3.1 Experimental results

Figure 3 shows representative examples of the distributions of particles in the tube cross-section obtained at various distances from the tube inlet with $Re=100{-}700$ and $d/D=0.1$ . Each dot in these figures represents the position of a particle centre. The evolutions of the particle distribution in the flow direction shown in figure 3(a,b) indicate prompt focusing of the particles on an annulus at low $Re$ . This annulus was the SS annulus. As $Re$ increased, the fraction of particles located on the SS annulus in the upstream cross-sections decreased, with more particles located in the inner region. These inner particles appeared to form a broad annulus inside the SS annulus. MMG called this annulus ‘the inner annulus’. They observed the inner annulus at $Re>600$ for $d/D=0.06{-}0.11$ and found that it was significantly broader than the SS annulus, with most particles spreading from 0.3 to 0.7 of the tube radius from the tube centre. At the highest $Re(=700)$ in figure 3, the left-most diagram of figure 3(d) shows that few particles were on the SS annulus, and only the inner annulus was clearly observed. The inner annulus, the remaining diagrams of figure 3(d) indicate that the fraction of particles on the inner annulus decreased farther downstream, accompanying an outward shift of the inner annulus towards the SS annulus and a decrease in its width, until almost all particles were focused on the SS annulus in the most downstream cross-section. Thus, in the range of $Re$ shown in figure 3, all particles were focused on the SS annulus, and the inner annulus disappeared in the downstream cross-sections. These results agree with the previous experimental study of MIS using macroscale tubes with the size ratio $d/D\approx 0.08$ . As shown in figure 3, the entry length after which radial migration was fully developed was larger for higher $Re$ , in accordance with the results of MIS.

Figure 3.

Particle distributions over the tube cross-section for $Re$ values of (a) 100, (b) 300, (c) 600 and (d) 700, with $d/D=0.10$ and $L/D=125$ , 250, 500 and 1000.

To clarify the characteristics of the particle distribution, we estimated the probability density function (p.d.f.) by using the obtained data for the particle positions. First, we calculated the radial probability of a particle being detected at $r$ , i.e. $P(r)$ , among all the obtained radial positions of particles. Because we observed all the particles passing through the cross-section during each measurement time (at least 300 particles), $P(r)$ represents the radial probability for the particle flux. Here, the suspending fluid is a Newtonian fluid, and the suspension is extremely dilute (volume fraction ${\sim}10^{-5}$ ); thus, the fluid is expected to have a Poiseuille velocity profile at the Reynolds numbers considered. Assuming that each particle travels at the same velocity with a Poiseuille flow at the position of the particle centre, we calculated the probability of a particle being located at a radial position $r$ as $p(r)=A^{-1}P(r)/(1-r^{2})$ , where $A=\int _{0}^{1}P(r)/(1-r^{2})\,\text{d}r$ . Our numerical simulation showed that, in the typical cases considered in this study, a particle located at an equilibrium radial position travels at an axial velocity approximately 3 % lower than the fluid velocity at the particle centre. The p.d.f., i.e. $p^{\ast }(r,\unicode[STIX]{x1D703})$ , associated with the probability of a particle being at a radial position $r$ and a given azimuthal angle $\unicode[STIX]{x1D703}$ , can be related to $p(r)$ by $p(r)=\unicode[STIX]{x03C0}^{-1}\int _{0}^{2\unicode[STIX]{x03C0}}p^{\ast }(r,\unicode[STIX]{x1D703})r\,\text{d}\unicode[STIX]{x1D703}$ . Assuming that $p^{\ast }$ depends only on $r$ , this yields $p^{\ast }(r)=p(r)/2r$ (MMG; MIS).

Figure 4 illustrates the $p^{\ast }(r)$ extracted from the particle distributions shown in figure 3(b,c). In figure 4(a), the p.d.f.s at various $L/D$ values exhibit a sharp peak at a radial position corresponding to the SS annulus, even in the most upstream cross-section at low $Re$ , and this position is almost independent of $L/D$ within the measurement error. The p.d.f.s shown in figure 4(b) also exhibit a sharp peak at the SS annulus. The p.d.f.s at $L/D=125$ and 250 have a second peak at $r\approx 0.6$ , which corresponds to the inner annulus reported by MMG.

Figure 4.

P.d.f. $p^{\ast }(r)$ for $Re$ values of (a) 300 and (b) 600, with $d/D=0.10$ and $L/D=125$ , 250, 500 and 1000.

The distributions of the particles at $Re>700$ are shown in figure 5, for $d/D=0.1$ . The p.d.f.s $p^{\ast }(r)$ corresponding to figures 5(b) and 5(d) are plotted in figures 6(a) and 6(b), respectively. In contrast to figure 3 (low $Re$ ), the left-most diagrams of figure 5 show no particles on the SS annulus in the most upstream cross-sections. Instead, the particles are located in the inner region. The absence of the SS annulus at the most upstream cross-section was reported by MIS. In their experiments, the SS annulus was absent in the cross-section at $L/D=65$ for $Re>700$ and $d/D\approx 0.08$ . In a downstream cross-section at $L/D\approx 310$ , MMG observed the absence of the SS annulus at high $Re$ close to the transition to the intermittency, e.g. $Re=1650$ for $d/D=0.059$ . At these high $Re$ , they claimed the possibility that the only equilibrium position could be the inner annulus, not the SS annulus.

Figure 5.

Particle distributions over the tube cross-section for $Re$ values of (a) 750, (b) 790, (c) 830, (d) 900 and (e) 1000, with $d/D=0.10$ and $L/D=125$ , 250, 500, 750 and 1000.

The second diagrams from the left in figure 5 show that particles located in the inner region at the upstream cross-section (at $L/D=125$ ) move outward, forming a thin annulus at $L/D=250$ . As shown in figure 6, the location of this annulus is close to or slightly outside of the peak position of the p.d.f. in the upstream cross-section at $L/D=125$ . The remaining diagrams of figure 5 show the evolution of the particle distribution in the flow direction, indicating the presence of two types of particle-focusing patterns in this $Re$ range. Two annuli are observed at the most downstream cross-sections in figure 5(a–c), whereas only one annulus is observed in figure 5(d,e). We call the $Re$ regime in figure 3 ‘regime (A)’, where particles are focused on the SS annulus. Figure 5 indicates the presence of two additional regimes for the particle-focusing pattern observed in the most downstream cross-section. In the regime of lower $Re$ , particles are focused on two annuli, as shown in figure 5(a–c) (regime (B)), whereas in the regime of higher $Re$ , particles are focused on one annulus, as shown in figure 5(d,e) (regime (C)).

Figure 6.

P.d.f. $p^{\ast }(r)$ at $Re$ values of (a) 790 and (b) 900, with $d/D=0.10$ and $L/D=125$ , 250, 500 and 1000.

In regime (B), the particle distributions in figure 5(a–c) suggest that particles located in the inner region of the upstream cross-section move outward, accumulating on an annulus, and then some of them continue moving towards the tube wall until they reach another annulus. The p.d.f.s at various $L/D$ values shown in figure 6(a) support such a process. In these figures, the outer annulus is more clearly observed, i.e. more particles are located on the outer annulus, at lower $Re$ . On the other hand, in regime (C) shown in figure 5(d,e), only one annulus is observed, regardless of the observation site. In both regimes, as shown in figure 6(a,b), the particle distributions are sharply peaked on the focusing annuli, and their radial positions are nearly constant, independent of $L/D$ . These properties differ from the broad inner annulus observed at low $Re$ in regime (A).

In figure 7, the measured radii of the particle-focusing annuli in regimes (A), (B) and (C) are indicated by circles, squares and triangles, respectively, for $d/D=0.1$ . In regime (A), the particles are focused on the SS annulus (closed circles). The closed squares, which indicate the radii of the outer annulus in regime (B), lie on the extension of the closed circles; thus, the outer annulus in regime (B) corresponds to the SS annulus. On the other hand, the open squares, which indicate the radii of the other (inner) annulus in regime (B), and open triangles, which indicate the radii of the annulus in regime (C), lie on a single line. In the present study, we call the annulus located closer to the tube centre ‘the inner annulus’, although it differs from the broad accumulation of particles observed at low $Re$ . The particle-focusing patterns in the most downstream cross-section are summarised as follows: the particles are focused on the SS annulus in regime (A), on the SS annulus and the inner annulus in regime (B) and on the inner annulus in regime (C).

Figure 7.

Radii of the particle-focusing annuli for $d/D=0.1$ and $L/D=1000$ , in regime (A) (circles), regime (B) (squares), and regime (C) (triangles). The closed symbols represent the radius of the SS annulus, $r_{s}$ , and the open symbols represent the radius of the inner annulus, $r_{i}$ . Numerical results for the stable and unstable equilibrium positions are indicated by solid and dotted lines, respectively. Previous experimental results are indicated by diamonds (MIS for $d/D=0.11$ at $L/D=250$ ), crosses and plus signs (MMG for $d/D=0.095$ and 0.11 at $L/D=313$ ) and asterisks (Segré & Silberberg (Reference Segré and Silberberg1962) for $d/D=0.11$ at $L/D=214$ ). The dashed line represents the numerical results of Yang et al. (Reference Yang, Wang, Joseph, Hu, Pan and Glowinski2005) for $d/D=0.1$ .

We express the radius of the SS annulus as $r_{s}$ and that of the inner annulus as $r_{i}$ . For comparison, the values of $r_{s}$ obtained in previous experiments with similar size ratios are plotted in figure 7. Previous numerical results for $r_{s}$ with $d/D=0.1$ are also plotted. Additionally, the values of $r_{i}$ obtained by MIS for $d/D=0.11$ at $L/D=250$ are plotted, although in their study, the term ‘the inner annulus’ referred to both the transient accumulation of particles observed at low $Re$ as well as particle focusing observed at high $Re$ . The present results for $r_{s}$ and $r_{i}$ agree well with the results of the previous studies. The inner annulus $r_{i}$ appeared at lower $Re$ in the study of MIS than in the present study because at $L/D=250$ , the transient accumulation of particles closer to the tube centre was observed in regime (A), as shown in the second diagrams from the left in figure 3(c,d). The most significant difference between the present study and previous macroscale experiments may be that we did not observe the SS annulus at $Re>850$ for $d/D=0.10$ , whereas MMG observed the SS annulus even beyond $Re=1000$ for comparable $d/D$ . In particular, they reported the presence of the SS annulus up to $Re\approx 1200$ for $d/D=0.095$ and up to $Re\approx 1400$ for $d/D=0.11$ at $L/D\approx 310$ . The reason for this difference is discussed in § 4.

In figure 7, with the increase of $Re$ , the SS annulus approaches the tube wall, while the inner annulus moves towards the tube centre. The former feature was first predicted theoretically by Asmolov (Reference Asmolov1999) and later confirmed in experiments by MMG. The latter feature was experimentally observed by MIS.

Next, we estimated the critical $Re$ between regimes (A) and (B), i.e. $Re_{c1}$ , and that between regimes (B) and (C), i.e. $Re_{c2}$ . To this end, we calculated the area between the curve $p(r)$ and the horizontal axis around the SS annulus, i.e. the hatched area shown in the inset of figure 8(a). This area, which is expressed as $P_{s}$ , represents the fraction of particles located on the SS annulus (MIS). In figure 8, $P_{s}$ is plotted with respect to $Re$ for $d/D=0.1$ and various values of $L/D$ . Figure 8(b) is an expanded graph of figure 8(a). As expected from the left-most diagrams of figures 3 and 5, the $P_{s}$ values at $L/D=125$ gradually decrease from ${\sim}1$ to 0 with the increase of $Re$ . As $L/D$ increases, the $P_{s}$ values decline more rapidly near $Re=800$ . If the values of $Re_{c1}$ and $Re_{c2}$ are approximated by the $Re$ values of intersections between the linear regression of $P_{s}$ for $L/D=1000$ and the horizontal lines at $P_{s}=1$ and 0, respectively, then we have $Re_{c1}\approx 757$ and $Re_{c2}\approx 845$ for $d/D=0.1$ (see figure 8 b). Here, we calculated the linear regression of $P_{s}$ for $0.05<P_{s}<0.95$ . As shown in figure 8(b), $P_{s}$ is not equal to unity in some range of $Re$ lower than the intersection between the linear regression of $P_{s}$ and the horizontal line of $P_{s}=1$ , indicating that the critical $Re$ could be lower than the intersection. In fact, two annuli are observed at $Re=750({<}757)$ in figure 5(a). However, as discussed later, the error of the $Re$ in our experiments, i.e. $\unicode[STIX]{x0394}Re$ , was estimated to be a few per cent; thus, this difference is within $\unicode[STIX]{x0394}Re$ .

Figure 8.

(a) Probability of the SS annulus, i.e. $P_{s}$ , for $d/D=0.1$ at $L/D=125$ (inverted triangles), 250 (diamonds), 500 (triangles), 750 (squares) and 1000 (circles). Previous experimental results are indicated by asterisks (MIS for $d/D=0.11$ at $L/D=250$ ), and crosses and plus signs (MMG for $d/D=0.095$ and 0.11 at $L/D=313$ ). (b) Expanded graph of (a).

For comparison, in figure 8(a), the $P_{s}$ values obtained by MIS for $d/D=0.11$ and $L/D=250$ ( $1-P_{i}$ in figure 9b of their paper) are indicated by asterisks, and the probability of finding a particle on the SS annulus reported by MMG ( $P_{out}$ in figure 7 of their paper) is indicated by crosses for $d/D=0.095$ and by plus signs for $d/D=0.11$ at $L/D\approx 310$ . The values of $P_{out}$ were reported to be almost independent of the size ratio $d/D$ for $d/D=0.059{-}0.11$ . The present results for $L/D=250$ agree well with the results of MIS. MMG reported that the probability of finding a particle on the inner annulus becomes higher than that of finding a particle on the SS annulus for $Re\approx 650$ , which is consistent with the present results, where $P_{s}$ at $L/D=250$ is nearly equal to 0.5 for $Re\approx 700$ . However, the values of $P_{out}$ do not vanish beyond $Re=1000$ , as previously noted.

Figure 9.

(a) Critical channel Reynolds numbers $Re_{c1}$ (closed circles) and $Re_{c2}$ (open circles). (b) Critical particle Reynolds numbers $Re_{c1}(d/D)^{2}$ (closed squares) and $Re_{c2}(d/D)^{2}$ (open squares), as well as $Re_{c1}(d/D)$ (closed triangles) and $Re_{c2}(d/D)$ (open triangles).

In figure 9, the critical Reynolds numbers, i.e. $Re_{c1}$ and $Re_{c2}$ , are plotted for $d/D=0.083$ , 0.1 and 0.133, and their values are connected by lines. The results indicate that with the increase of the size ratio, the critical $Re$ decreases for both $Re_{c1}$ and $Re_{c2}$ . Additionally, the $Re$ range in regime (B) decreases. The decrease of $Re_{c1}$ and $Re_{c2}$ with the increase of $d/D$ is reasonable, because an increase of $d/D$ can increase the inertial effect at a constant $Re$ . In figure 9(b), we plot the corresponding particle Reynolds numbers $Re_{c1}(d/D)^{2}$ and $Re_{c2}(d/D)^{2}$ , as well as $Re_{c1}(d/D)$ and $Re_{c2}(d/D)$ . The latter values appear to be nearly horizontal in this range of $d/D$ , even if the reason is not yet clear.

3.2 Numerical results

Figure 10 presents the profiles of the non-dimensional lateral (radial) force, i.e. $F/\unicode[STIX]{x1D70C}U^{2}d^{2}$ , exerted on a particle for $d/D=0.10$ , at $Re=100$ , 300, 600 and 700 (figure 10 a) and at $Re=750$ , 800, 900 and 1000 (figure 10 b). Figure 10(a) shows that, in general, the non-dimensional lateral force decreases with the increase of $Re$ and has a local minimum between two local maxima, except at the lowest $Re$ . Each curve intersects the horizontal axis once at $r>0.6$ . By expressing this intersection as $r_{e}$ , we observe that the lateral force is positive (outward direction) for $r<r_{e}$ and negative (inward direction) for $r>r_{e}$ , indicating that the equilibrium position is stable. Additionally, the equilibrium radial position, i.e. $r_{e}$ , approaches the tube wall as $Re$ increases. These properties of the non-dimensional lateral force are in accordance with those obtained via theoretical analyses based on the method of matched asymptotic expansions for a spherical particle in a two-dimensional Poiseuille flow by Asmolov (Reference Asmolov1999) and for a particle in a circular tube flow by Matas et al. (Reference Matas, Morris and Guazzelli2009).

Figure 10.

Non-dimensional lateral force exerted on a spherical particle for $d/D=0.1$ (a) at $Re=100$ (diamonds), 300 (squares), 600 (triangles) and 700 (crosses), and (b) at $Re=750$ (diamonds), 800 (squares), 900 (circles) and 1000 (triangles). In (b), the force curve for $Re=700$ is indicated by a dotted line, for comparison.

Figure 10(b) shows the lateral-force profiles at $Re>700$ . The corresponding curve for $Re=700$ is also plotted (dashed line), for comparison. As $Re$ increases, the non-dimensional force curves continue to decrease, particularly in the region adjacent to the tube wall. Consequently, the local minimum becomes negative, and the force curves for $Re=750$ and 800 intersect the horizontal axis three times. Although all of these intersections represent equilibrium positions, the inner and outer ones are stable, whereas the middle one is unstable. Thus, it can be deduced that the outer intersection corresponds to the SS annulus, and the inner intersection corresponds to the inner annulus (regime (B)). A further increase in $Re$ decreases the outer local maximum below zero (see the force curves for $Re=900$ and 1000). Thus, the force curve intersects the horizontal axis at one point again. The intersection is located at a radial position corresponding to the inner annulus, as indicated by the force curves for $Re=900$ and 1000. At these $Re$ values, the equilibrium position corresponding to the SS annulus vanishes, and the inner annulus is the only equilibrium position (regime (C)).

The numerically obtained radial coordinates of these equilibrium positions are plotted by lines in figure 7, where the unstable equilibrium position in regime (B) is indicated by a dotted line. Figure 7 indicates that the numerical results agree well with the present experimental results, as well as with the results of previous studies. In summary, our numerical results support the presence of three regimes for the focusing pattern of spherical particles in circular tube flows. According to our computations, for $d/D=0.1$ , $Re_{c1}$ lies between 740 and 750, and $Re_{c2}$ lies between 840 and 850. These values are comparable to the present experimental results.