the weaker shear-induced lift force toward the shorter walls. The opposite is true if the particle starts close to the centre and focuses at the equilibrium position at the centre of a shorter wall, see the purple line in the figure, in which case the initial lateral motion toward the longer wall is significant. Secondly, we report that the focusing length is longer for particles in a rectangular duct, about three times longer than that in square duct if the particle starts at the same lateral distance from the equilibrium position, i.e. see figure 3(b). This is attributed to the lower average shear rate in the rectangular duct, $\overline{\dot{\unicode[STIX]{x1D6FE}}}_{rec}=2U_{b}/D_{h}|_{rec}=3U_{b}/2H$ , with respect to that in the square duct, $\overline{\dot{\unicode[STIX]{x1D6FE}}}_{squ}=2U_{b}/D_{h}|_{squ}=2U_{b}/H$ . This induces weaker lateral forces on a particle in the former case.

3.2.1 Oblate particles in square ducts

As for spherical particles, we first simulate the motion of the oblate spheroid in a square duct. The majority of the simulations have been performed for an aspect ratio of the oblate, polar over equatorial radius, $AR=1/3$ (in appendix B we report the effect of aspect ratio on the particle inertial migration). In addition we set the ratio between the duct height and the effective diameter of the oblate, the diameter of a sphere with the same volume, equal to $H/D_{o}^{e}=5$ . This corresponds to a ratio between the duct height and the diameter of the oblate, the largest dimension of the oblate, equal to $H/D_{o}\approx 3.466$ . We keep $Re_{b}=100$ , $\overline{Re_{p}}=\overline{\dot{\unicode[STIX]{x1D6FE}}}(D_{o}^{e})^{2}/\unicode[STIX]{x1D708}=8$ , as for the simulations of a sphere discussed above and vary the initial position of the oblate centre over one-eighth of the domain. The orientation of the oblate is monitored by the unit vector $\boldsymbol{n}=[n_{x},n_{y},n_{z}]$ parallel to the oblate symmetry axis. Similarly, the particle rotation vector is defined by $\unicode[STIX]{x1D714}=[\unicode[STIX]{x1D714}_{x},\unicode[STIX]{x1D714}_{y},\unicode[STIX]{x1D714}_{z}]$ . The particle initial orientation is $\boldsymbol{n}=[0,1,0]$ for all simulations, indicating that its axis of symmetry is parallel to the streamwise direction whereas the initial velocity and rotation of the particle are zero.

By symmetry, we show in figure 4(a) the lateral trajectory of the oblate only over one-eighth of the duct cross-section. The initial orientation and the actual size of the oblate are depicted by the light blue dashed circle. The results of the simulations reveal features similar to those observed for spherical particles. In particular, the oblate particle experiences a lateral motion towards a face-centred equilibrium position so that four face-centred equilibrium positions are obtained as for spheres. We also observe an initial lateral motion of the oblates towards a side wall, if they are positioned close to the centre, and the collapse of the trajectories on the equilibrium manifold before reaching the equilibrium position. At the final equilibrium position, the orientation vector is $n=[1,0,0]$ for the face-centred equilibrium position shown in the figure regardless of the initial position. This is in agreement with the experimental finding by Di Carlo et al. (Reference Di Carlo, Irimia, Tompkins and Toner2007) on inertial focusing of particles in microchannels and can be related to energy dissipation, as this is a configuration of lowest dissipation for oblate particles in shear flow. The oblate centre focuses approximately $0.26H$ from the duct centre, slightly closer to the centre than a sphere at the same bulk Reynolds number.

Figure 4.

(a) Lateral trajectories of a rigid oblate particle over one-eighth of a square duct at $Re_{b}=100$ . The effective particle diameter is one-fifth of the duct height (width), see definition in the text. The open circles and triangles show the initial position and final equilibria, respectively. The initial and final orientation of the oblate are shown for a particular case with the light blue solid line. (b) Orientation of the oblate symmetry axis as a function of time scaled by $D_{o}^{e}/U_{b}$ . (c) Focusing length of oblate spheroids from the different simulations: the symbols represent the particle centre position at equal intervals of time.

The streamwise motion of an oblate spheroid, from the initial to the final equilibrium position, is accompanied by rotation around its main symmetry axis (polar axis) and tumbling around an equatorial axis. Initially, particle exhibits a Jeffery-like orbit (Jeffery Reference Jeffery1922) with two distinct motions. First, a long-time motion without tumbling where the main symmetry axis is positioned perpendicular to the flow stream. Second, a tumbling motion in the form of a sudden rotation around an equatorial axis. As particle gets closer to the equilibrium position the tumbling motion vanishes gradually and particles just rotates around their axis of symmetry while travelling downstream. In figure 4(b) we report the time evolution of the $x$ -component of the orientation vector, $n_{x}$ , for some of the simulated cases. In this plot as well as the following plots, time is scaled by $D_{o}^{e}/U_{b}$ . For a particular trajectory, indicated by the pink line, the first stage of the motion displays significant tumbling toward the equilibrium manifold, $0<t<110$ , while the second has negligible tumbling along the manifold, $110<t<t_{final}\approx 500$ . The data in the figure indicate that longer time is needed to reach to the final orientation for particles starting close to the centre. Finally, we note that at the equilibrium position an oblate particle rotates around its symmetry axis. For the configuration in the figure, $\unicode[STIX]{x1D714}=[-0.395,0,0]$ where the values are normalised by $U_{b}/D_{o}^{e}$ .

The focusing length of oblate particles in square ducts are displayed in figure 4(c). Comparing to spherical particles of same size ratio, the focusing length is slightly longer for oblate particles when both are released from the same initial lateral position. This is mainly attributed to the tumbling motion in the cross-flow direction which effectively reduces the lateral force on the oblate. The symbols in the plot indicate the position of the particle centre at equal time intervals showing the two-stage particle migration, the fast and slow motions, similar to the case of spherical particles.

3.2.2 Oblate particles in rectangular ducts

Next, we discuss the motion of an oblate spheroid in a rectangular duct of aspect ratio 2 as for the simulations of spherical particles discussed above. We also use the same particle size ratio and bulk Reynolds number, $H/D_{e}=5$ and $Re_{b}=100$ . The lateral motion of the oblate is displayed in figure 5(a) over one-quarter of the cross-section. As in the case of a square duct, the particle reaches a steady configuration on the duct symmetry line at the centre of the different faces. In addition, the particle symmetry axis, which is initially set parallel to the flow stream, $n=[0,1,0]$ , finds its final orientation as $n=[1,0,0]$ for all the cases if trajectories end at the face centre of the long wall, see figure. The distance of the oblate centre from the duct centre is $0.22H$ and $0.78H$ if it focuses at the equilibrium at centre of the long and short wall. The rotation vector at the two equilibrium positions shown in the figure are $\unicode[STIX]{x1D714}=[-0.34,0,0]$ and $\unicode[STIX]{x1D714}=[0,0.31,0]$ for the coordinate system adopted here.

Figure 5.

(a) Lateral trajectories of a rigid oblate particle transported in a rectangular duct, displayed over one-quarter of the cross-section. The bulk Reynolds number $Re_{b}=100$ and the effective particle diameter is one-fifth of the duct height. Open circles and triangles show the initial positions and final equilibria from different realisations. The light blue dashed line indicates the equilibrium manifold over the entire cross-section, that is where all the trajectories collapse before ending at an equilibrium position. The dotted red and green lines are isolevels of the total shear rate for the values of 0.67 and 0.74, chosen to be tangential to the equilibrium manifold (only shown on one side). The black dashed parabola defines two regions in the cross-section according to the final equilibrium position reached by particles released on each side of it. The two equilibria are at the face centre of the long or short wall as indicated by the individual particle trajectories (only shown in one side). (b) Focusing length for different particle trajectories. The dashed lines represent the horizontal ( $z=2.5$ ) and vertical ( $x=5$ ) duct symmetry line, as the particle focuses on either of them.

Comparing these observations with the trajectories of a sphere in the same rectangular duct, we conclude that it is more likely for the oblate particle to focus close to the long wall, i.e. a larger portion of the cross-section will take the particle to the symmetry line orthogonal to the duct long wall, see black dashed line in the figure. In other words, the region of the cross-section delimiting initial positions corresponding to the equilibrium position at the centre of the short sides is significantly smaller for oblate particles than for spheres. We also note that, in analogy to the case of spherical particles, the equilibrium manifolds along which all trajectories collapse before reaching the final state are tangential to isolines of the total shear rate for the values 0.67 and 0.74. These are slightly lower than the corresponding values for spherical particles as the oblate spheroids focus closer to the duct centre. Finally, we show in figure 5(b) the focusing length corresponding to some of the particle trajectories. The focusing length of the oblate in rectangular duct is shorter than that of a sphere in the same configuration. This is opposite to what we have reported for the particle motion in a square duct and is attributed to the higher lateral velocity of the oblate in the second stage of the migration, that with negligible tumbling.

Figure 6.

(a) Lateral trajectories of oblate rigid particles of different sizes in a square duct at $Re_{b}=100$ . The effective particle diameters are $1/10$ , $1/5$ and $1/3.5$ of the duct height (width); open circles and triangles show the initial and final equilibrium position respectively. The initial and final orientation of the oblates are shown with dashed and solid lines. (b) Streamwise rotation rate versus streamwise particle displacements for the three particles in (a), indicated by lines of the same colour. (c) Streamwise particle displacements versus the vertical distance to the target symmetry line.

3.2.3 Size effect

As discussed in § 3.1.1 particle size is an important parameter used in many biological applications. Here we therefore investigate how the particle size affects the particle trajectory and the ensuing equilibrium position for an oblate particle in a square duct. Three different size ratios, $H/D_{e}=[3.5,5,10]$ are considered while keeping the bulk Reynolds number and oblate aspect ratio constant to $Re_{b}=100$ and $AR=1/3$ . The lateral migration of the oblate spheroids in the square duct is depicted in figure 6(a). Note that for each particle size, we performed several simulations, differing for the initial position over one-eighth of the cross-section. In the figure, however, we only report the trajectories pertaining particles of different size released from the same position (shown in different quadrants of the cross-section for the sake of clarity). The initial and final configuration of the oblates are shown with dashed and solid lines respectively.

First, we observe that the smallest oblate under consideration, $H/D_{e}=10$ , exhibits a behaviour similar to that of the reference oblate, $H/D_{e}=5$ , discussed in the previous section. Regardless of the starting point, the oblate particle focuses at one of the four face-centred equilibrium positions with the symmetry axis, $\boldsymbol{n}$ , perpendicular to the duct symmetry line as shown in the figure. Interestingly, the behaviour is completely different for the largest oblate studied, $H/D_{e}=3.5$ . Indeed, the equilibrium position is along the diagonal symmetry line, between the corner and the centre of the duct. In addition, we note that the oblate particle does not focus on the diagonal closest to its initial position but on that on the opposite side. This behaviour is consistently observed, for any initial condition considered over the cross-section of the channel.

In order to explain the peculiar behaviour of the large oblate in square duct, we examine the rotation rate of the oblate with respect to the streamwise direction during the migration process, see figure 6(b). Typically, the rotation rate during the migration is initially increasing and then decreasing. The former is associated with the fast motion of the particle towards the equilibrium wall and the latter to the particle slow motion towards the equilibrium position under the action of the rotation-induced lift force. This rotation-induced lift force reduces when decreasing the rotation rate and the particle eventually focuses at a face-centred equilibrium position. This behaviour is however not observed for the largest oblate under investigation. In this case, the streamwise rotation does not reduce to zero as the particle moves towards the vertical symmetry line, at $y/H\approx 100$ in the figure, which accelerates the particle motion. This behaviour might occur because the large oblate is more susceptible to tumbling, something that we do not observe for spherical particles. As a results of this acceleration, the particle crosses the vertical symmetry line and focuses on the diagonal where the streamwise rotation rate becomes zero and the shear-induced and wall-induced lift forces balance each other.

To conclude the analysis, we report the focusing length of oblate particles of different sizes in figure 6(c). Note that here we display the streamwise length as a function of the vertical distance to the target symmetry line for each case (vertical symmetry line for $H/D_{o}^{e}=[5,7]$ and diagonal symmetry line for $H/D_{o}^{e}=3.5$ ). It is evident from the figure that the smaller particles need longer times, and therefore travel longer in the streamwise direction, to reach their final equilibrium, similarly to what observed for spherical particles. It is also interesting to note that the centre of the large oblate cuts the target symmetry line several times while focusing, a behaviour always observed, regardless of the initial position of the particle. To summarise, we report in table 1 the final equilibrium position, orientation and rotation rate of the oblate particles in square duct at $Re_{b}=100$ as a function of the size ratio, $H/D_{o}^{e}$ . We note that even though the smallest oblate, of size ratio $H/D_{o}^{e}=7$ , focuses closer to the wall, where the magnitude of the background shear is higher than that of the larger oblate ( $H/D_{o}^{e}=5$ ), its final rotation rate is lower due to the hydrodynamic interactions with the wall.

Table 1.

Final equilibrium position, orientation and rotation of an oblate rigid particle at $Re_{b}=100$ as a function of the size ratio, $H/D_{o}^{e}$ .

3.2.4 Reynolds number effect

In this section we study the effect of the Reynolds number on the migration of an oblate particle in a square duct. We first note that we have run simulations at $Re_{b}<100$ and observed the same behaviour as at $Re_{b}=100$ , with the only difference that the final equilibrium position is closer to the centre as observed for spherical particles, see as an example the data pertaining $Re_{b}=50$ in table 2. Thus we will focus here on the results obtained at $Re_{b}=[100,150,200]$ , see figure 7. Note that we have performed more than two simulations for each Reynolds number changing the particle initial position, but we report in the figure only the trajectories originating from the same initial position and pertaining the different Reynolds numbers, displaying them over different quadrants of the cross-section for the sake of clarity.

Figure 7.

(a) Lateral trajectories of a rigid oblate particle, size ratio $H/D_{o}^{e}=5$ , in a square duct at different bulk Reynolds numbers. Open circles and triangles show the initial and final equilibrium position, respectively. The final orientation of the oblate spheroid is shown with solid and dashed (for unstable cases) lines. (b) Zoom of a particle trajectory at $Re_{b}=200$ . (c) The $x$ -component of the orientation vector, $n_{x}$ , as a function of time for the different Reynolds numbers under investigations (same colour coding as in panel a).

At $Re_{b}=150$ , the oblate moves laterally toward a face-centred equilibrium position and eventually focuses near (not exactly at) the duct symmetry line. In addition, at the final configuration, the particle remains tilted, at an angle $\unicode[STIX]{x1D713}\approx 21^{\circ }$ with respect to the duct symmetry line, see figure 7(a). This configuration is independent of the initial position of the oblate and mimics the inclined rolling dynamic of an oblate in a simple shear flow, see table 2 for the particle specification. It is described by a pitchfork bifurcation in the dynamical system analysis of the motion of an oblate spheroid in a simple shear flow (see for more details Ding & Aidun (Reference Ding and Aidun2000) and Rosen et al. (Reference Rosen, Do-Quang, Aidun and Lundell2015)). We note that the same behaviour is observed when we increase the streamwise box length from $6h$ to $8h$ and therefore we believe that this effect is not related to the periodicity of the domain in the streamwise direction. Finally, we report that the focusing length of the oblate spheroid at $Re_{b}=100$ and $Re_{b}=150$ , for the trajectories shown in figure 7, are $190H$ and $140H$ , respectively. Therefore, the reduction in the focusing length by increasing the Reynolds number is also evident for the oblate particles.

Table 2.

Final equilibrium position, orientation and rotation of an oblate particle of size ratio $H/D_{o}^{e}=5$ transported in a square duct as a function of the bulk Reynolds number,  $Re_{b}$ .

At $Re_{b}=200$ however the particle dynamics changes again. The oblate particle exhibits time-dependent rolling and tumbling motions around an equilibrium position at the face centre. Therefore, we cannot define a steady orientation and rotation rate at $Re_{b}=200$ . We display one of the orientations of the oblate in the lower part of figure 7(a) using dotted line. The zoom on the particle trajectory is shown in figure 7(b) where its chaotic dynamic is evident. A similar behaviour is obtained for particles transported at $Re_{b}>200$ , not shown here. The occurrence of a combined rolling and tumbling motion of an oblate at particle Reynolds number above a critical threshold is reported among others in the recent work by Rosen et al. (Reference Rosen, Do-Quang, Aidun and Lundell2015) who considered homogeneous shear flow. Here we observe the chaotic motion of the oblate for $\overline{Re_{p}}\approx 16$ . This may appear in contradiction with the results of the previous section for the large oblate, $H/D_{o}^{e}=3.5$ at $Re_{b}=100$ , where the $\overline{Re_{p}}$ shares similar value. We believe that two following reasons could possibly prevent the large oblate from tumbling at its final configuration: (i) the geometrical confinement when the large oblate focuses close to the corner and (ii) the weaker background shear (the background flow shear rate is much lower at the corners) which induces a lower local (effective) particle Reynolds number than $\overline{Re_{p}}\approx 16$ .

Finally we show in figure 7(c) the time evolution of the $x$ -component of the oblate symmetry vector, $n_{x}$ , for each Reynolds number under consideration. It is evident that at the final focusing $n_{x}$ is constant but smaller that unity for $Re_{b}=150$ , whereas it does not converge to the a particular value when $Re_{b}=200$ . For the specification of the oblate particle orientation and velocities at the final position we refer the readers to table 2.