Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow

Abstract

A new boundary-integral algorithm for the motion of a particle between two parallel plane walls in Poiseuille flow at low Reynolds number was developed to study the translational and rotational velocities for a broad range of particle sizes and depths in the channel. Instead of the free-space Green’s function more commonly employed in boundary-integral equations, we used the Green’s function for the domain between two infinite plane walls [Liron and Mochon, J. Eng. Math. 10, 287 (1976)]. This formulation allows us to directly incorporate the effects of the wall interactions into the stress tensor, without discretizing the bounding walls, and use well-established iterative methods. Our results are in good agreement with previous computations [Ganatos et al., J. Fluid Mech. 99, 755 (1980)] and limiting cases, over their range of application, with additional results obtained for very small particle–wall separations of less than 1% of the particle radius. In addition to the boundary-integral solution in the mobility formulation, we used the resistance formulation to derive the near-field asymptotic forms for the translational and rotational velocities, extending the results to even smaller particle–wall separations. The decrease in translational velocity from the unperturbed fluid velocity increases with particle size and proximity of the particle to one or both of the walls. The rotational velocity exhibits a maximum magnitude between the centerline and either wall, due to the competing influences of wall retardation and the greater fluid velocity gradient near the walls. The average particle velocity for a uniform distribution of particles was generally found to exceed the average fluid velocity, due in large part to exclusion of the particle centers from the region of slowest fluid near the walls. The maximum average particle velocity is 18% greater than the average fluid velocity and occurs for particle diameters that are 42% of the channel height; particles with diameters greater than 82% of the channel height have smaller average velocities than does the fluid, due to the retarding effects of the nearby walls. Additionally, the translational and rotational velocities of oblate and prolate ellipsoids were calculated using the boundary-integral algorithm. The proximity of the walls to the ellipsoids was found to have a strong effect on particle velocity, so that a prolate spheroid aligned on the channel centerline moves faster than an oblate spheroid of the same volume, because the edge of the latter spheroid is closer to the channel walls. For off-centerline locations for a prolate spheroid with its major axis at an angle to the walls, a second translational velocity component normal to the walls is present. For each lower-wall gap examined, the maximum normal translational velocity occurs for smaller gaps from the upper wall (i.e., larger angle). The direction of this velocity component changes sign for mid-range angles studied, due to increased interactions with the lower wall that prevent the ellipsoid from rotating upward and, hence, yield a negative velocity perpendicular to the walls. The rotational velocity changes direction for particles at the smallest angles studied, due to the competition between the Poiseuille fluid velocity profile, which pushes the particle clockwise, and the lubrication forces, which impede this rotation.

Boundary-integral method for drop deformation between parallel plates

“…To include the effects of the walls, we have used modified Green's functions, so G = G ϱ + G 2W , where G 2W includes all corrections due to the presence of two walls. Contrary to Staben et al, 37 who used the Green's functions derived by Liron and Mochon, 40 we have used those derived by Jones. 42 For completeness, these functions are reproduced here, All these Green's functions, as well as those derived by Liron and Mochon, 40 consist of infinite integrals with Bessel function weights.…”

“…Since the calculation of the Green's functions includes cumbersome integrals over quite complicated functions, we have modified the formulation in a similar way as has been done by Staben et al 37 to reduce the computational effort. For high q, some approximations can be made to simplify the expressions.…”

“…However, we have used the formulation for the Green's functions as derived by Jones, 42 which results in a convenient symmetric form and does not impose special demands on the location of the pole and the integration point relative to each other. Furthermore, in a similar fashion to Staben et al, 37 we subtract slow decaying terms and handle them analytically to speed up the computations of the Green's functions. This paper is organized as follows.…”

“…The solid particle contribution was handled by a Hebeker representation 36 by Zinchenko and Davis. 33 Staben et al [37][38][39] used the Green's functions associated with a parallel plate configuration, derived by Liron and Mochon, 40 and studied the movement of solid spheres and spheroids through channels. Those same Green's functions were also used to examine the movement and deformation of drops in a Poiseuille flow.…”

Microconfined equiviscous droplet deformation: Comparison of experimental and numerical results

“…27,28 Boundary integral methods have been used successfully to study drop dynamics in bulk flow conditions [29][30][31][32][33] or the movement of solid spheres and more recently drops between parallel walls. [34][35][36] In the boundary integral formulation, the velocity u at a point x 0 on the interface is given by…”