Effect of particle collisions in dense suspension flows

We study nonlocal effects associated with particle collisions in dense suspension flows, in the context of the Affine Solvent Model, known to capture various aspects of the jamming transition. We show that an individual collision changes significantly the velocity field on a characteristic volume ${\mathrm{\Omega }}_c\sim 1/\delta z$ that diverges as jamming is approached, where $\delta z$ is the deficit in coordination number required to jam the system. Such an event also affects the contact forces between particles on that same volume ${\mathrm{\Omega }}_c$, but this change is modest in relative terms, of order $f_{\mathrm{coll}}\sim {\overline{f}}^{0.8}$, where $\overline{f}$ is the typical contact force scale. We then show that the requirement that coordination is stationary (such that a collision has a finite probability to open one contact elsewhere in the system) yields the scaling of the viscosity (or equivalently the viscous number) with coordination deficit $\delta z$. The same scaling result was derived [E. DeGiuli, G. Düring, E. Lerner, and M. Wyart, Phys. Rev. E 91, 062206 (2015)] via different arguments making an additional assumption. The present approach gives a mechanistic justification as to why the correct finite size scaling volume behaves as $1/\delta z$ and can be used to recover a marginality condition known to characterize the distributions of contact forces and gaps in jammed packings.

Figure 1

Snapshot a of suspension flowing under simple shear using the ASM at the instant of a collision between two particles (in blue). (a) Black lines represent the contact network; the width of the lines are proportional to the magnitude of contact forces $f$ immediately after the collision. (b) The width of the lines connecting centers of particles are proportional to the magnitude of the instantaneous variations in contact forces induced by the collision. Red (black) lines correspond to negative (positive) variations in the contact forces. The dashed circle is a visualization of the typical volume ${\mathrm{\Omega }}_c$ as defined in the text. (c) The vector field represents the instantaneous variations in the particles' velocity induced by the collision. The dashed circle is a visualization of the typical volume ${\mathrm{\Omega }}_v$ as defined in the text.

Figure 2

Data from simulations of the ASM in three dimensions, for systems of $N=2000$ particles, and pressures of $10,30,100,300,$ and 1000 (see Sec. D of the Appendix for more details on the numerical method). Solid lines are a guideline to compare data with the theoretical predictions. (a) Forces $f_{\text{open}}$ in contacts that open due to a collision, just before the collision takes place, vs the force in the newly created contact $f_{\text{coll}}$. Data are binned according to the measured $f_{\text{coll}}$ for all different pressures. (b) Relative radial velocity $v_{\text{coll}}$ of pairs of colliding particles just before a collision takes place, vs the mean nonaffine velocity of the particles $V_{\text{n.a}}$, taken over all particles in the system. Data are averaged over runs at a certain pressure.

Figure 3

Data from simulations of the ASM in three dimensions, for systems of $N=2000$ particles, and pressures of $10,30,100,300,$ and 1000. Solid lines are a guideline to compare data with the theoretical predictions. (a) The correlation volume ${\mathrm{\Omega }}_c$ vs the coordination $\delta z$. Data are averaged over runs at a certain pressure. (b) The correlation volume ${\mathrm{\Omega }}_v$ vs the coordination $\delta z$. Data are averaged over runs at a certain pressure. (c) Mean contact force created at collisions $f_{\text{coll}}$ vs the product of the correlation volume ${\mathrm{\Omega }}_v$ and the relative radial velocity at the collision $v_{\text{coll}}$. Data are binned according to the measured ${\mathrm{\Omega }}_c{v}_{\text{coll}}$ for all different pressures. (d) Contact force created upon collisions normalized by the pressure $f_{\text{coll}}/p$ vs the correlation volume ${\mathrm{\Omega }}_c$. The exponent $\theta =0.44$. Data are averaged over runs at a certain pressure.