Slow viscous flow of two porous spherical particles translating along the axis of a cylinder

Abstract

We describe the motion of two freely moving porous spherical particles located along the axis of a cylindrical tube with background Poiseuille flow at low Reynolds number. The stream function and a framework based on cylindrical harmonics are adopted to solve the flow field around the particles and the flow within the tube, respectively. The two solutions are employed in an iterated framework using the method of reflections. We first consider the case of two identical particles, followed by two particles with different dimensions. In both cases, the drag force coefficients of the particles are solved as functions of the separation distance between the particles and the permeability of the particles. The detailed flow field in the vicinity of the two particles is investigated by plotting the streamlines and velocity contours. We find that the particle–particle interaction is dependent on the separation distance, particle sizes and permeability of the particles. Our analysis reveals that when the permeability of the particles is large, the streamlines are more parallel and the particle–particle interaction has less effect on the particle motion. We further show that a smaller permeability and bigger particle size generally tend to squeeze the streamlines and velocity contour towards the wall.

Creeping flow of a sphere nearby a cylinder

“…The unknown coefficients in cylindrical harmonics can be solved by constructing equations from boundary conditions. The details have been included in our works [29,30]. By utilizing the spherical harmonics and cylindrical harmonics for the sphere and cylinder, respectively, the total velocity flow field can be obtained using Eq.(3a).…”

“…By repeating the process and adding more reflections, the results will converge, as the magnitude of higher reflections decays to zero [29,30]. We can apply any arbitrary number of reflections in our framework when needed for convergence (details of the convergence test are provided in section 2.1.3).…”

“…s and set R = R 0[29,30]. In order to obtain an analytical solution, we firstly express the boundary condition (v(0) s | R=R0 ) into a parabolic function A(Φ)Z 2 + B(Φ)Z + C(Φ) and further into a periodic form Ã(Z) cos Φ + B(Z) or Ã(Z) sin Φ + B(Z).Since it is an infinite cylinder, the configuration is symmetric with respect to XY plane.…”

Interaction between two spheres under a uniform electric field in a porous medium

“…The mathematical details of coordinate transformation between the two spheres in spherical coordinates system can be found in our previous work (Yao et al, 2019).…”

Shielding Effect of Fractal Nanoparticle Agglomerates Aligned Streamwise

“…They found that increasing Re enhances the effect of the leading sphere on the trailing sphere. Yao et al 33 solved the flow of porous particles moving along the axis of a cylindrical tube and discussed the effect of L and permeability on the interparticle interaction. Zhang et al 35 numerically investigated the interaction between porous particles with the same diameter.…”