Normal stress and diffusion in a dilute suspension of hard spheres undergoing simple shear

The complete set of normal stresses in a dilute suspension of hard spheres undergoing simple shear at low Reynolds number is calculated using a path integration approach for the cases where the concentration is uniform and where a small gradient in concentration is present. As expected, the normal stresses are seen to be a strong function of epsilon (s)=2(b-a)/a, where b is the hard sphere radius and a is the particle radius. The normal stress differences N-1 and N-2, are negative while the osmotic pressure is large and positive, with Pi > parallel toN(2)parallel to and N-1-->0 as epsilon (s)--> infinity. For epsilon (s)much less than1, the asymmetry in the pair distribution function due to a depletion of particles in the extensional side of a pair interaction leads to \N-1\> \N-2\. On the other hand, for epsilon (s)--> infinity, the additional stresslet induced when hard sphere radii touch dominates the stress generated in the suspension, and N-2 becomes the prevailing normal stress difference. The self and gradient diffusivities are calculated using da Cunha and Hinch's [J. Fluid Mech. 309, 211 (1996)] trajectory method. Numerical results for the diffusivities are in agreement with those obtained by da Cunha and Hinch for epsilon (s)less than or equal to0.08 while matching the analytically obtained diffusivities for large epsilon (s). Finally, we calculate the normal stress in the presence of a small concentration gradient and compare two models of migration for this case, namely the suspension balance model of Nott and Brady [J. Fluid Mech. 275, 157 (1994)] and the diffusive flux model first introduced by Leighton and Acrivos [J. Fluid Mech. 181, 415 (1987)]. The results show that although the two models equally describe migration in the presence of a concentration gradient for the case where b much greater thana (or epsilon (s)--> infinity), the two models are shown to be quantitatively different when near-field hydrodynamics are relevant. (C) 2001 American Institute of Physics.