Solid-body collisions between smooth particles in a gas would not occur if the lubrication force for a continuum incompressible fluid were to hold at all particle separations. When the gap between the particles is of the order of the mean free path λ0 of the gas, the discrete molecular nature of the gas becomes important. For particles of radii a smaller than about 50 μm colliding in air at a relative velocity comparable to their terminal velocity, the effects of compressibility of the gas in the gap are not important.The nature of the flow in the gap depends on the relative magnitudes of the minimum gap thickness h0 ≡ aε, the mean-free path λ0, and the distance aε1/2 over which the effects of curvature become important. The slip-flow regime, a[Gt ]λ0, was analysed by Hocking (1973) using the Maxwell slip boundary condition at the particle surface. To find the lubrication force in the transition regime (aε ∼ O(λ0)), we use the results of Cercignani & Daneri (1963) for the flux as a function of the pressure gradient in a Poiseuille channel flow. When aε[Lt ]λ0[Lt ]aε1/2, one might expect the local flow in the gap to be governed by Knudsen diffusion. However, an attempt to calculate the Knudsen diffusivity between parallel plates leads to a logarithmic divergence, which is cut off by intermolecular collisions, and the flux is therefore proportional to h0c log(λ0/h0), where c is the mean molecular speed. The non-continuum lubrication force is shown to have a weak, log - log divergence as the particle separation goes to zero. As a result, the energy dissipated in the collision is finite. In the limit of large particle inertia, the energy dissipated is 6πμU0a2(log h0/λ0 – 1.28), where 2U0 is the relative velocity of the particles.When λ0[Gt ]aε1/2, we have a free molecular flow in the gap. In this case, owing to the curvature of the particles, the flux versus pressure gradient relation is non-local. We analyse the free molecular flow between two cylinders and obtain scalings for the lubrication force.
The sedimentation of a small dense sphere through a suspension of neutrally buoyant fibres is investigated via a numerical simulation technique that includes both fibre–fibre contact forces and long-range hydrodynamic interactions. In situations where the diameter of the sphere is smaller than the length of the fibres, calculations that exclude the effect of contacts between fibres severely underestimate the drag force on the sphere measured in experiments. By including fibre–fibre contacts in our simulations we are to able to account for this discrepancy, and also the strong dependence of the drag on the initial orientation of the fibres. At low and moderate values of nL3, where n is the number of fibres per unit volume and L the fibre length, hydrodynamic interactions are found to be important in moderating the effect of contacts between fibres.An asymptotic solution is presented for the limit when the sphere diameter is much smaller than both the fibre length and inter-fibre spacing, but large compared to the fibre thickness. This is found to be in good agreement with the simulations.Results of calculations on sedimentation through a monolayer of fibres are also presented, as a model of a semi-concentrated suspension. Collisions between fibres are much more frequent, due to the geometric confinement.
This paper discusses the properties of a semidilute suspension of disks, for which nl3≫1 and φ≪1, where n is the number of disks per unit volume, l is their large dimension, and φ is their volume fraction. The effective conductivity of a dispersion of aligned, highly conducting disks is shown to be O[k(nl3)2], where k is the conductivity of the matrix. The extensional viscosity is shown to be O[μ(nl3)2], where μ is the viscosity of the fluid. In addition, similar scaling results are shown to hold for the case of a semidilute suspension of aligned, two-dimensional slabs which are of infinite extent in the direction perpendicular to their plane of cross section. Specifically, for nl2≫1, the effective conductivity and the extensional viscosity are shown to be O[k(nl2)2] and O[μ(nl2)2] respectively, where n is now the number of slabs per unit area and l is the width of the slab. Planar extensional flow simulations of a periodic array of aligned slabs confirm the quadratic scaling for stress in the semidilute regime. The simulations also show the crossover from a linear dependence of the stress on particle concentration in the dilute regime to the quadratic, semidilute scaling.
The electrical conductivity of a quiescent isotropic fibre suspension is measured for a concentration range nL 3 = 0.1-4, where n is the number density of fibres and L is the fibre length. In the dilute concentration regime, for which nL 3 1, our experiments agree with available theoretical predictions.
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