Self-diffusion in sheared suspensions

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either raacroscopically quiescent or in linear flow at arbitrary Peclet number Pe = ẏa^2/2D, where ẏ is the shear rate, a is the particle radius, and D = k_BT/6πηa is the diffusion coefficient of an isolated particle. Here, k_B is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by k_BT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction ø = 4/3πa^3n and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, D_∞^s, is given by the sum of D_0^s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D_0^s and D_∞^s are anisotropic, in general, with the anisotropy of D_0^s due solely to that of the steady microstructure. The influence of flow upon D_∞^s is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations. The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(φPe^(3/2)) for Pe « 1 and ø « 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D_0^s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with ẏ. Regardless of whether or not the particles interact hydrodynamically, flow influences D_∞^s at O(øPe) and O(øPe^(3/2)). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of O(øPe^(3/2)), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe^(3/2)DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor. At high ø a scaling theory based on the approach of Brady (1994) is used to approximate D_∞^s. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of Pe = ẏa^2/2D^s_0(φ)$. At small Pe the dependence on Pe is the same as at low ø

Microstructure of strongly sheared suspensions and its impact on rheology and diffusion

The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe[rightward arrow][infty infinity] under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe=[gamma][ogonek]a2/2D is the Péclet number with [gamma][ogonek] the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe[minus sign]1) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe0.78) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a3Pe[minus sign]1) layer volume, vanish as Pe[minus sign]0.22 as Pe[rightward arrow][infty infinity]

The pressure moments for two rigid spheres in low-Reynolds-number flow

The pressure moment of a rigid particle is defined to be the trace of the first moment of the surface stress acting on the particle. A Faxén law for the pressure moment of one spherical particle in a general low-Reynolds-number flow is found in terms of the ambient pressure, and the pressure moments of two rigid spheres immersed in a linear ambient flow are calculated using multipole expansions and lubrication theory. The results are expressed in terms of resistance functions, following the practice established in other interaction studies. The osmotic pressure in a dilute colloidal suspension at small Péclet number is then calculated, to second order in particle volume fraction, using these resistance functions. In a second application of the pressure moment, the suspension or particle-phase pressure, used in two-phase flow modeling, is calculated using Stokesian dynamics and results for the suspension pressure for a sheared cubic lattice are reported

Collective diffusion in sheared colloidal suspensions

Collective diffusivity in a suspension of rigid particles in steady linear viscous flows is evaluated by investigating the dynamics of the time correlation of long-wavelength density fluctuations. In the absence of hydrodynamic interactions between suspended particles in a dilute suspension of identical hard spheres, closed-form asymptotic expressions for the collective diffusivity are derived in the limits of low and high Péclet numbers, where the Péclet number Pe = gamma-dot a^2/D0 with gamma-dot being the shear rate and D0 = kB T/6πη a is the Stokes–Einstein diffusion coefficient of an isolated sphere of radius a in a fluid of viscosity η. The effect of hydrodynamic interactions is studied in the analytically tractable case of weakly sheared (Pe « 1) suspensions. For strongly sheared suspensions, i.e. at high Pe, in the absence of hydrodynamics the collective diffusivity Dc = 6 Ds∞, where Ds∞ is the long-time self-diffusivity and both scale as φ gamma-dot a^2$, where φ is the particle volume fraction. For weakly sheared suspensions it is shown that the leading dependence of collective diffusivity on the imposed flow is proportional to D0 φPe Ê, where Ê is the rate-of-strain tensor scaled by gamma-dot, regardless of whether particles interact hydrodynamically. When hydrodynamic interactions are considered, however, correlations of hydrodynamic velocity fluctuations yield a weakly singular logarithmic dependence of the cross-gradient-diffusivity on k at leading order as ak → 0 with k being the wavenumber of the density fluctuation. The diagonal components of the collective diffusivity tensor, both with and without hydrodynamic interactions, are of O(φPe2), quadratic in the imposed flow, and finite at k = 0. At moderate particle volume fractions, 0.10 ≤ φ ≤ 0.35, Brownian Dynamics (BD) numerical simulations in which there are no hydrodynamic interactions are performed and the transverse collective diffusivity in simple shear flow is determined via time evolution of the dynamic structure factor. The BD simulation results compare well with the derived asymptotic estimates. A comparison of the high-Pe BD simulation results with available experimental data on collective diffusivity in non-Brownian sheared suspensions shows a good qualitative agreement, though hydrodynamic interactions prove to be important at moderate concentrations