Variation of focusing patterns of laterally migrating particles in a square-tube flow due to non-Newtonian elastic force

The elasto-inertial effects on particle focusing in a square-tube flow were investigated experimentally and numerically. Microscale experiments using spherical particles in dilute polymer solutions demonstrated that the particles are focused on the midline and/or the diagonal in a downstream cross section, depending on the polymer concentration. Numerical computations based on the finitely extensible nonlinear elastic–Peterlin (FENE-P) model for the viscoelastic flow reproduced these focusing patterns. It was revealed that the transitions among the patterns are accounted for by the elastic forces due to the first normal stress difference and the polymer elongation, which are the essentials of the viscoelastic fluid.

Figure 1

Focusing positions in the PVP-solution experiments. (a) Experimental setup (schematic). (b)–(d) Existence probability of particles on a cross section near the outlet of the tube with the PVP concentrations (b) $\varphi =1$ wt%, (c) $\varphi =1.9$ wt%, and (d) $\varphi =2.5$ wt%. ($\mathrm{Re}=50$ and $\kappa =0.15$.)

Figure 2

Radial component of the lift exerted on the particle located (a) on the midline and (b) on the diagonal. Azimuthal component of the lift exerted on the particle located (c) on the line $z=ytan(\pi /8)$.

Figure 3

Lift and equilibrium positions. Stable, saddle, and unstable equilibrium positions are, respectively, represented by $●$, $◯$, and $\square$. The solid line and the dashed lines represent the contour of $F_r=0$ and those of $F_{\theta }=0$, respectively. Each arrow represents the lift direction with its magnitude shown by color. (a) $\mathrm{Wi}=0.01$, (b) $\mathrm{Wi}=0.08$, and (c) $\mathrm{Wi}=0.16$.

Figure 4

(a),(c) Alternative elastic lift ${\tilde{\mathit{F}}}_{\mathrm{E}}={\mathit{F}}_{\mathrm{Wi}}-{\mathit{F}}_0$. (b),(d) Contours of $N_1$ for every (b) $6.85\times {10}^{-3}$ and (d) $1.31\times {10}^{-2}$. The arrows represent $-\mathbf{\nabla }{N}_1$. (a),(b) $\mathrm{Wi}=0.08$ and (c),(d) $\mathrm{Wi}=0.16$.

Figure 5

(a)–(c) Conformation tensor on the $z=0$ plane and the elastic force acting on the particle placed at (a) $(y_{\mathrm{C}},z_{\mathrm{C}})\approx (0.25,0)$ and (b),(c) $(y_{\mathrm{C}},z_{\mathrm{C}})\approx (0.4,0)$. The ellipsoids colored by $l_{\mathrm{s}}$ represent the eigenvectors and eigenvalues of $\mathsf{C}$. (d),(e) Isosurfaces of $l_{\mathrm{s}}=15$ (yellow) and 55 (red) and the elastic force when the particle center is approximately at $(y_{\mathrm{C}},z_{\mathrm{C}})\approx (0.31,0.17)$. $\mathrm{Wi}=0.08$, except (c) $\mathrm{Wi}=0.16$.