Inertial migration of a deformable particle in pipe flow

We perform fully Eulerian numerical simulations of an initially spherical hyperelastic particle suspended in a Newtonian pressure-driven flow in a cylindrical straight pipe. We study the full particle migration and deformation for different Reynolds numbers and for various levels of particle elasticity, to disentangle the interplay of inertia and elasticity on the particle focusing. We observe that the particle deforms and undergoes a lateral displacement while traveling downstream through the pipe, finally focusing at the pipe centerline. We note that the migration dynamics and the final equilibrium position are almost independent of the Reynolds number, while they strongly depend on the particle elasticity; in particular, the migration is faster as the elasticity increases (i.e., the particle is more deformable), with the particle reaching the final equilibrium position at the centerline in shorter times. Our simulations show that the results are not affected by the particle initial conditions, position, and velocity. Finally, we explain the particle migration by computing the total force acting on the particle and its different components, viscous and elastic.

Figure 1

Visualization of the motion of a single deformable particle, initially spherical with radius $r$, in a cylindrical straight pipe of radius $R$, for the case $\text{Re}=200$ and $\text{We}=0.5$. The initial radial particle position is at $r=0.7487R$. The particle and the pipe are shown at the actual scale.

Figure 2

The time history of the particle radial position $r$ at different Weber numbers We for various Reynolds numbers Re, as indicated in the legend. The radial position $r$ is normalized by the radius of the pipe $R$ whereas time is scaled by $D/U_b$.

Figure 3

The time history of the particle radial position $r$ at different Reynolds numbers Re for various Weber numbers We, as indicated in the legend.

Figure 4

Normalized time $t_{\mathrm{cen}}$ needed to reach the final equilibrium position vs (a) the bulk Reynolds number Re for different $\text{We}$ and (b) the Weber number We for different Re.

Figure 5

Normalized period of the particle oscillatory motion, $T_{\mathrm{osc}}{U}_b/D$, as a function of the Weber number We for different Re.

Figure 6

The time history of the radial position of the deformable particle released at (a) three different initial positions and (b) with two different initial velocities at the same Reynolds and Weber numbers: $\text{Re}=400$ and $\text{We}=1$. (c) Trajectory of the particle in the pipe cross section ($y-z$ plane) for the cases reported in panel (b).

Figure 7

Trajectory of the particle in the pipe cross section ($y-z$ plane) at different Reynolds numbers Re and for various Weber numbers We. The black squares and purple circles show the initial and the final equilibrium positions, respectively.

Figure 8

Cross section of the deformed shape of an hyperelastic particle at $\text{Re}=200$ and $\text{We}=0.5$ at seven different time instants, marked in the time history of the radial position with black circles, from the initial to the equilibrium position. The red and green lines represent the change of particle shape during an oscillation.

Figure 9

Cross section of the deformed shape of an hyperelastic particle for (top) three different Reynolds numbers and at the same Weber number $\text{We}=0.5$ and for (bottom) three different Weber numbers and at the same Reynolds number $\text{Re}=400$ at the final equilibrium state.

Figure 10

The time history of the normalized total radial forces $F_r$ acting on the particle at different Reynolds numbers Re and for various Weber numbers We.

Figure 11

Normalized time needed to reverse the sign of the total force acting on the particle, $t_{\mathrm{rev}}$, vs the Weber number We for different Re.

Figure 12

The time history of the normalized radial force components $F_r^i$ acting on the particle for different Weber numbers and a fixed Reynolds number $\text{Re}=200$.

Figure 13

Time history of the radial position of the deformable particle for three different solid/fluid viscosity ratios at the same Reynolds and Weber numbers: $\text{Re}=400$ and $\text{We}=4$.

Figure 14

The time history of the particle radial position $r$ at different Weber numbers $\mathrm{We}$ at constant Reynolds numbers $\mathrm{Re}=400$, as indicated in the legend. The radial position $r$ is normalized by the radius of the pipe $R$ whereas time is scaled by $D/U_b$.