The Effects of Viscosity on the Deformation Behavior of Rotating Liquid Drop

Previous Space-based experiments (Wang et al. 1994a) showed that a rotating liquid drop bifurcates into a two-lobed shape at a lower critical angular velocity, if it is flattened acoustically by the leviating sound field. In this work, we undertake a systematic experimental study of the effect of acoustic flattening on the rotational bifurcation of a liquid drop. We also look into the complementary effect of rotation on the equilibrium of an acoustically drastically flattened drop. Theoretical models are developed for each of the two effects and then woven into a unified picture. The first effect concerns neutral equilibrium, while the second concerns loss of equilibrium, neither of them involving instability. The theories agree well with the experiments.

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Experimental study of the oscillations of a rotating drop

Journal of Fluid Mechanics ◽

10.1017/s002211208500266x ◽

1985 ◽

Vol 158 ◽

pp. 317-327 ◽

Cited By ~ 20

Author(s):

P. Annamalai ◽

E. Trinh ◽

T. G. Wang

Keyword(s):

Experimental Study ◽

Analytical Study ◽

Liquid Drop ◽

Angular Speed ◽

Immiscible Fluid ◽

Drop Shape ◽

Free Decay ◽

Resonance Frequencies ◽

Rotating Liquid ◽

Relative Change

Two- and three-lobed oscillations of a rotating liquid drop immersed in an immiscible fluid of comparable density and the same angular velocity were studied experimentally. Using acoustically suspended drops, it has been found that the relative change in the resonance frequencies of the axisymmetric drop-shape oscillations Δωl/ωl(0) is proportional to the square of the normalized angular speed (Ω/ωl(0))2 when ωl > 2Ω. This is in agreement with a recent analytical study of the same problem. Some preliminary results regarding the effect of rotation on the free-decay rate of the two-lobed oscillations are also presented.

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Oscillations of a rotating liquid drop

Journal of Fluid Mechanics ◽

10.1017/s0022112084000963 ◽

1984 ◽

Vol 142 ◽

pp. 1-8 ◽

Cited By ~ 69

Author(s):

F. H. Busse

Keyword(s):

Surface Tension ◽

Circular Polarization ◽

Coriolis Force ◽

Liquid Drop ◽

Centrifugal Distortion ◽

Axis Of Rotation ◽

First Approximation ◽

Rotating Liquid ◽

Surrounding Fluid

The effect of rotation on the frequencies of oscillations of a liquid drop is investigated. It is assumed that the drop is imbedded in a fluid of the same or different density and that a constant surface tension acts on the interface. Rotation influences the oscillations through the Coriolis force and through the centrifugal distortion of the drop. For non-axisymmetric oscillations only the Coriolis force is important in first approximation and causes the expected splitting of the frequency for the two modes differing in their sign of circular polarization with respect to the axis of rotation. In the case of axisymmetric oscillations the centrifugal distortion and the Coriolis force combine to increase the frequency whenever the density ρi of the drop exceeds the density of ρ° of the surrounding fluid. For ρi < ρ° a decrease of the frequency of oscillation is possible for some modes of higher degree.

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On the shape of a rotating liquid drop in an electric field

Acta Mechanica ◽

10.1007/bf01291092 ◽

1967 ◽

Vol 4 (1) ◽

pp. 107-114 ◽

Cited By ~ 3

Author(s):

Lui M. Habip ◽

Julius Siekmann ◽

Shih-Chih Chang

Keyword(s):

Electric Field ◽

Liquid Drop ◽

Rotating Liquid

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Stability of a rotating liquid drop immersed in a corotating fluid with different density

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1987.0133 ◽

1987 ◽

Vol 414 (1846) ◽

pp. 59-82 ◽

Cited By ~ 1

Keyword(s):

Integral Method ◽

Liquid Drop ◽

Second Harmonic ◽

Equations Of Motion ◽

Prolate Spheroid ◽

Dynamical Instability ◽

Angular Momenta ◽

Rotating Liquid ◽

The Stability ◽

Spheroidal Coordinates

The stability of a uniformly rotating, incompressible drop with density ρ i and immersed in a corotating fluid with different density ρ e is investigated. The equilibrium figure is approximated by an oblate or prolate spheroid. The linearized equations of motion are solved by means of ‘modified’ spheroidal coordinates. A dispersion relation is derived with the aid of the energy integral method. The curves of overstability are calculated for the second harmonic modes of oscillation and for the mode n = m = 3. The points of bifurcation for the n = m modes are independent of the presence of the external fluid. It appears, however, that dynamical instability may initiate before the point of bifurcation is attained. This occurs when z = ρ e /ρ i > 0.192 for the n = m = 2 mode and when z > 0.207 for the n = m = 3 mode. In several cases we observed a bending back of the curve of overstability in the ( z, e ) or ( z, h ) plane, where e and h are the oblate and prolate eccentricities, respectively. This indicates stability for low or high eccentricities (or angular momenta) and instability for intermediate eccentricities (or angular momenta).

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Normal oscillations of a rotating liquid drop

Fluid Dynamics ◽

10.1007/bf01051253 ◽

1980 ◽

Vol 14 (4) ◽

pp. 535-543

Author(s):

N. K. Radyakin

Keyword(s):

Liquid Drop ◽

Rotating Liquid

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The stability of a rotating liquid drop

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1965.0127 ◽

1965 ◽

Vol 286 (1404) ◽

pp. 1-26 ◽

Cited By ~ 97

Keyword(s):

Surface Tension ◽

Liquid Drop ◽

Remarkable Similarity ◽

Interfacial Surface ◽

Rotating Liquid ◽

Figures Of Equilibrium ◽

Dissipative Mechanism ◽

Parameter Sequence ◽

Neutral Mode ◽

The Stability

In this paper, the stability of a rotating drop held together by surface tension is investigated by an appropriate extension of the method of the tensor virial. Consideration is restricted to axisymmetric figures of equilibrium which enclose the origin. These figures form a one parameter sequence; and a convenient parameter for distinguishing the members of the sequence is Σ = ρΩ 2 a 3 /8 T , where Ω is the angular velocity of rotation, a is the equatorial radius of the drop, ρ is its density, and T is the interfacial surface tension. It is shown that Σ ⩽ 2.32911 ( not 1 + √2 as is sometimes supposed) if the drop is to enclose the origin. It is further shown that with respect to stability, the axisym metric sequence of rotating drops bears a remarkable similarity to the Maclaurin sequence of rotating liquid masses held together by their own gravitation. Thus, at a point along the sequence (where Σ = 0.4587) a neutral mode of oscillation occurs without in stability setting in at that point (i.e. provided no dissipative mechanism is present); and the in stability actually sets in at a subsequent point (where Σ = 0.8440) by overstable oscillations with a frequency Ω. The dependence on Σ of the six characteristic frequencies, belonging to the second harmonics, is determined (tables 3 and 4) and exhibited (figures 3 and 4).

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