Sedimentation of an ellipsoid inside an infinitely long tube at low and intermediate Reynolds numbers

Abstract

The motion of a heavy rigid ellipsoidal particle settling in an infinitely long circular tube filled with an incompressible Newtonian fluid has been studied numerically for three categories of problems, namely, when both fluid and particle inertia are negligible, when fluid inertia is negligible but particle inertia is present, and when both fluid and particle inertia are present. The governing equations for both the fluid and the solid particle have been solved using an arbitrary Lagrangian-Eulerian based finite-element method. Under Stokes flow conditions, an ellipsoid without inertia is observed to follow a perfectly periodic orbit in which the particle rotates and moves from side to side in the tube as it settles. The amplitude and the period of this oscillatory motion depend on the initial orientation and the aspect ratio of the ellipsoid. An ellipsoid with inertia is found to follow initially a similar oscillatory motion with increasing amplitude. Its orientation tends towards a flatter configuration, and the rate of change of its orientation is found to be a function of the particle Stokes number which characterizes the particle inertia. The ellipsoid eventually collides with the tube wall, and settles into a different periodic orbit. For cases with non-zero Reynolds numbers, an ellipsoid is seen to attain a steady-state configuration wherein it falls vertically. The location and configuration of this steady equilibrium varies with the Reynolds number.

Nanoparticle Brownian motion and hydrodynamic interactions in the presence of flow fields

“…For this purpose, a weak formulation that incorporates both the fluid and particle equations (1), (2), (6), and (7) is considered. 22,23,27,29 Let, V be the function space given by…”

“…This feature also keeps the overall mesh size computationally reasonable even with a nanoparticle moving in a very large domain. 22,23,[27][28][29] Thermal fluctuations are included in the equations of linearized hydrodynamics by adding stochastic components to the stress tensor as white noise in space and time. 4,30 As noted in Español et al, 30 "even though the original equations of fluctuating hydrodynamics are written in terms of stochastic partial differential equations, at a very fundamental level, the inclusion of thermal fluctuations requires always the notion of a "mesoscopic cell" in order to define the fluctuating quantities."…”

Respiratory Deposition of Fibers in the Non-Inertial Regime—Development and Application of a Semi-Analytical Model

“…As a result, the Euler angles inevitably bring problems with singularities in the evaluation of their time rate of change, which makes them inconvenient for rigid body simulations of fibers undergoing full rotations. An alternative is to calculate the fiber rotation using quaternions (Fan and Ahmadi 2000;Zhang, et al 2001;Swaminathan et al 2006), which are algebraic constructs composed of four parameters describing evolvement of fiber orientation utilizing a unit vector for the axis of rotation and an angle of rotation around that axis. In addition to singularity-free equations of motion, the use of quaternions offers easy control of the numerical drift and may lead to reduced computational costs since only arithmetic operations are involved.…”

Generalized Langevin dynamics of a nanoparticle using a finite element approach: Thermostating with correlated noise

“…Even though the Brownian motion of a nanoparticle using fluctuating hydrodynamics has been extensively studied using the LBM technique, 6,7 the advantage of the FEM approach is that it is versatile for flow description through arbitrary geometries. 13 In this paper, we pursue the Langevin approach, in which the thermal fluctuations from the fluid are incorporated as random forces and torques in the particle equation of motion. [14][15][16][17][18][19] The primary objective here is to build a robust thermostat, which preserves equilibrium distributions at constant temperatures (i.e., adheres to the equipartition theorem) and enables the evaluation of free energy landscapes of nanoparticle adhesion with surfaces in future applications.…”