Roles of solid effective stress and fluid-particle interaction force in modeling shear-induced particle migration in non-Brownian suspensions

The mixture, or suspension balance, model is the most frequently used to describe shear-induced particle migration in Newtonian fluids. The model suggests that neutrally buoyant particles migrate only if the effective stress tensor of the solid phase is nonuniform. In moderately dense suspensions, where direct particle contacts and interparticle forces are negligible, this tensor originates from the velocity fluctuations of the particles. Using Buyevich's constitutive equation, we show that the time required by these fluctuations to induce significant particle migration exceeds considerably the timescale of the process, obtained experimentally. In particular, the ratio between these two timescales is proportional to the reciprocal of the Reynolds number; hence, for vanishingly small values of the Reynolds number (the case on which this work focuses), particle velocity fluctuations cannot be responsible for particle migration. We conclude that, if direct particle contacts and interparticle forces are absent, particle migration must have another driver. In the literature, it has been suggested that this is the lubrication forces between the particles. On this assumption, Morris and Boulay advanced a closure for the solid effective stress tensor. This closure seems to predict well the migration process, but as discussed by P. R. Nott, E. Guazzelli, and O. Pouliquen, The suspension balance model revisited, Phys. Fluids 23, 043304 (2011), it presents some conceptual issues. We investigate this matter and show that lubrication forces can induce particle migration, and the tensor quantifying their effect can be closed by Morris and Boulay's equation; however, this tensor is not part of the effective stress tensor of the solid phase.

Figure 1

Solid volume fraction profiles in a wide-gap Couette flow. The outer cylinder is stationary, while the inner one rotates at angular velocity $\mathrm{\Omega }$. The suspensions, of neutrally buoyant particles with mean radius $a$, are initially uniform with solid volume fraction ${\overline{\varepsilon }}_s^{}$. Model curves from Eqs. (22) and (24). Experimental data from Phillips et al. [22].

Figure 2

Solid volume fraction profiles in a wide-gap Couette flow. The outer cylinder is stationary, while the inner one rotates at angular velocity $\mathrm{\Omega }$. The suspensions, of neutrally buoyant particles with mean radius $a$, are initially uniform with solid volume fraction ${\overline{\varepsilon }}_s^{}$. Model curves from Eqs. (22) and (24). Experimental data from Phillips et al. [22].

Figure 3

Steady-state solid volume fraction profile in a plate-plate rheometer for a dense suspension with initial homogenous concentration ${\overline{\varepsilon }}_s^{}=0.4$ obtained using Eq. (25) and by assuming ${\varepsilon }_s^{★}=0.68$. This equation predicts an inward flux of particles that contradicts experimental observations. The horizontal line at ${\varepsilon }_s^{}=0.4$ is plotted to guide the eye.