3.1.1 Viscoelasticity-induced migration at a low Reynolds number $Re=1$

We now examine the particle migration in square channel flow at a low Reynolds number $Re=1$ . Note that the particle Reynolds number based on the particle size and the shear rate is around $(d/H)^{2}Re=0.0225$ . Therefore, the fluid inertial effect on the particle motion is expected to be small. Initially, both the particle and fluid are at rest. Figure 3 shows the trajectories of particles released at different positions projected in the channel cross-section for $Re=1$ , $d/H=0.15$ and different Weissenberg numbers $Wi=0.01$ , 0.1, 0.5 and 1.0 (the elasticity number being 0.01, 0.1, 0.5 and 1.0, respectively). For $Wi=0.01$ , almost all particles migrate towards the channel centreline, except those released in a very small corner region (figure 3 a). For $Wi=0.1$ , the particles released near the wall move towards the corner along the wall, and the other particles migrate towards the channel centreline (figure 3 b). As $Wi$ increases from 0.01 to 0.5, the corner attractive region is enlarged, and as $Wi$ further increases to 1.0, the corner attractive region begins to shrink, as shown in figure 3 where the corner attractive region is marked by the red closed lines. From our results, most particles initially distributed homogeneously in a square channel filled with a viscoelastic fluid with constant viscosity even at high Weissenberg numbers and low Reynolds numbers would migrate to the channel centre region, consistent with previous experimental observations (Del Giudice et al. Reference Del Giudice, D’Avino, Greco, Netti and Maffettone2015; Liu et al. Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ). It has been recognized that the shear-thinning effect of the viscoelastic fluid tends to make the particles move towards the wall or the corner (Villone et al. Reference Villone, D’Avino, Hulsen, Greco and Maffettone2013; Del Giudice et al. Reference Del Giudice, D’Avino, Greco, Netti and Maffettone2015; Liu et al. Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ). From figure 3, the particle migration rate is overall increased with increasing Weissenberg number.

In addition, figure 3 shows that the particles released near the corner attractive region first migrate towards the diagonal line and then move towards the channel centreline along the diagonal line. Particles migrate towards the channel centreline normally along curved lines particularly for $Wi\geqslant 0.1$ as if they were attracted by the diagonal line, except for the particles released directly on the midline or the diagonal line, which migrate along a straight line due to symmetry confinement.

3.1.2 Elasto-inertial migration at $Re=10$ , $Re=50$ and $Re=100$

Figure 4 shows the particle trajectories projected in the channel cross-section at $Re=10$ , and $Wi=0.01$ , 0.05, 0.1 and 0.5 (the elasticity number being 0.001, 0.005, 0.01 and 0.05, respectively). For $Wi=0.01$ ( $El=0.001$ ), the elasticity effect is expected to be very small, and the particle migration is dominated by the fluid inertial effect. It was shown in the experiments of Choi et al. (Reference Choi, Seo and Lee2011) and Abbas et al. (Reference Abbas, Magaud, Gao and Geoffroy2014) that the particles suspended in Newtonian square channel flow at $Re\approx 100$ migrated towards the channel walls along the cross-stream direction at the first stage and then migrated slowly towards the face centres at the second stage, with the migration rate being around one order smaller compared to that at the first stage, and at $Re\approx 10$ the second-stage migration did not occur and only the particle ring (annulus) could be observed. In our simulations for $Re=10$ and $Wi=0.01$ , the particles migrate along the cross-stream direction to an annulus region located at around 0.28 $H$ away from the channel centreline (the shaded area in figure 4 a) and the second-stage migration is not observed, consistent with the experiments of Choi et al. (Reference Choi, Seo and Lee2011) and Abbas et al. (Reference Abbas, Magaud, Gao and Geoffroy2014). Figure 4(b) shows that the particles migrate towards an equilibrium position located on the diagonal line of the channel cross-section for $Wi=0.05$ ( $El=0.005$ ). For $Wi=0.1$ ( $El=0.01$ ), besides the diagonal equilibrium position, which is shifted closer to the channel centreline compared to $Wi=0.05$ , the corner equilibrium position with small attractive region appears and the particles released near the wall move towards the corner along the wall, as shown in figure 4(c). From figure 4(d), the channel centreline equilibrium position appears for $Wi=0.5$ ( $El=0.05$ ), with the diagonal-line equilibrium position shifted to the centreline. The presence of both centreline and corner equilibrium positions is a feature of viscoelasticity-dominated migration.

Figure 4. Particle trajectories for different initial positions projected in the channel cross-section at $Re=10$ and $\unicode[STIX]{x1D6FD}=0.15$ : (a) for $Wi=0.01$ , $El=0.001$ , particles migrate to the shaded ring area; (b) for $Wi=0.05$ , $El=0.005$ , there exists only a diagonal equilibrium position; (c) for $Wi=0.1$ , $El=0.01$ , there exist both the diagonal equilibrium position and the corner equilibrium position with small attractive region; and (d) for $Wi=0.5$ , $El=0.05$ , there exist both the centreline and corner equilibrium positions. The big dots denote the particle initial positions, and the small dots represent the particle positions for every 100 dimensionless time units.

Figure 5. Particle trajectories for different initial positions projected in the channel cross-section at $Re=50$ and $d/H=0.15$ : (a) for $Wi=0.01$ , $El=0.0002$ , particles migrate to the ring marked with the yellow dotted line along the cross-stream direction, and then migrate slowly towards the midline equilibrium position; (b) for $Wi=0.25$ , $El=0.005$ , only the diagonal equilibrium position exists; (c) for $Wi=0.5$ , $El=0.01$ , both diagonal and corner equilibrium positions exist; and (d) for $Wi=1.0$ , $El=0.02$ , both centreline and corner equilibrium positions exist. The big dots denote the particle initial positions, and the small dots represent the particle positions for every 100 dimensionless time units.

Figure 6. Particle trajectories for different initial positions projected in the channel cross-section at $Re=100$ and $a/H=0.15$ : (a) for $Wi=0.02$ , $El=0.0002$ , particles migrate towards the midline equilibrium position; (b) for $Wi=0.1$ , $El=0.001$ , both the midline and diagonal equilibrium positions exist; (c) for $Wi=0.5$ , $El=0.005$ , only the diagonal equilibrium position exists; and (d) for $Wi=1.5$ , $El=0.015$ , both centreline and corner equilibrium positions exist. The big dots denote the particle initial positions, and the small dots represent the particle positions for every 100 dimensionless time units.

The particle trajectories projected in the channel cross-section at $Re=50$ and $Wi=0.01$ , 0.25, 0.5 and 1.0 (the elasticity number being 0.0002, 0.005, 0.01 and 0.02, respectively) are shown in figure 5. For $Re=50$ and $Wi=0.01$ ( $El=0.0002$ ), the fluid inertia dominates the particle migration, and one can see from figure 5(a) that the particles migrate to the ring and then migrate slowly towards the midline equilibrium position. For $Wi=0.25$ ( $El=0.005$ ), only the diagonal equilibrium position exists (figure 5 b). For $Wi=0.5$ ( $El=0.01$ ), both diagonal and corner equilibrium positions occur (figure 5 c). For $Wi=1.0\;(El=0.02)$ , both centreline and corner equilibrium positions exist (figure 5 d).

The particle trajectories projected in the channel cross-section at $Re=100$ and $Wi=0.02$ , 0.1, 0.5 and 1.5 (the elasticity number being 0.0002, 0.001, 0.005 and 0.015, respectively) are shown in figure 6. As the fluid elasticity is enhanced, the equilibrium position undergoes the transition from midline (figure 6 a), both midline and diagonal line (figure 6 b), diagonal line (figure 6), to both centreline and corner (figure 6 d).

The comparison between figures 4(a), 5(a) and 6(a) shows that the inertia-induced migration towards the midline equilibrium position becomes faster as the Reynolds number increases. Our results in figures 3, 4(d), 5(d) and 6(d) indicate that the rate of the elasticity-induced migration towards the channel centreline depends much more strongly on the Weissenberg number than on the elasticity number, although the Reynolds number plays a negative role for the elasticity-induced migration. In addition, the corner attractive area for the same Weissenberg number becomes smaller for a higher Reynolds number, as expected.

3.1.3 Phase diagram for the equilibrium positions

We have shown that the midline, diagonal line, corner and centreline equilibrium positions occur successively as the fluid elasticity is enhanced, when the fluid inertia is significant. The diagonal-line EP (abbreviation for ‘equilibrium position’ or ‘equilibrium positions’) can exist alone or coexist with the midline and corner EP. The numerical results of Li et al. (Reference Li, McKinley and Ardekani2015) showed that the transition between the off-centreline (or midline) and centreline EP depended on both elasticity and Reynolds numbers (but more strongly on the elasticity number than on the Reynolds number) for the particle elasto-inertial migration in the midplane of a square channel. Here, we examine the dependence of our EP on the elasticity and Reynolds numbers, and the phase diagram is plotted in figure 7(a). It can be seen that the equilibrium position depends strongly on the elasticity number and weakly on the Reynolds number. For  $Re=10{-}100$ , the diagonal-line EP occurs largely at $El=0.001{-}0.15$ . For $El=0.015$ , the equilibrium position for $Re=10$ is located on the diagonal line, whereas those for $Re=50$ and 100 are located on the channel centreline, indicating that the critical elasticity number for the centreline EP decreases with increasing Reynolds number, in qualitative agreement with the observation of Li et al. (Reference Li, McKinley and Ardekani2015). A schematic diagram for the equilibrium positions changing with $Re$ and $El$ is presented in figure 7(b). The arrows therein guide the transition of the equilibrium position from midline, diagonal line and to centre and corner of the channel cross-section, as $El$ increases.

Figure 7. (a) Phase diagram for the equilibrium positions as a function of $Re$ and $El$ . ‘EP’ is the abbreviation of ‘equilibrium position’. (b) Schematic diagram for the equilibrium positions changing with $Re$ and $El$ . The arrows guide the transition of the equilibrium position from midline, diagonal line and to centre and corner of the channel cross-section, as $El$ increases.

Figure 8. The distances away from the channel centreline of the diagonal-line equilibrium positions as a function of $El$ at different Reynolds numbers.

The distances away from the channel centreline of the diagonal-line equilibrium positions as a function of $El$ at $Re=10$ , 50 and 100 are plotted in figure 8. The midline EP and the diagonal-line EP at very low elasticity numbers are located farther away from the channel centreline at a larger Reynolds number. However, the diagonal-line equilibrium positions are shifted towards the channel centreline more rapidly with increasing elasticity number at a larger Reynolds number, leading to a lower critical transition elasticity number for a higher Reynolds number.

Li et al. (Reference Li, McKinley and Ardekani2015) compared their numerical prediction of the critical transition elasticity number to the theoretical value from the balance between the elastic force and the inertia-induced force. We here conduct similar analysis, following their work. The elastic force derived by Ho & Leal (Reference Ho and Leal1976) has the form for two-dimensional Poiseuille flow of an Oldroyd-B fluid (Li et al. Reference Li, McKinley and Ardekani2015):

(3.1) $$\begin{eqnarray}F_{e}=\frac{40}{3}\unicode[STIX]{x03C0}\unicode[STIX]{x1D70C}_{f}U_{0}^{2}d^{2}\frac{d}{H}El(1-\unicode[STIX]{x1D702}_{r})y^{\ast },\end{eqnarray}$$

where $y^{\ast }$ is the dimensionless lateral position in the range of $-0.5$ to $0.5$ . The inertia-induced force derived by Ho & Leal (Reference Ho and Leal1974) for the two-dimensional Poiseuille flow at low particle Reynolds numbers is

(3.2) $$\begin{eqnarray}F_{i}=C(y^{\ast })\unicode[STIX]{x1D70C}_{f}U_{0}^{2}d^{2}\left(\frac{d}{H}\right)^{2},\end{eqnarray}$$

where $C$ is a function of $y^{\ast }$ and has the maximum value of around 0.24 at $y^{\ast }\approx 0.15$ . The balance between the elastic and inertial forces at $y^{\ast }=0.15$ yields the critical elasticity number $El_{c}\approx 0.038(d/H)(1/(1-\unicode[STIX]{x1D702}_{r}))$ , which is around 0.011 for $d/H=0.15$ and $\unicode[STIX]{x1D702}_{r}=0.5$ , comparable to our prediction for $Re=100$ in figure 8.

3.1.4 Effects of channel length, blockage ratio and Giesekus model

The diagonal-line EP for a rigid spherical particle in a viscoelastic channel flow has not been reported previously, to the best of our knowledge. The experiments on particle migration in viscoelastic square channel flow were conducted for relatively large elasticity numbers (typically $El>0.1$ ). We attempt to examine whether the presence of the diagonal-line EP is dependent on other factors that have not been considered so far, such as the effects of the channel length, the particle size and the fluid rheological property. The channel length of $L=H$ is relatively small, and the hydrodynamic interactions between the particle and its periodic images might be important. In order to examine the effect of the channel length, simulations for $L=2H$ at $Re=50$ and $Wi=0.5$ are performed, and the results on the particle trajectories are shown in figure 9(b). The particle trajectories for $L=H$ at $Re=50$ and $Wi=0.5$ are shown in figure 9(a) for comparison. For $L=2H$ , the diagonal-line EP still exists, although its position is shifted a little closer to the channel centreline, compared to the case of $L=H$ .

Figure 9. Particle trajectories for different initial positions projected in the channel cross-section at $Re=50$ and $Wi=0.5$ (i.e.  $El=0.01$ ): (a)  $L=H$ , $d/H=0.15$ ; (b)  $L=2H$ , $d/H=0.15$ ; (c)  $L=H$ , $d/H=0.1$ ; and (d)  $L=H$ , $d/H=0.15$ and Giesekus model with $\unicode[STIX]{x1D6FC}=0.2$ . The big dots denote the particle initial positions, and the small dots represent the particle positions for every 100 dimensionless time units.

Figure 10. Particle trajectories for different initial positions projected in the channel cross-section: (a) at $Re=1$ , $Wi=0.5$ , $El=0.5$ , the fluid viscoelasticity drives particles towards the cross-section corner or shaded area; (b) at $Re=50$ , $Wi=0.5$ , $El=0.01$ , particles migrate towards the diagonal-line EP (equilibrium position); (c) at $Re=50$ , $Wi=1.0$ , $El=0.02$ , an off-centre long-midline EP occurs; and (d) at $Re=100$ , $Wi=0.5$ , $El=0.005$ , particles migrate towards the diagonal-line and short-midline EP. Here $d/H=0.15$ .

Figure 9(c) shows the particle trajectories for the blockage ratio of 0.1 (the blockage ratio being defined as the ratio of the particle diameter to the channel width, $d/H$ ), and the particles migrate towards the diagonal-line EP, which is closer to the centreline compared to the case of $d/H=0.15$ . This is consistent with the theoretical prediction (Ho & Leal Reference Ho and Leal1974, Reference Ho and Leal1976) that the inertial effect decreases more significantly than the elastic effect as the particle size decreases, since the elastic force is proportional to $d^{3}$ (equation (3.1)), while the inertial force is proportional to $d^{4}$ (equation (3.2)).

The Oldroyd-B model is characterized by a constant viscosity. The Giesekus model is a typical constitutive model for various polymer solutions with the shear-thinning property and a non-zero second normal stress difference, which leads to the secondary flow in a square channel (Villone et al. Reference Villone, D’Avino, Hulsen, Greco and Maffettone2013; Li et al. Reference Li, McKinley and Ardekani2015). The secondary flow is characterized by eight circulations in the cross-section with the flow direction pointing to the channel centreline along the diagonal line. The Giesekus constitutive equation can be written as follows:

(3.3) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D63D}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D63D}-\unicode[STIX]{x1D63D}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}-(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}\boldsymbol{\cdot }\unicode[STIX]{x1D63D}+\frac{\unicode[STIX]{x1D6FC}}{Wi}(\unicode[STIX]{x1D63D}-\unicode[STIX]{x1D644})^{2}+\frac{\unicode[STIX]{x1D63D}-\unicode[STIX]{x1D644}}{Wi}=0\quad \text{in }\unicode[STIX]{x1D6FA},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}$ is a parameter in the Giesekus model, which reduces to the Oldroyd-B model for $\unicode[STIX]{x1D6FC}=0$ . The particle trajectories projected in the cross-section of a channel filled with a Giesekus fluid with $\unicode[STIX]{x1D6FC}=0.2$ for $L=H$ , $d/H=0.15$ , $Re=50$ and $Wi=0.5$ are plotted in figure 9(d). The diagonal-line EP can also be observed in the Giesekus fluid, which is located farther away from the channel centreline compared to the Oldroyd-B case. Li et al. (Reference Li, McKinley and Ardekani2015) also observed that the equilibrium position in the midline for the Giesekus fluid was closer to the wall than that for the Oldroyd-B fluid at the same flow conditions. They attributed the reason to the effects of the shear-thinning and secondary flow. The secondary flow points to the wall on the midline and thus tends to push the particle on the midline towards the wall. By contrast, the secondary flow points to the channel centreline on the diagonal line and consequently tends to push the particle on the diagonal line towards the centreline. Therefore, the shear-thinning effect should be the primary reason for the equilibrium position being shifted farther away from the channel centreline. In addition, the elastic effect of the Giesekus fluid for the same Reynolds and Weissenberg numbers is weaker than that of the Oldroyd-B fluid, which is expected to be partly responsible for the shift of the equilibrium position observed above. We note that this section (§ 3.1.4) is the only one in the present study to discuss the effect of the Giesekus fluid.

3.2.1 Elasto-inertial migration equilibrium positions

Our results have revealed a diagonal-line EP for the particle migration in square channel flow. We now further investigate the particle migration in a rectangular channel with an aspect ratio of 2, to see whether similar EP would exist in rectangular channel flow. The results on the particle trajectories for $(Re,Wi,El)=(1,0.5,0.5)$ , $(50,0.5,0.01)$ , $(50,1.0,0.02)$ and $(100,0.5,0.005)$ are shown in figure 10. It is interesting that similar ‘diagonal-line’ EP does occur for the cases of $(Re,Wi,El)=(50,0.5,0.01)$ (figure 10 b) and $(100,0.5,0.005)$ (figure 10 d), the same parameters at which the diagonal-line EP occur for the square channel flow (see figures 5 c and 6 c). For $(Re,Wi,El)=(50,1.0,0.02)$ , the ‘diagonal-line’ EP disappears in figure 10(c), as the $El$ number is increased to 0.02. The diagonal-line EP is shifted to the channel centreline for the square channel (figure 5 d), whereas it is shifted to an off-centre position located on the long midline. The channel centreline EP is expected if the elasticity number is further increased above a critical value. Since a relatively large $Wi$ number is limited by our code, the viscoelasticity-dominated particle migration behaviour is examined at a low-Reynolds-number case of $(Re,Wi,El)=(1,0.5,0.5)$ . The results are shown in figure 10(a), from which the particles released near the wall or the corner migrate towards the corner, and the other particles migrate towards a region on the long midline with the distance away from the centreline being less than roughly $0.35H$ . Further migration along the long midline cannot be identified clearly with our code, and it is not clear whether this is a numerical artifact. Migration towards the centreline is expected if the fluid elasticity is further enhanced. In addition, it is shown later that the particle size has significant effects on the particle migration along the long midline at low Reynolds numbers, and smaller particles with $d/H=0.1$ migrate towards the channel centreline (figure 12 a).

Figure 11. (a) Contour map for $(\text{d}u/\text{d}x)^{2}+(\text{d}u/\text{d}y)^{2}$ . (b) Contour lines for $(\text{d}u/\text{d}x)^{2}-(\text{d}u/\text{d}y)^{2}$ , with values marked on the lines, showing that the particle equilibrium positions are located on the line where $(\text{d}u/\text{d}x)^{2}=(\text{d}u/\text{d}y)^{2}$ . The red and violet circles represent the particle equilibrium positions for $(Re,Wi)=(50,0.5)$ and $(Re,Wi)=(100,0.5)$ , respectively.

For $Wi=0.5$ , as $Re$ is increased from 50 to 100, the ‘diagonal-line’ EP is shifted farther away from the centreline (see figure 10 b and 10 d), and, on the other hand, an EP on the short midline appears (figure 10 d). It is known that, for Newtonian rectangular channel flow, the fluid inertia drives particles to move towards the EP located on the short midline (Zhou & Papautsky Reference Zhou and Papautsky2013), and also the EP on the long midline if $Re$ exceeds a critical value (Liu et al. Reference Liu, Hu, Jiang and Sun2015a ). Our results indicate that the important elasto-inertia EP for relatively small particles undergoes the transition from midline, ‘diagonal line’, off-centre long midline and channel centreline. The corner EP may appear together with the off-centre long-midline EP and the centreline EP, but is not so important for the non-shear-thinning fluids since its attractive region is always small compared to the area of the channel cross-section.

For the non-square rectangular channel, the ‘diagonal-line’ EP is actually not located on the diagonal line, as shown in figure 10. Considering that the shear of the flow is the driving factor for the particle lateral migration, we conjecture that the ‘diagonal-line’ EP is related to the competition between the shear effects from two lateral directions. Figure 11(b) shows the contours of $(\text{d}u/\text{d}x)^{2}-(\text{d}u/\text{d}y)^{2}$ (here $u$ being the mainstream velocity of the undisturbed channel flow for $Wi=0.5$ ) and the particle equilibrium positions for $(Re,Wi)=(50,0.5)$ and $(100,0.5)$ . One can see that both equilibrium positions are located on the line where the shear rates in two lateral directions are equal to each other, i.e.  $(\text{d}u/\text{d}x)^{2}=(\text{d}u/\text{d}y)^{2}$ . The contour map for $(\text{d}u/\text{d}x)^{2}+(\text{d}u/\text{d}y)^{2}$ is shown in figure 11(a), which indicates that the equilibrium positions are located in the region of low total shear rates. It has been well recognized that the elastic force drives particles from the high-shear-rate region to the low-shear-rate region (Ho & Leal Reference Ho and Leal1976; Leshansky et al. Reference Leshansky, Bransky, Korin and Dinnar2007; Seo et al. Reference Seo, Kang and Lee2014; Liu et al. Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ). The inertial force pushes particle away from the channel centreline. The ‘diagonal-line’ EP can be easily understood by considering the combined effects of the elastic and inertial forces.

Figure 12. Particle trajectories for different initial positions projected in the channel cross-section for (a) $d/H=0.1$ and (b) $d/H=0.3$ , showing the particle-size-dependent focusing behaviour. Here $Re=1$ , $Wi=0.5$ and $El=0.5$ .

3.2.2 Particle-size-dependent focusing behaviour

Liu et al. (Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ) observed in their experiments that the particles with the blockage ratio of 0.1 in rectangular microchannels with the aspect ratio of $1{-}4$ were focused along the channel centreline, whereas the focusing position of the particles with the blockage of 0.3 was shifted away from the centreline in the direction of the long midline for Reynolds numbers of order 1 and large elasticity numbers. Di et al. (Reference Di, Lu and Xuan2016) and Xiang et al. (Reference Xiang, Dai and Ni2016) also observed off-centreline focusing positions in planes parallel to the shorter walls at $Re<1$ . The presence of the off-centreline equilibrium position at low Reynolds numbers is surprising, since the inertial effect that drives particles away from the centreline is expected to be small. Here, simulations are performed for $d/H=0.1$ and 0.3 at $Re=1$ and $Wi=0.5$ , in an attempt to check whether the observation of Liu et al. (Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ) can be reproduced numerically. The results on the particle trajectories are presented in figure 12. It is interesting that the observation of Liu et al. (Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ) is reproduced successfully: the particles with $d/H=0.1$ migrate towards the channel centreline, while the particles with $d/H=0.3$ migrate towards an off-centreline EP located in the long midline, as shown in figure 12. Liu et al. (Reference Liu, Xue, Chen, Shan, Tian and Hu2015b ) provided an explanation for their observation, which was proposed by Huang et al. (Reference Huang, Feng, Hu and Joseph1997) to account for the effect of the blockage ratio on the viscoelasticity-induced particle migration in plane Poiseuille flow. The numerical results of Huang et al. (Reference Huang, Feng, Hu and Joseph1997) indicated that the particle in plane Poiseuille flow with $d/H=0.025$ migrated all the way towards the channel centre, whereas the particle with $d/H=0.25$ migrated towards the wall for $Re=0$ and different Weissenberg numbers. Huang et al. (Reference Huang, Feng, Hu and Joseph1997) assumed that for a small blockage ratio the migration of the particle is controlled by the normal stresses through the curvature of the velocity profile which push the particle towards the centreline, as predicted by the perturbation solution of Ho & Leal (Reference Ho and Leal1976), and in contrast, for a large blockage ratio, the strong effects of the wall blockage overwhelm the effects of the curvature of the velocity profile and increase the effects of the compressive normal stresses acting on the particle surface so as to drive the particle towards the wall. This mechanism cannot fully explain the off-centre position on the long midline for the three-dimensional rectangular channel flow, because according to this mechanism the particle released near the short midline (the pink trajectory in figure 12 b for $d/H=0.3$ ) should move towards the long wall, instead of moving towards the long midline. Nevertheless, the numerical results of Huang et al. (Reference Huang, Feng, Hu and Joseph1997) indicated that the elastic effects on the particle migration can be strongly modulated by the blockage ratio, and therefore it is not impossible that the elastic effects drive a large particle in rectangular channel flow with bidirectional shear actions towards an off-centre position on the long midline, rather than on the short midline. The inertial effect could push the particle away from the channel centreline, but the results are similar when we decrease the Reynolds number from 1 to 0.2.

3.2.3 Force analysis

Finally, we attempt to determine which type of force is mainly responsible for the particle migration. For this purpose, we fix the lateral position $(x_{0},y_{0})$ of a particle which moves in the streamwise direction and rotates freely, and then calculate the different types of forces on the particle. Our fictitious domain method is a non-boundary-fitted method, and thus interpolation is needed to compute the stresses on the particle surface. For a particle with a fixed lateral position $(x_{0},y_{0})=(0.2,0)$ , the total hydrodynamic force in the $x$ direction, $F_{x}$ , can be decomposed into the pressure force, $F_{p}$ , the viscous force, $F_{V}$ , and the polymer force, $F_{E}$ , which have the following forms:

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle F_{p}=\int _{\unicode[STIX]{x2202}P}-n_{x}p\,\text{d}S, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle F_{V}=\unicode[STIX]{x1D702}_{s}\int _{\unicode[STIX]{x2202}P}\left[2n_{x}\frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}x}+n_{y}\left(\frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}y}+\frac{\unicode[STIX]{x2202}u_{y}}{\unicode[STIX]{x2202}x}\right)+n_{z}\left(\frac{\unicode[STIX]{x2202}u_{x}}{\unicode[STIX]{x2202}z}+\frac{\unicode[STIX]{x2202}u_{z}}{\unicode[STIX]{x2202}x}\right)\right]\,\text{d}S, & \displaystyle\end{eqnarray}$$

(3.6) $$\begin{eqnarray}\displaystyle & \displaystyle F_{E}=\int _{\unicode[STIX]{x2202}P}(n_{x}\unicode[STIX]{x1D70F}_{xx}+n_{y}\unicode[STIX]{x1D70F}_{xy}+n_{z}\unicode[STIX]{x1D70F}_{xz})\,\text{d}S. & \displaystyle\end{eqnarray}$$

The polymer force, $F_{E}$ , can be further decomposed into the polymer normal force, $F_{En}$ , and the polymer shear force, $F_{Es}$ :

(3.7) $$\begin{eqnarray}\displaystyle & \displaystyle F_{En}=\int _{\unicode[STIX]{x2202}P}n_{x}\unicode[STIX]{x1D70F}_{xx}\,\text{d}S, & \displaystyle\end{eqnarray}$$

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle F_{Es}=\int _{\unicode[STIX]{x2202}P}(n_{y}\unicode[STIX]{x1D70F}_{xy}+n_{z}\unicode[STIX]{x1D70F}_{xz})\,\text{d}S. & \displaystyle\end{eqnarray}$$

Note that the pressure force in (3.4) is calculated from the pressure that serves to make the fluid satisfy the incompressibility constraint. For the viscoelastic fluid, the average polymer normal stresses or the polymer isotropic stress $(\unicode[STIX]{x1D70F}_{xx}+\unicode[STIX]{x1D70F}_{yy}+\unicode[STIX]{x1D70F}_{zz})/3$ can also be regarded as ‘pressure’, and the elasticity-induced force is related to the normal stress differences (Ho & Leal Reference Ho and Leal1976). Therefore, we decompose the polymer normal force, $F_{En}$ , into the polymer ‘pressure’ force, $F_{Enp}$ , the force from the first normal stress difference, $F_{En1}$ , and the force from the second normal stress difference, $F_{En2}$ :

(3.9) $$\begin{eqnarray}F_{En}=\underbrace{\int _{\unicode[STIX]{x2202}P}n_{x}\frac{(\unicode[STIX]{x1D70F}_{xx}+\unicode[STIX]{x1D70F}_{yy}+\unicode[STIX]{x1D70F}_{zz})}{3}\,\text{d}S}_{F_{Enp}}+\underbrace{\int _{\unicode[STIX]{x2202}P}n_{x}\frac{(\unicode[STIX]{x1D70F}_{xx}-\unicode[STIX]{x1D70F}_{zz})}{3}\,\text{d}S}_{F_{En1}}+\underbrace{\int _{\unicode[STIX]{x2202}P}n_{x}\frac{(\unicode[STIX]{x1D70F}_{xx}-\unicode[STIX]{x1D70F}_{yy})}{3}\,\text{d}S}_{F_{En2}}.\end{eqnarray}$$

Table 1. Forces on the particle with a fixed lateral position $(x_{0},y_{0})$ for different cases, normalized by $\unicode[STIX]{x1D702}_{0}U_{0}H$ . Here $F_{p}$ , $F_{V}$ , $F_{E}$ , $F_{Es}$ , $F_{En}$ , $F_{Enp}$ , $F_{En1}$ and $F_{En2}$ represent the pressure force, viscous force, polymer force, polymer shear force, polymer normal force, polymer pressure force, first normal stress difference force and second normal stress difference force defined in (3.4)–(3.9), respectively. RecWi05Re1d03X: rectangular channel, $Wi=0.5$ , $Re=1$ , $d/H=0.3$ and $(x_{0},y_{0})=(0,0.2)$ . RecWi05Re1d03X: rectangular channel, $Wi=0.5$ , $Re=1$ , $d/H=0.3$ and $(x_{0},y_{0})=(0.2,0.0)$ . SquWi05Re1CS: square channel, $Wi=0.5$ , $Re=1$ , $d/H=0.15$ and $(x_{0},y_{0})=(0.2,0.0)$ . SquWi15Re100CS: square channel, $Wi=1.5$ , $Re=100$ , $d/H=0.15$ and $(x_{0},y_{0})=(0.2,0.0)$ . SquWi0Re100CS: square channel, $Wi=0.0$ , $Re=100$ , $d/H=0.15$ and $(x_{0},y_{0})=(0.2,0.0)$ . SquWi0Re100CL: square channel, $Wi=0.0$ , $Re=100$ , $d/H=0.15$ and $(x_{0},y_{0})=(0.2,0.31)$ . SquWi05Re100CL: square channel, $Wi=0.5$ , $Re=100$ , $d/H=0.15$ and $(x_{0},y_{0})=(0.2,0.28)$ .

Our results on the forces on the particle with a fixed lateral position $(x_{0},y_{0})$ at steady state for some cases are presented in table 1. The forces are normalized by $\unicode[STIX]{x1D702}_{0}U_{0}H$ . Note that for the case of ‘RecWi05Re1d03Y’, where the particle lateral position is fixed at $(0,0.2)$ , the forces in the $y$ direction are calculated with the subscripts $x$ and $y$ exchanged in (3.4)–(3.9). Our results indicate that for the elasticity-induced rapid cross-stream migration (RecWi05Re1d03Y, SquWi05Re1CS and SquWi15Re100CS), the pressure force is the most important driving force for the migration, compared to the viscous force and the polymer force, particularly for the low-Reynolds-number cases (RecWi05Re1d03Y and SquWi05Re1CS). Nevertheless, if one adds the polymer pressure force $F_{Enp}$ to the pressure $F_{p}$ , then the total pressure force plays a negative role in the particle migration, and the force from the first normal stress difference becomes the driving force for the rapid cross-stream migration, consistent with the theory of Ho & Leal (Reference Ho and Leal1976). The second normal stress difference is zero for the pure shear flow of the Oldroyd-B fluid, and thus its contribution to the total force is always small, though it is interesting that its direction is always the same as that of the first normal stress difference force. For the inertia-induced rapid cross-stream migration (SquWi0Re100CS), the pressure is also the most important driving force. For the inertia-induced slow cross-lateral migration to the midline EP (SquWi0Re100CL), the pressure plays a negative role, and the viscous shear force becomes the driving force. For the elasticity-induced slow cross-lateral migration to the diagonal-line EP (SquWi05Re100CL), the pressure is not important and both the first normal stress difference and the polymer shear force are important. For the elasticity-induced anomalous slow off-centre migration in the rectangular channel (RecWi05Re1d03X), both the pressure and the first normal stress difference still tend to drive the particle to move towards the channel centreline, and the shear forces including both the viscous and polymer shear forces become the driving forces for the particle off-centre migration, unlike the elasticity-induced rapid cross-stream migration (RecWi05Re1d03Y).

The contours of the pressure and the lateral velocity in the symmetry plane for the particles with the fixed lateral positions of $(x_{0},y_{0})=(0,0.2)$ (RecWi05Re1d03Y) and $(x_{0},y_{0})=(0.2,0)$ (RecWi05Re1d03X) are compared in figure 13. Owing to the significantly different distributions of the shear rate (see figure 11 a), there are differences in the distribution pattern of the pressure and the lateral velocity for the two cases. However, it is not clear why the pressure force is important in the inward migration (RecWi05Re1d03Y) and not important in the off-centre migration (RecWi05Re1d03X). We leave the complete understanding of the off-centre EP on the long midline at low Reynolds numbers to a future study.

Figure 13. Flow fields in the symmetry plane for the particle with a fixed lateral position in a rectangular channel: (a) pressure, $(x_{0},y_{0})=(0,0.2)$ ; (b) pressure, $(x_{0},y_{0})=(0.2,0)$ ; (c) lateral fluid velocity, $(x_{0},y_{0})=(0,0.2)$ ; and (d) lateral fluid velocity, $(x_{0},y_{0})=(0.2,0)$ . The arrows indicate the flow direction. Here $Re=1$ , $Wi=0.5$ and $d/H=0.3$ .