Integral representation of channel flow with interacting particles

We construct a boundary integral representation for the low-Reynolds-number flow in a channel in the presence of freely suspended particles (or droplets) of arbitrary size and shape. We demonstrate that lubrication theory holds away from the particles at horizontal distances exceeding the channel height and derive a multipole expansion of the flow which is dipolar to the leading approximation. We show that the dipole moment of an arbitrary particle is a weighted integral of the stress and the flow at the particle surface, which can be determined numerically. We introduce the equation of motion that describes hydrodynamic interactions between arbitrary, possibly different, distant particles, with interactions determined by the product of the mobility matrix and the dipole moment. Further, the problem of three identical interacting spheres initially aligned in the streamwise direction is considered and the experimentally observed “pair exchange” phenomenon is derived analytically and confirmed numerically. For nonaligned particles, we demonstrate the formation of a configuration with one particle separating from a stable pair. Our results suggest that in a dilute initially homogenous particulate suspension flowing in a channel the particles will eventually separate into singlets and pairs.

Figure 1

Schematic configuration of spherical particles flowing in the pressure-driven (Poiseuille) flow in a channel.

Figure 2

Depth-averaged disturbance flow around a sphere from the numerical simulations. The heavy arrows indicate the magnitude and light arrows the direction. The color in the background depicts the decay of the vorticity outside the sphere. The particle travels in the $x^{\prime }$ direction and is located at the midchannel ($z_p/h=0.5$), with $h/\left(2a\right)=1.5$. Only half of the plane is shown due to symmetry.

Figure 3

Spatial variation of the normalized streamwise depth-averaged disturbance velocity, $\delta u_d/U_b$, along the streamwise ($y=0$, circle) and spanwise ($x=0$, triangle) directions away from the particle center. The particle is located at the midchannel ($z_p/h=0.5$), with $h/\left(2a\right)=1.125$. The collapse of the disturbance velocity away from the particle confirms the leading-order dipolar decay (dashed line).

Figure 4

Pair exchange phenomenon as obtained from numerical simulations. Initially particles 1 are 2 are close and particle 3 is trailing behind the pair. As a result of hydrodynamic interactions, the trailing particle is catching up with the pair, the leading particle breaks away from the newly formed pair, and the trailing particles 2 and 3 are separated by the same distance as 1 and 2 were initially.