Tip streaming from slender drops in a nonlinear extensional flow

1984 ◽  
Vol 144 ◽  
pp. 281-295 ◽  
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).

A numerical study of the deformation and burst of a viscous drop in an extensional flow

1978 ◽  
Vol 89 (1) ◽  
pp. 191-200 ◽  
Author(s):  
J. M. Rallison ◽  
A. Acrivos
Keyword(s):  
Numerical Study ◽  
Extensional Flow ◽  
Small Deformation ◽  
Surface Velocity ◽  
Deformation Analysis ◽  
Slender Body Theory ◽  
Spherical Drop ◽  
Viscous Drop ◽  
Freely Suspended ◽  
Body Theory

We study the deformation and conditions for breakup of a liquid drop of viscosity λμ freely suspended in another liquid of viscosity μ with which it is immiscible and which is being sheared. The problem at zero Reynolds number is formulated exactly as an integral equation for the unknown surface velocity, which is shown to reduce to a particularly simple form when Δ = 1. This equation is then solved numerically, for the case in which the impressed shear is a radially symmetric extensional flow, by an improved version of the technique used, for Δ = 0, by Youngren & Acrivos (1976) so that we model the time-dependent distortion of an initially spherical drop. It is shown that, for a given Δ, a steady shape is attained only if the dimensionless group Ω ≡4πGμa/γ lies below a critical value Ωc(Δ), where G refers to the strength of the shear field, a is the radius of the initial spherical drop and γ is the interfacial tension. On the other hand, when Ω > Ωc the drop extends indefinitely along its long axis. The numerical results for Δ = 0·3, 0·5, 1, 2, 10 and 100 are in good agreement with the predictions of the small deformation analysis by Taylor (1932) and Barthès-Biesel & Acrivos (1973) and, at the smaller Δ, with those of slender-body theory (Taylor 1964; Acrivos & Lo 1978).

Asymmetric creeping motion of an open torus

1980 ◽  
Vol 101 (1) ◽  
pp. 97-110 ◽  
Author(s):  
Simon L. Goren ◽  
Michael E. O'Neill
Keyword(s):  
Fluid Motion ◽  
Slip Boundary Conditions ◽  
Slender Body Theory ◽  
Slip Boundary ◽  
Open Hole ◽  
Axis Of Symmetry ◽  
Exact Calculations ◽  
Body Theory ◽  
Creeping Motion ◽  
Quiescent Fluid

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.

The Bursting of Pointed Drops in Slow Viscous Flow

1973 ◽  
Vol 40 (1) ◽  
pp. 18-24 ◽  
Author(s):  
J. Buckmaster
Keyword(s):  
Surface Tension ◽  
Viscous Fluid ◽  
Viscous Flow ◽  
Small Drop ◽  
Shape Preserving ◽  
Slender Body Theory ◽  
Slow Viscous Flow ◽  
Viscous Drops ◽  
Steady Solutions ◽  
Body Theory

Viscous drops, confined by the slow axisymmetric straining motion of a viscous fluid, are considered when the surface tension is weak. The shape of the drops is determined using slender-body theory, and it is found that steady solutions only exist for sufficiently small drop viscosities. Nonuniqueness exists, with bifurcation from a simple quadratic solution. At high drop viscosities, when there are no steady solutions, a description of the unsteady elongation of shape-preserving drops is obtained. This is the bursting phenomenon described experimentally by Taylor [1].

A higher-order slender-body theory for axisymmetric flow past a particle at moderate Reynolds number

2018 ◽  
Vol 855 ◽  
pp. 421-444 ◽  
Author(s):  
Aditya S. Khair ◽  
Nicholas G. Chisholm
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Slender Body ◽  
Reciprocal Theorem ◽  
Hydrodynamic Drag ◽  
Slender Body Theory ◽  
Moderate Reynolds Number ◽  
Body Theory ◽  
Uniform Stream

Slender-body theory is utilized to derive an asymptotic approximation to the hydrodynamic drag on an axisymmetric particle that is held fixed in an otherwise uniform stream of an incompressible Newtonian fluid at moderate Reynolds number. The Reynolds number, $Re$ , is based on the length of the particle. The axis of rotational symmetry of the particle is collinear with the uniform stream. The drag is expressed as a series in powers of $1/\text{ln}(1/\unicode[STIX]{x1D716})$ , where $\unicode[STIX]{x1D716}$ is the small ratio of the characteristic width to length of the particle; the series is asymptotic for $Re\ll O(1/\unicode[STIX]{x1D716})$ . The drag is calculated through terms of $O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$ , thereby extending the work of Khayat & Cox (J. Fluid Mech., vol. 209, 1989, pp. 435–462) who determined the drag through $O[1/\text{ln}^{2}(1/\unicode[STIX]{x1D716})]$ . The calculation of the $O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$ term is accomplished via the generalized reciprocal theorem (Lovalenti & Brady, J. Fluid Mech., vol. 256, 1993, pp. 561–605). The first dependence of the inertial contribution to the drag on the cross-sectional profile of the particle is at $O[1/\text{ln}^{3}(1/\unicode[STIX]{x1D716})]$ . Notably, the drag is insensitive to the direction of travel at this order. The asymptotic results are compared to a numerical solution of the Navier–Stokes equations for the case of a prolate spheroid. Good agreement between the two is observed at moderately small values of $\unicode[STIX]{x1D716}$ , which is surprising given the logarithmic error associated with the asymptotic expansion.

Note on the swimming of slender fish

1960 ◽  
Vol 9 (2) ◽  
pp. 305-317 ◽  
Author(s):  
M. J. Lighthill
Keyword(s):  
Swimming Speed ◽  
Critical Value ◽  
Slender Body ◽  
Vortex Wake ◽  
Frictional Drag ◽  
Slender Body Theory ◽  
Propulsive Efficiency ◽  
Deformable Bodies ◽  
Work Done ◽  
Body Theory

The paper seeks to determine what transverse oscillatory movements a slender fish can make which will give it a high Froude propulsive efficiency, $\frac{\hbox{(forward velocity)} \times \hbox{(thrust available to overcome frictional drag)}} {\hbox {(work done to produce both thrust and vortex wake)}}.$ The recommended procedure is for the fish to pass a wave down its body at a speed of around $\frac {5} {4}$ of the desired swimming speed, the amplitude increasing from zero over the front portion to a maximum at the tail, whose span should exceed a certain critical value, and the waveform including both a positive and a negative phase so that angular recoil is minimized. The Appendix gives a review of slender-body theory for deformable bodies.

Slender-body theory for slow viscous flow

1976 ◽  
Vol 75 (4) ◽  
pp. 705-714 ◽  
Author(s):  
Joseph B. Keller ◽  
Sol I. Rubinow
Keyword(s):  
Integral Equation ◽  
Incompressible Fluid ◽  
Viscous Incompressible Fluid ◽  
Unit Length ◽  
The Body ◽  
Slender Body ◽  
Slow Flow ◽  
Slender Body Theory ◽  
Slow Viscous Flow ◽  
Body Theory

Slow flow of a viscous incompressible fluid past a slender body of circular crosssection is treated by the method of matched asymptotic expansions. The main result is an integral equation for the force per unit length exerted on the body by the fluid. The novelty is that the body is permitted to twist and dilate in addition to undergoing the translating, bending and stretching, which have been considered by others. The method of derivation is relatively simple, and the resulting integral equation does not involve the limiting processes which occur in the previous work.

A numerical study of the sedimentation of fibre suspensions

1998 ◽  
Vol 376 ◽  
pp. 149-182 ◽  
Author(s):  
MICHAEL B. MACKAPLOW ◽  
ERIC S. G. SHAQFEH
Keyword(s):  
Monte Carlo ◽  
Steady State ◽  
Monte Carlo Simulations ◽  
Numerical Study ◽  
Point Particle ◽  
Hydrodynamic Interactions ◽  
Dynamic Simulations ◽  
Slender Body Theory ◽  
Fibre Suspensions ◽  
The Mean

The sedimentation of fibre suspensions at low Reynolds number is studied using two different, but complementary, numerical simulation methods: (1) Monte Carlo simulations, which consider interparticle hydrodynamic interactions at all orders within the slender-body theory approximation (Mackaplow & Shaqfeh 1996), and (ii) dynamic simulations, which consider point–particle interactions and are accurate for suspension concentrations of nl3=1, where n and l are the number density and characteristic half-length of the fibres, respectively. For homogeneous, isotropic suspensions, the Monte Carlo simulations show that the hindrance of the mean sedimentation speed is linear in particle concentration up to at least nl3=7. The speed is well predicted by a new dilute theory that includes the effect of two-body interactions. Our dynamic simulations of dilute suspensions, however, show that interfibre hydrodynamic interactions cause the spatial and orientational distributions to become inhomogeneous and anisotropic. Most of the fibres migrate into narrow streamers aligned in the direction of gravity. This drives a downward convective flow within the streamers which serves to increase the mean fibre sedimentation speed. A steady-state orientation distribution develops which strongly favours fibre alignment with gravity. Although the distribution reaches a steady state, individual fibres continue to rotate in a manner that can be qualitatively described as a flipping between the two orientations aligned with gravity. The simulation results are in good agreement with published experimental data.

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