Shear-induced particle migration in viscous suspensions with continuous size distribution

We present an approach to study the shear-induced diffusion of particles in suspensions with continuous particle-size distribution by addressing the migration of local moments of the size distribution. This approach replaces studying fluxes of particles of particular sizes. The physical problem is redefined in terms of fluxes of moments and their conservation. Particle-size distribution at each point is consequently obtained from the resulting moments, by solving inverse problems locally. The approach is applicable to any composite suspension, depending only on the initial size distribution in it. A particular example of migration in a circular tube is described. Results include concentration inhomogeneity, moments’ distribution, and the consequent local continuous particle-size distributions. We present stationary physical flow parameters and the similarity and difference from cases of monodispersed suspensions. For example, while in most cases of particle concentration, the general phenomena resemble the migration directions observed in monodisperse suspensions, there are cases, associated with extreme input high or low concentrations, where further accumulation of specific sizes at locations close to domain boundaries are encountered. This basic study will enhance better understanding of the complex behavior and properties of multiphase systems.

Figure 1

A normal continuous particle-size distribution normalized by the average value.

Figure 2

Profiles of stationary moments obtained for high, low, and intermediate input particle concentration: (a) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.65$; (b) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.25$; (c) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.4$; (d) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.35$. The profiles are obtained by minimizing (17) with respect to the coefficients in the approximation (18).

Figure 3

The constant of integration, $C$, in Eq. (9) for the range of input concentration ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}$.

Figure 4

Stationary profiles of ${\mathrm{\Phi }}^{\left(0\right)}, {\mathrm{\Phi }}^{\left(1\right)}, \overline{a}={\mathrm{\Phi }}^{\left(1\right)}/{\mathrm{\Phi }}^{\left(0\right)}, \gamma , \mu$, and $u$ in the tube cross section. Cases (a)–(c) correspond to initial ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}$ values of 0.65, 0.6, and 0.55; 0.5, 0.45, and 0.4; and 0.35, 0.3, and 0.25, respectively.

Figure 5

Stationary velocity profiles: (a) Flattening profiles at ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}\ge 0.45$ and (b) Comparison of at high, intermediate, and low particle concentration.

Figure 6

A logarithmic plot of stationary moments obtained for high-input particle concentration at ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.65$.

Figure 7

The deviation from unity for sequential moment ratios.

Figure 8

Stationary particle-size distributions of, $\varphi \left(a\right)$, in various radial locations. Cases (a)–(h) correspond to initial uniform ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}$ values of 0.65, 0.6, 0.55, 0.5, 0.45, 0.4, 0.35, and 0.3, respectively.

Figure 9

Particles-size profiles at positions close to the tube wall: (a) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.6$ and (b) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.65$.

Figure 10

Particle-size profiles at positions close to the tube center: (a) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.3$ and (b) ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.35$.

Figure 11

Particle-size distributions about $a_{max}\approx 1.2$ at intermediate $r$ positions, ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.45$.

Figure 12

Particle sizes at $r=0$ for the various cases of ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}$. The minimum obtained for $S$ in Eq. (21) is of $O$ (${10}^{-11}$).

Figure 13

The concentration distributions of sizes 1.95, 1,909, 1.85 near the tube center $r\to 0$ in the case ${\widehat{\mathrm{\Phi }}}^{\left(0\right)}=0.65$.