Asymptotic behavior of a suspension of oriented particles in a viscous incompressible fluid

Abstract

A viscous incompressible fluid with a large number of small axially symmetric solid particles is considered. It is assumed that the particles are identically oriented and under the influence of the fluid move translationally or rotate around a symmetry axis with the direction of their symmetry axes unchanged. The asymptotic behavior of oscillations of the system is studied, when the diameters of particles and distances between the nearest particles are decreased. The equations, describing the homogenized model of the system, are derived. It is shown that the homogenized equations correspond to a non-standard hydrodynamics. Namely, the homogenized stress tensor linearly depends not only on the strain tensor but also on the rotation tensor.

Non-standard dynamics of elastic composites

“…Analogously to [1], [2], [3], [4] and [6], it may be shown that the family of solutions u ε (x, λ) of the problem (4.1)-(4.5) is analytic in the domain {Reλ > 0}. Moreover, in this domain the following estimate holds:…”

“…Due to the estimates (6.3) and (6.4), we can apply the inverse Laplace transform (see, for example, [14] and [8]) and prove, thereby, the statement of Theorem 2 (see details in [1], [2], [3], [4] and [6]).…”

“…where lim h→0 lim ε→0 δ εh L2(Ω) = 0 (for more details see, for example, [3] and [6]). Then, using (5.7) and (5.11), similarly to [1], [2], [3], [4] and [6] we can show that…”

Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles

“…The influence of particle concentration was also considered in [20,21]. Rigorous asymptotical homogenization strategies for suspensions were used for example in [12,3]. The work by Junk & Illner [11] deals with the strict asymptotic derivation of Jeffery's equation, using expansions in the small size ratio (ratio between the characteristic length scale associated to the particle and the one associated to the fluid).…”