Asymmetric creeping motion of an open torus

This paper presents exact solutions using toroidal co-ordinates to the equations of creeping fluid motion with the no-slip boundary conditions for a toroidal particle translating in a direction normal to the axis of symmetry or rotating about an axis normal to the axis of symmetry through an otherwise infinite expanse of quiescent fluid. The associated resisting force and resisting torque are computed for toroids of various geometrical ratios b/a, b being the smallest radius of the open hole and (b + 2a) being the radius to the outermost rim of the torus. These results are compared with approximate calculations based on slender-body theory and on the theory for interacting beads. The exact and approximate calculations become asymptotically equal as b/a becomes very large, but departures from the exact calculations are apparent for b/a less than 10−100 depending on the mode of motion and the method of approximation and the approximations are unreliable for b/a less than 2·0.

Download Full-text

Secondary fluid flows driven electromagnetically in a two-dimensional extended duct

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1454 ◽

2005 ◽

Vol 461 (2058) ◽

pp. 1659-1683 ◽

Cited By ~ 7

Author(s):

Zhi-Min Chen ◽

W.G Price

Keyword(s):

Steady State ◽

Laboratory Experiments ◽

Fluid Motion ◽

Fluid Flows ◽

Two Dimensional ◽

Slip Boundary Conditions ◽

Slip Boundary ◽

Wave Numbers ◽

Motion Problem ◽

Secondary Fluid

This study focuses on two-dimensional fluid flows in a straight duct with free-slip boundary conditions applied on the channel walls y =0 and y =2 πN with N >1. In this extended wall-bounded fluid motion problem, secondary fluid flow patterns resulting from steady-state and Hopf bifurcations are examined and shown to be dependent on the choice of longitudinal wave numbers. Some secondary steady-state flows appear at specific wave numbers, whereas at other wave numbers, both secondary steady-state and self-oscillation flows coexist. These results, derived through analytical arguments and truncation series approximation, are confirmed by simple numerical experiments supporting the findings observed from laboratory experiments.

Download Full-text

Tip streaming from slender drops in a nonlinear extensional flow

Journal of Fluid Mechanics ◽

10.1017/s0022112084001609 ◽

1984 ◽

Vol 144 ◽

pp. 281-295 ◽

Cited By ~ 38

Author(s):

J. D. Sherwood

Keyword(s):

Steady State ◽

Extensional Flow ◽

Axisymmetric Flow ◽

Rayleigh Instability ◽

Slender Body Theory ◽

Viscous Drops ◽

Roll Mill ◽

Viscous Drop ◽

Axis Of Symmetry ◽

Body Theory

The deformation of inviscid and slightly viscous drops is studied using slender-body theory. The imposed axisymmetric flow is a combination of a linear extensional flow, with velocity uz = G1 z along the axis of symmetry, together with a cubic flow uz = G3z3. When G3/G1 is sufficiently small the viscous drop breaks in a manner similar to that described by Acrivos & Lo (1978). For larger G3 > 0 the drop breaks by a rapid growth at its end. Steady-state experiments in a 4-roll mill show the ejection of a column of liquid from the tip of the drop, though this is probably caused by a change in the pressure gradient rather than the mechanism described above. The ejected column then breaks into droplets via the Rayleigh instability. It is hypothesized that one or other of these mechanisms corresponds to tip streaming as observed by Taylor (1934).

Download Full-text

Regularized Stokeslets Lines Suitable for Slender Bodies in Viscous Flow

Fluids ◽

10.3390/fluids6090335 ◽

2021 ◽

Vol 6 (9) ◽

pp. 335

Author(s):

Boan Zhao ◽

Lyndon Koens

Keyword(s):

Regularization Parameter ◽

Near Field ◽

The Body ◽

Slender Body ◽

Slender Body Theory ◽

Slip Boundary ◽

Slender Bodies ◽

Angular Dependency ◽

Regularized Stokeslets ◽

Body Theory

Slender-body approximations have been successfully used to explain many phenomena in low-Reynolds number fluid mechanics. These approximations typically use a line of singularity solutions to represent flow. These singularities can be difficult to implement numerically because they diverge at their origin. Hence, people have regularized these singularities to overcome this issue. This regularization blurs the force over a small blob and thereby removing divergent behaviour. However, it is unclear how best to regularize the singularities to minimize errors. In this paper, we investigate if a line of regularized Stokeslets can describe the flow around a slender body. This is achieved by comparing the asymptotic behaviour of the flow from the line of regularized Stokeslets with the results from slender-body theory. We find that the flow far from the body can be captured if the regularization parameter is proportional to the radius of the slender body. This is consistent with what is assumed in numerical simulations and provides a choice for the proportionality constant. However, more stringent requirements must be placed on the regularization blob to capture the near field flow outside a slender body. This inability to replicate the local behaviour indicates that many regularizations cannot satisfy the no-slip boundary conditions on the body’s surface to leading order, with one of the most commonly used blobs showing an angular dependency of velocity along any cross section. This problem can be overcome with compactly supported blobs, and we construct one such example blob, which can be effectively used to simulate the flow around a slender body.

Download Full-text