Taylor dispersion of orientable Brownian particles in unbounded homogeneous shear flows

Abstract

The physical- and orientation-space transport of non-spherical , generally non-neutrally buoyant, Brownian particles in unbounded homogeneous shear flows is analysed with the goal of studying the respective effects of the orientational degrees of freedom of such particles upon their sedimentation and dispersion rates. In particular, the present contribution concentrates on the interaction between the Taylor dispersion mechanism (arising from coupling between the orientational dependence of the particle's translational velocity and the stochastic sampling of the orientation space via rotary Brownian diffusion) and the shear velocity field. Making use of a recent extension of generalized Taylor dispersion theory to homogeneous (unbounded) shear flows, the mean transport process in physical space is modelled by a convection–diffusion problem characterized by a pair of constant phenomenological coefficients, provided that the eigenvalues of the (constant) undisturbed velocity gradient are purely imaginary. The latter phenomenological coefficients – namely, U * , the average ‘slip velocity’ vector (of the particles relative to the ambient fluid), and D * , the dispersivity dyadic or, equivalently, the pair of dyadics $\bar{\bm M}$ and D C (or $\bar{\bm D}^c$ ), the average mobility and the Taylor (or modified Taylor) dispersivity, respectively – are evaluated both asymptotically (in the respective limits of small and large rotary Peclet numbers) as well as numerically (for arbitrary Peclet numbers). It is established that (up to a scalar multiplication factor, independent of Peclet number) the anisotropic portion of the average mobility is formally equivalent to the direct diffusive contribution to the particle stress in the context of suspension rheology. The analysis focuses mainly on the case of simple shear flow. The approximate calculation in the limit of large Peclet numbers, Pe [Gt ] 1, which makes extensive use of the ‘natural coordinates’ along Jeffery orbits previously introduced by Leal & Hinch, verifies that, if the external force is non-orthogonal to the direction along which (undisturbed) fluid velocity variations occur, two of the eigenvalues of D c are proportional to Pe ; moreover, one of these O ( Pe ) eigenvalues is negative . When the external force is parallel to the latter direction, the negative eigenvalue corresponds to the principal direction of contraction in the shear velocity field; this thus relates the non-positive nature of D C to the interaction between the Taylor dispersion mechanism and the (deterministic) convection within the shear field. Explicit results for the variation of the dyadics $\bar{\bm M}$ , D c and $\bar{\bm D}^{c}$ jointly with the respective magnitude of the shear rate and the deviation of the particle geometry from a spherical shape are presented for spheroidal particles. Among other things, it is demonstrated that the proposed definition of the modified Taylor dispersivity coefficient, $\bar{\bm D}^{c}$ , does indeed yield a non-negative dyadic.