The stability of a rotating liquid drop

1965 ◽  
Vol 286 (1404) ◽  
pp. 1-26 ◽  
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).

Oscillations of a rotating liquid drop

1984 ◽  
Vol 142 ◽  
pp. 1-8 ◽  
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.

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

1987 ◽  
Vol 414 (1846) ◽  
pp. 59-82 ◽  
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).

scholarly journals Further consideration of the stability of the pear-shaped figure of a rotating mass of liquid

1908 ◽  
Vol 80 (537) ◽  
pp. 166-167 ◽  
Keyword(s):  
Rigorous Solution ◽  
Opposite Conclusion ◽  
Rotating Liquid ◽  
Figures Of Equilibrium ◽  
Imperial Academy ◽  
The Stability

In vol. 17, No. 3 (1905), of the ‘Memoirs of the Imperial Academy of St. Petersburg,' M. Liapounoff has published an abstract of his work on figures of equilibrium of rotating liquid under the title “Sur un Problème de Tchebychef.” In this paper he explains how he has obtained a rigorous solution for the figure and stability of the pear-shaped figure, and he pronounces it to be unstable. In my paper in the ‘Philosphical Transactions’ I had arrived at an opposite conclusion. The stability or instability depends, in fact, on whether the sign of a certain function, which M. Liapounoff calls A, is negative or positive.

scholarly journals The Stability of a Rotating Liquid Mass Held Together by Surface Tension

1968 ◽  
Vol 21 (6) ◽  
pp. 837 ◽  
Author(s):  
DK Ross
Keyword(s):  
Surface Tension ◽  
Incompressible Fluid ◽  
Angular Speed ◽  
Linear Series ◽  
Original Series ◽  
Surface Of Revolution ◽  
Liquid Mass ◽  
Rotating Liquid ◽  
The Stability

The stability of a drop of incompressible fluid held together by the action of surface tension and made to rotate rigidly about an axis is determined, the effect of gravity being neglected. Two distinct problems are investigated. In the first is considered an isolated drop in the form of a surface of revolution and the manner in which its stability changes with angular speed is investigated. At zero angular speed, where the drop is spherical, infinitesimal disturbances are shown to be stable and beyond a certain critical angular speed a new linear series of equilibrium forms emerges, the original series becoming unstable.

scholarly journals I. Further consideration of the stability of the pear-shaped figure of a rotating mass of liquid

1908 ◽  
Vol 208 (427-440) ◽  
pp. 1-19
Keyword(s):  
Rigorous Solution ◽  
Opposite Conclusion ◽  
Rotating Liquid ◽  
Figures Of Equilibrium ◽  
Imperial Academy ◽  
The Stability

In vol. 17, No. 3 (1905), of the ‘Memoirs of the Imperial Academy of St. Petersburg’ M. Liapounoff has published an abstract of his work on figures of equilibrium of rotating liquid under the title “Sur un Probleme de Tchebychef.” In this paper he explains how he has obtained a rigorous solution for the figure and stability of the pear-shaped figure, and he pronounces it to be unstable. In my paper in the ‘Philosophical Transactions I had arrived at an opposite conclusion. The stability or instability depends, in fact, on the sign of a certain function which M. Liapounoff calls A, and which I denote A 0 +∑(B s i where A 0 is equal to A 3 [1/3(σ 2 ) 2 - 1/3σ 4 ∑[ i, s ].

scholarly journals A pH-Responsive Foam Formulated with PAA/Gemini 12-2-12 Complexes

2021 ◽  
Vol 5 (3) ◽  
pp. 37
Author(s):  
Hernán Martinelli ◽  
Claudia Domínguez ◽  
Marcos Fernández Leyes ◽  
Sergio Moya ◽  
Hernán Ritacco
Keyword(s):  
Surface Tension ◽  
Dynamic Surface Tension ◽  
Foam Formation ◽  
Ph Responsive ◽  
Dynamic Surface ◽  
X Ray ◽  
Equilibrium Surface Tension ◽  
Potential Applications ◽  
The Stability ◽  
Air Interfaces

In the search for responsive complexes with potential applications in the formulation of smart dispersed systems such as foams, we hypothesized that a pH-responsive system could be formulated with polyacrylic acid (PAA) mixed with a cationic surfactant, Gemini 12-2-12 (G12). We studied PAA-G12 complexes at liquid–air interfaces by equilibrium and dynamic surface tension, surface rheology, and X-ray reflectometry (XRR). We found that complexes adsorb at the interfaces synergistically, lowering the equilibrium surface tension at surfactant concentrations well below the critical micelle concentration (cmc) of the surfactant. We studied the stability of foams formulated with the complexes as a function of pH. The foams respond reversibly to pH changes: at pH 3.5, they are very stable; at pH > 6, the complexes do not form foams at all. The data presented here demonstrate that foam formation and its pH responsiveness are due to interfacial dynamics.

Electrohydrodynamic stability: Taylor–Melcher theory for a liquid bridge suspended in a dielectric gas

2002 ◽  
Vol 452 ◽  
pp. 163-187 ◽  
Author(s):  
C. L. BURCHAM ◽  
D. A. SAVILLE
Keyword(s):  
Surface Tension ◽  
Dielectric Constant ◽  
Electric Fields ◽  
Liquid Bridge ◽  
Numerical Solutions ◽  
Specific Conductivity ◽  
Axisymmetric Deformation ◽  
Aspect Ratios ◽  
The Stability ◽  
Theory And Experiment

A liquid bridge is a column of liquid, pinned at each end. Here we analyse the stability of a bridge pinned between planar electrodes held at different potentials and surrounded by a non-conducting, dielectric gas. In the absence of electric fields, surface tension destabilizes bridges with aspect ratios (length/diameter) greater than π. Here we describe how electrical forces counteract surface tension, using a linearized model. When the liquid is treated as an Ohmic conductor, the specific conductivity level is irrelevant and only the dielectric properties of the bridge and the surrounding gas are involved. Fourier series and a biharmonic, biorthogonal set of Papkovich–Fadle functions are used to formulate an eigenvalue problem. Numerical solutions disclose that the most unstable axisymmetric deformation is antisymmetric with respect to the bridge’s midplane. It is shown that whilst a bridge whose length exceeds its circumference may be unstable, a sufficiently strong axial field provides stability if the dielectric constant of the bridge exceeds that of the surrounding fluid. Conversely, a field destabilizes a bridge whose dielectric constant is lower than that of its surroundings, even when its aspect ratio is less than π. Bridge behaviour is sensitive to the presence of conduction along the surface and much higher fields are required for stability when surface transport is present. The theoretical results are compared with experimental work (Burcham & Saville 2000) that demonstrated how a field stabilizes an otherwise unstable configuration. According to the experiments, the bridge undergoes two asymmetric transitions (cylinder-to-amphora and pinch-off) as the field is reduced. Agreement between theory and experiment for the field strength at the pinch-off transition is excellent, but less so for the change from cylinder to amphora. Using surface conductivity as an adjustable parameter brings theory and experiment into agreement.

THE STABILITY ANALYSIS OF HEAT TRANSFER IN SATURATED LIQUID FILM OF THE SPECIAL MATERIALS

2005 ◽  
Vol 19 (28n29) ◽  
pp. 1547-1550
Author(s):  
YOULIANG CHENG ◽  
XIN LI ◽  
ZHONGYAO FAN ◽  
BOFEN YING
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Surface Tension ◽  
Stability Analysis ◽  
Liquid Film ◽  
Nonlinear Relationship ◽  
Saturated Liquid ◽  
Nonlinear Surface ◽  
Isothermal Wall ◽  
The Stability

Representing surface tension by nonlinear relationship on temperature, the boundary value problem of linear stability differential equation on small perturbation is derived. Under the condition of the isothermal wall the effects of nonlinear surface tension on stability of heat transfer in saturated liquid film of different liquid low boiling point gases are investigated as wall temperature is varied.

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