The stress system in a suspension of heavy particles: antisymmetric contribution

Abstract

The nature of the stress in a suspension of equal homogeneous spheres all subject to the same force, such as weight, is considered; inertial effects are neglected. This study builds upon some of the well-known work devoted to this problem by the founder of the Journal of Fluid Mechanics, Professor George K. Batchelor. After developing a general theory, the antisymmetric part of the stress tensor is considered in detail. It is shown that, in addition to a term already found by Batchelor and characterized by an axial vector, the antisymmetric stress contains another term characterized by the curl of a polar vector. As a consequence, a suspension will possess, in addition to an axial vortex viscosity, a polar vortex viscosity. Appendix C presents a calculation of the hindrance function for rotation correct to the first order in the particle volume fraction.

Heterogeneous dispersions as microcontinuum fluids

“…To validate the model, we apply it to the case of viscous droplets and Navier-slip spheres. The results correct and extend several prior results for constitutive equations of rigid spheres (Einstein 1906(Einstein , 1911Prosperetti 2004;Prosperetti et al 2006;Jin et al 2018;Zhou & Prosperetti 2020) and viscous droplets (Taylor 1932;Raja, Subramanian & Koch 2010), and are shown to be consistent with experimental observations. The study is concluded with a discussion of results and future prospects.…”

“…These limitations can be overcome by the phase-averaging method of Zhang & Prosperetti (1994a, 1997, which, in principle, can be used to compute the ensemble average of any field in a suspension of arbitrary composition. However, their methods require the use of high-rank tensors, and hence have largely been limited to dilute suspensions of inviscid spherical droplets (Zhang & Prosperetti 1994b) or rigid, solid spheres (Zhang & Prosperetti 1997;Prosperetti 2004;Prosperetti, Zhang & Ichiki 2006) due to the mathematical complexity involved in separating field averages in the fluid from those in the particles, which are most tractable as moments of the hydrodynamic surface traction.…”

Physics-based analysis of the hydrodynamic stress in a fluid-particle system

“…Both terms vanish in Stokes flow as shown in Ref. 5. The third term gives rise to the vector V defined in Eq.…”

“…This is the procedure followed in our earlier paper. 5 Bardet and Verdoulakis 51 have calculated the stress in a granular medium by applying the principle of virtual work and found it to have an antisymmetric component when the forces on the particles have nonzero moment about their centers. This is similar to our result ͑20͒ and, in fact, their Eq.…”

“…4,5 The main contribution of the present paper consists in its direct physical approach which gives a deeper insight into these matters addressing, among others, the question of how the averaging process results in a nonsymmetric stress in spite of the symmetry prevailing at the microscopic level ͑Secs. VI and X; in particular, the relation with the earlier ensembleaverage derivation is discussed at the end of Sec.…”