Constant Surface Tension Research Articles

Stability of Axisymmetric Figures of Equilibrium of a Rotating Charged Liquid Drop

The method of the tensor virial is used to investigate the equilibrium and stability of a rotating, uniformly charged (with a total charge Q), incompressible liquid drop (of volume V) held together by a constant surface tension T. The tensor virial theorems provide relations which define the sequences of equilibrium figures and perturbation equations which govern the oscillations. Since spheroids satisfy the virial theorems of orders one, two, and three, the stability of exact axisymmetric equilibrium figures with respect to second- and third-harmonic deformations can be inferred from the nature of the characteristic oscillation frequencies of spheroids. As the rotation increases for a given fission-ability parameter x = Q2/10TV (0 ≤ x < 1), a sequence of oblate spheroids representing initially stable rotating ground states exhibits a neutral point (where an ellipsoidal sequence bifurcates), but remains stable (in the absence of dissipation) until one of the three second-harmonic modes of vibration (the toroidal or γ mode) becomes overstable. Later, for third harmonics, it exhibits a second neutral point (where a sequence of asymmetric figures bifurcates), but again remains stable until the onset of the associated overstability. A third neutral point does not indicate the bifurcation of pear shapes. The error in using oblate spheroids is less than 5% up to the first overstable point. One class of saddle shapes can be represented by prolate spheroids (which must rotate about the symmetry axis) in the neighborhood of x = 1. They are initially unstable with respect to the other two second-harmonic modes of vibration (the pulsation or β mode and the transverse-shear mode).

Surface-Energy Tensors

In this paper the surface-energy tensors are defined and general expressions for their first variations are obtained. These quantities are appropriate for an investigation of the equilibrium and the stability of a charged liquid drop (held together by a constant surface tension) using the method of the tensor virial, which was developed by Chandrasekhar and Lebovitz for the theory of self-gravitating masses. The definitions are shown to be consistent with the conservation laws.

Particle motions in sheared suspensions XI. Internal circulation in fluid droplets (experimental)

The circulation pattern inside fluid drops suspended in a liquid subjected to hyperbolic and to shear flow has been studied and compared with that predicted from a fluid mechanical theory which assumes that the drops remain spherical and that shear stresses are transmitted without diminution across the drop interface. Observations of liquid drops and air bubbles showed that the streamline pattern predicted for hyperbolic flow was established. The intensity of circulation in the drops was found to be decreased by impurities and added surface-active agents. Details of streamlines inside and outside drops and periods of circulation have been determined for a variety of liquid pairs in laminar shear flow. Impurities and surface-active agents exercise a pronounced inhibitory effect and often completely prevent internal circulation. In several vigorously purified systems the extent of internal circulation was less than predicted, but one liquid pair was found in which the theory was followed in all measurable details. It is shown theoretically that a viscoelastic film at the interface can reduce the shear stresses transmitted inside the drop and thus attenuate internal circulation. Calculations show that films due to surface-active materials of representative surface compressibilities are capable of completely suppressing internal circulation under the experimental conditions employed. Whereas the theory based on a nonviscous interface of zero elasticity, i.e., of constant surface tension, held for one vigorously purified system, disagreement with the theory in several others may have been due to interfacial viscosity.

An analysis of the shapes of a cell during division with particular reference to the role of surface tension

An expression is given for the contribution to the relative elongation by any volume element or its surface. The case in which only a constant surface tension is acting is considered in greater detail. From measurements of an egg in several stages, the empirical variation of an expression proportional to the above function is obtained. In the constricting region and near the ends of the cells, the change of shape is opposite to that which would be due to a constant surface tension alone. The effect of streaming which may arise from a variable surface tension is considered. The effect of forces which arise in this manner may be sufficient to explain the discrepancy in the constricting region if the streaming is considered due largely to such a variable surface tension.

Elongation and constriction in cell division

The deformation of a constant reaction ellipsoidal cell by diffusion and constant surface tension forces is studied. The critical size of a spherical cell at which it becomes unstable to ellipsoidal deformations is found to be the same as that obtained previously by N. Rashevsky from energy considerations. It is shown that such a cell once unstable will elongate to a finite amount, and that it will tend to constrict in the center and round up at the poles.

On the mechanics of viscous bodies and elongation in ellipsoidal cells

After deriving some auxiliary equations for the average elongation of a viscous body under the action of forces derived from a potential, the diffusion problem for an ellipsoidal cell with a constant rate of reaction is solved for the case of an infinite permeability. The equation of elongation of such a cell under the influence of diffusion forces is derived, and compared with the, approximate expression obtained by N. Rashevsky for any kind of oblong cell. The two equations are in fair agreement. Effects of constant and variable surface tension are studied.