Resistance and mobility functions for the near-contact motion of permeable particles

Abstract

A lubrication analysis is presented for the resistances between permeable spherical particles in near contact, $h_0/a\ll 1$ , where $h_0$ is the minimum separation between the particles, and $a=a_1 a_2/(a_1+a_2)$ is the reduced radius. Darcy's law is used to describe the flow inside the permeable particles and no-slip boundary conditions are applied at the particle surfaces. The weak permeability regime $K=k/a^{2} \ll 1$ is considered, where $k=\frac {1}{2}(k_1+k_2)$ is the mean permeability. Particle permeability enters the lubrication resistances through two functions of $q=K^{-2/5}h_0/a$ , one describing axisymmetric motions, the other transverse. These functions are obtained by solving an integral equation for the pressure in the near-contact region. The set of resistance functions thus obtained provide the complete set of near-contact resistance functions for permeable spheres and match asymptotically to the standard hard-sphere resistances that describe pairwise hydrodynamic interactions away from the near-contact region. The results show that permeability removes the contact singularity for non-shearing particle motions, allowing rolling without slip and finite separation velocities between touching particles. Axisymmetric and transverse mobility functions are presented that describe relative particle motion under the action of prescribed forces and in linear flows. At contact, the axisymmetric mobility under the action of oppositely directed forces is $U/U_0=d_0K^{2/5}$ , where $U$ is the relative velocity, $U_0$ is the velocity in the absence of hydrodynamic interactions and $d_0=1.332$ . Under the action of a constant tangential force, a particle in contact with a permeable half-space rolls without slipping with velocity $U/U_0=d_1(d_2+\log K^{-1})^{-1}$ , where $d_1=3.125$ and $d_2=6.666$ ; in shear flow, the same expression holds with $d_1=7.280$ .

A pairwise hydrodynamic theory for flow-induced particle transport in shear and pressure-driven flows

“…The results in figure 5 of Reboucas & Loewenberg (2022) show that the contact value of the transverse relative mobility for permeable particles, B * , has the upper and lower bounds, B(s * eq ) < B * ≤ B( 2), (C43)…”

“…where B( 2) is the contact value of the hard-sphere mobility function, corresponding to rough particles with tangentially locking particle contacts; the equality holds for κ = 1 because permeability has no effect on the transverse relative mobility for equal-size particles (Reboucas & Loewenberg 2022). The quantity B(s * eq ) corresponds to frictionless contact between particles with equivalent roughness, where s * eq = 2 + δeq with δeq defined by (4.1).…”

“…The results in figure 5 of Reboucas & Loewenberg (2022) show that the contact value of the transverse relative mobility for permeable particles, , has the upper and lower bounds, where is the contact value of the hard-sphere mobility function, corresponding to rough particles with tangentially locking particle contacts; the equality holds for because permeability has no effect on the transverse relative mobility for equal-size particles (Reboucas & Loewenberg 2022). The quantity corresponds to frictionless contact between particles with equivalent roughness, where with defined by .…”

“…The prototypical example of a short-range mechanism for contact interactions is small-amplitude surface roughness, , that prevents surface-to-surface particle separations less than (Smart & Leighton 1989; Smart, Beimfohr & Leighton 1993; da Cunha & Hinch 1996; Rampall, Smart & Leighton 1997; Wilson 2005; Ingber & Zinchenko 2012). Other examples of particles with short-range contact interactions include particles with weak permeability (Reboucas & Loewenberg 2021 a , b , 2022), particles stabilized by screened electrostatic interactions (Zinchenko 1984; Kremer, Robbins & Grest 1986) and emulsion drops under small-deformation conditions (Loewenberg & Hinch 1997; Ramachandran et al. 2010).…”