Effect of fluid inertia on the dynamics and scaling of neutrally buoyant particles in shear flow

Abstract

The basic dynamics of a prolate spheroidal particle suspended in shear flow is studied using lattice Boltzmann simulations. The spheroid motion is determined by the particle Reynolds number (${\mathit{Re}}_{p} $) and Stokes number ($\mathit{St}$), estimating the effects of fluid and particle inertia, respectively, compared with viscous forces on the particle. The particle Reynolds number is defined by ${\mathit{Re}}_{p} = 4G{a}^{2} / \nu $, where $G$ is the shear rate, $a$ is the length of the spheroid major semi-axis and $\nu $ is the kinematic viscosity. The Stokes number is defined as $\mathit{St}= \alpha \boldsymbol{\cdot} {\mathit{Re}}_{p} $, where $\alpha $ is the solid-to-fluid density ratio. Here, a neutrally buoyant prolate spheroidal particle ($\mathit{St}= {\mathit{Re}}_{p} $) of aspect ratio (major axis/minor axis) ${r}_{p} = 4$ is considered. The long-term rotational motion for different initial orientations and ${\mathit{Re}}_{p} $ is explained by the dominant inertial effect on the particle. The transitions between rotational states are subsequently studied in detail in terms of nonlinear dynamics. Fluid inertia is seen to cause several bifurcations typical for a nonlinear system with odd symmetry around a double zero eigenvalue. Particle inertia gives rise to centrifugal forces which drives the particle to rotate with the symmetry axis in the flow-gradient plane (tumbling). At high ${\mathit{Re}}_{p} $, the motion is constrained to this planar motion regardless of initial orientation. At a certain critical Reynolds number, ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $, a motionless (steady) state is created through an infinite-period saddle-node bifurcation and consequently the tumbling period near the transition is scaled as $\vert {\mathit{Re}}_{p} - {\mathit{Re}}_{c} {\vert }^{- 1/ 2} $. Analyses in this paper show that if a transition from tumbling to steady state occurs at ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $, then any parameter $\beta $ (e.g. confinement or particle spacing) that influences the value of ${\mathit{Re}}_{c} $, such that ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $ as $\beta = {\beta }_{c} $, will lead to a period that scales as $\vert \beta - {\beta }_{c} {\vert }^{- 1/ 2} $ and is independent of particle shape or any geometric aspect ratio in the flow.

Rotation of a spheroid in a simple shear at small Reynolds number

“…This explains why stable log-rolling is not observed in DNS [3][4][5][6] at the smallest shear Reynolds numbers accessible in the simulations. Moreover, we find that tumbling in the flow-shear plane is stable for prolate particles.…”

“…The question is currently of great interest: several recent papers have reported results of direct numerical simulations (DNSs) of the problem, using "lattice Boltzmann" methods. [3][4][5][6] These studies reveal that fluid and particle inertias affect the orientational dynamics of a neutrally buoyant spheroid in a simple shear in intricate ways. The DNSs are performed at moderate and large shear Reynolds numbers, defined as Re s = sa 2 /ν, where a is the largest particle dimension, s is the shear strength, and ν the kinematic viscosity of the suspending fluid.…”

Measurements of the coupling between the tumbling of rods and the velocity gradient tensor in turbulence

“…Leal (1980) provides a review of the older literature, and a wide range of work has followed, for example: Koch & Shaqfeh (1989); Szeri & Leal (1993); Herzhaft et al (1996); Olson & Kerekes (1998); Parsa et al (2011); Rosen et al (2014); Andersson & Soldati (2013). Turbulent flows advecting anisotropic particles provide a compelling test case, both because of the many applications and because of the nearly universal statistics of the velocity gradients experienced by small particles in many turbulent flows at large Reynolds number.…”

Motion of a spherical capsule in branched tube flow with finite inertia

“…The effect of inertia on path selection of a capsule at a bifurcation remains unknown. In other systems, for example non-spherical particles in shear flows, it has been shown that inertial effect could fundamentally change the dynamics of particles even at low Reynolds number (Rosén, Lundell & Aidun 2014;Dabade, Marath & Subramanian 2016). Furthermore, when a capsule approaches the bifurcation, it can sustain high shear stresses, which may damage the capsule membrane.…”