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An exact pairwise hydrodynamic theory is developed for the flow-induced spatial distribution of particles in dilute polydisperse suspensions undergoing two-dimensional unidirectional flows, including shear and planar Poiseuille flows. Coupled diffusive fluxes and a drift velocity are extracted from a Boltzmann-like master equation. A boundary layer is predicted in regions where the shear rate vanishes with thickness set by the radii of the upstream collision cross-sections for pair interactions. An analysis of this region yields linearly vanishing drift velocities and non-vanishing diffusivities where the shear rate vanishes, thus circumventing the source of the singular particle distribution predicted by the usual models. Outside of the boundary layer, a power-law particle distribution is predicted with exponent equal to minus half the exponent of the local shear rate. Trajectories for particles with symmetry-breaking contact interactions (e.g. rough particles, permeable particles, emulsion drops) are analytically integrated to yield particle displacements given by quadratures of hard-sphere (or spherical drop) mobility functions. Using this analysis, stationary particle distributions are obtained for suspensions in Poiseuille flow. The scale for the particle distribution in monodisperse suspensions is set by the collision cross-section of the particles but its shape is almost universal. Results for polydisperse suspensions show size segregation in the central boundary layer with enrichment of smaller particles. Particle densities at the centreline scale approximately with the inverse square root of particle size. A superposition approximation reliably predicts the exact results over a broad range of parameters. The predictions agree with experiments in suspensions up to approximately 20 % volume fraction without fitting parameters. 

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$$d_1=7.280$$. 

An analysis is presented for the axisymmetric lubrication resistance between permeable spherical particles. Darcy's law is used to describe the flow in the permeable medium and a slip boundary condition is applied at the interface. The pressure in the near-contact region is governed by a non-local integral equation. The asymptotic limit$$K=k/a^{2} \ll 1$$is considered, where$$k$$is the arithmetic mean permeability, and$$a^{-1}=a^{-1}_{1}+a^{-1}_{2}$$is the reduced radius, and$$a_1$$and$$a_2$$are the particle radii. The formulation allows for particles with distinct particle radii, permeabilities and slip coefficients, including permeable and impermeable particles and spherical drops. Non-zero particle permeability qualitatively affects the axisymmetric near-contact motion, removing the classical lubrication singularity for impermeable particles, resulting in finite contact times under the action of a constant force. The lubrication resistance becomes independent of gap and attains a maximum value at contact$$F=6{\rm \pi} \mu a W K^{-2/5}\tilde {f}_c$$, where$$\mu$$is the fluid viscosity,$$W$$is the relative velocity and$$\tilde {f}_c$$depends on slip coefficients and weakly on permeabilities; for two permeable particles with no-slip boundary conditions,$$\tilde {f}_c=0.7507$$; for a permeable particle in near contact with a spherical drop,$$\tilde {f}_c$$is reduced by a factor of$$2^{-6/5}$$. 

Binary collision rates are calculated for the permeable particles undergoing (i) Brownian motion, (ii) gravity sedimentation, (iii) uniaxial straining flow, and (iv) shear flow. Darcy's law is used to describe the flow inside the permeable particles, and no-slip boundary conditions are applied at particle surfaces. A leading-order asymptotic solution of the problem is developed for the weak permeability regime K=k/a2≪1, where k=12(k1+k2) is the mean permeability and a=a1a2/(a1+a2) is the reduced radius; ai, ki (i = 1, 2), respectively, is the radius and permeability of each particle. The resulting collision rates are given by the quadrature of the pair mobility functions for permeable particles in the near-contact lubrication region and size-ratio-dependent parameters obtained from standard hard-sphere pair mobility functions. Collision rates in shear flow vanish below a critical value of the permeability parameter K* that increases with diminishing size ratio. The analogous problem of pair collision rates of particles with small-amplitude surface roughness δa is also analyzed. The formulas for the collision rates of rough particles provide accurate analytical approximations for the collision rates of permeable particles for all four aggregation mechanisms and a wide range of size ratios using an equivalent roughness δ=0.72K2/5.