The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number

Abstract

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t-f behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as r2.

Forces on a boiling bubble in a developing boundary layer, in microgravity with g-jitter and in terrestrial conditions

“…Due to the small size of the bubbles, the measuring accuracy of F 2 p + F 2 σ with (21) or with other expressions is not high. Moreover, in (18) or similar expressions for the h-equation, for example, the one employing (22), the estimates of drag and inertia appear to be counteracting although drag is not particularly small at early times (Fig. 15).…”

“…The above drag acts in a direction parallel to the approaching flow. In the present study, drag perpendicular to the wall, connected to W 22 value of 12 π μ R(t) for a sphere in an unbounded fluid. We may therefore postulate that the drag on a growing truncated sphere at a plane wall is approximately represented by an equation of the form…”

“…This is because far from the particle, the advection terms neglected in the derivation of f 1 and f 2 become significant in the transport of the vorticity field. 22 Application is therefore recommended only for Re V ≥ 1. If Re 1 1, Reynolds number Re 1 is the relevant ratio of inertia to viscous effects involved, and drag may be 20 close to the viscous potential drag whatever the value of Re V .…”

“…IV B it stands to reason to consider the overpressure as determined from (16), or from p = -σ 2H , as the most accurate one. The resulting overpressure can be put into the h-equation (17), which also contains p as the only unknown, yielding (22). However, the sum of the dominant forces, F 2 p + F 2 σ , is alternatively assessed with the aid of (18) or (21).…”

Particle migration and sorting in microbubble streaming flows

“…The precise nature of the hydrodynamic forces acting on the particles is potentially very complex: in the past 50þ years, since the pioneering work of Segre and Silberberg, [26][27][28] descriptions of forces on particles in many types of flows have been developed, from simple shear flows 29 to more general background flows, 30 sometimes explicitly taking into account the presence of a nearby wall, 31,32 or the explicit time dependence of the flow. 33,34 Even the behavior of common microparticles in the transport flow through ordinary, ubiquitous microfluidic channels still reveals novel insights today. 35,36 Because the bubbles are driven acoustically in our experiments, one may suspect acoustic radiation forces at work; however, using our typical parameters to evaluate these forces 37 and translating them to particle displacements during passage near the bubble, we find that even our largest (a p ¼ 5 lm) particles would not be displaced perpendicular to streamlines by more than $100 nm if the particle and fluid densities were not matched, and far less under the density-matched conditions of our experiment.…”