Shear flow over a wall with suction and its application to particle screening

Abstract

In this paper, an extension of Miyazaki & Hasimoto's (1984) Green function for the slow flow created by a point force of arbitrary direction above an infinite plane wall with a circular hole was used to formulate a set of boundary-integral equations for the motion, at low Reynolds and Stokes numbers, of a finite rigid sphere in a simple shear flow with suction past an infinite wall containing a circular side hole. The equations were solved numerically by discretizing the surface of the sphere into a finite number of elements and then using a constant-density approximation for the unknown surface force distribution and a boundary collocation technique to satisfy the no-slip boundary condition at the centre of each element. Numerical tests and comparisons with available exact and numerical results show that convergence to three or four significant figures can be achieved for all the 21 independent unknown force and torque coefficients. Numerical values for these coefficients were obtained throughout the flow field for sphere–hole radii ratios of $\frac{1}{10}, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}$ and 1, and the neutrally buoyant velocities and trajectories of individuals spheres were then computed for a range of initial upstream positions, and for various values of a suction parameter defined in Yan et al. (1991 a) which refers to the relative strengths of the suction and shear flows. In turn, these trajectories were used to map out the particle capture tube and its upstream cross-section and thereby determine the particle screening effect, one of the underlying mechanisms responsible for the well-known exit concentration defect observed when particles enter a side pore. The other mechanism, the fluid skimming effect due to the presence of a particle-free layer on the upstream wall, was considered recently in a companion paper (Yan et al. 1991 a). It is shown here that the fluid skimming effect provides a lower bound for this concentration defect under the conditions of this analysis. The theoretical predictions exhibit features that are qualitatively similar to the experimental observations of the hematocrit (red cell) defect in the microcirculation, although the dilute suspension limit considered herein is well below the observed hematocrit in the microcirculation and the particles are modelled as rigid spheres.