Chapter V: Shape Analysis and Energy Stability of Conductive Liquids
5.1 Energy Stability of Electrically Charged Conductive Liquid
In 1882, Lord Rayleigh (1882) p resented a concise derivation on the theoretical estimate of the maximum amount of charge a nearly spherical liquid drop could carry while still maintaining stable, which is now known as the Rayleigh charge limit
QRa= 8πqσε0R3c. (5.1)
Here Rc is the characteristic radius, σ is the surface tension coefficient and ε0 is the vacuum permittivity. Rayleigh’s prediction that a droplet reaching this limit would be- come unstable has been widely quoted since then and confirmed by careful experimental studies for many liquids and droplet sizes (Doyle, Moffett, and Vonnegut,1964; Taflin, Ward, and E. J. Davis, 1989; Duft et al., 2003). In Rayleigh’s original approach, the surface of a near-spherical liquid drop is explicitly parametrized by a series of azimuthally symmetric spherical harmonics multiplied by time-dependent coefficients. Under suit- able assumptions, velocity and electric fields are expressed as the gradients of harmonic potentials, which can again be expanded in terms of spherical harmonics multiplied by some radial functions. These harmonic coefficients are then coupled through the kine- matic condition at the free surface, exchanging their amplitudes provided the global conservation of kinematic and potential energies. To alleviate the overabundance of the omitted steps in Rayleigh’s short communication, Hendricks and Schneider (1963) provided a detailed derivation for an inviscid and incompressible liquid drop, albeit still being ambiguous on the truncation of the first and second order terms. Along similar veins, Rayleigh’s theory on the oscillation of electrified liquid drops was later extended to include various effects such as a uniform external electric field (G. I. Taylor, 1964), vis- cous dissipation (Morrison, Leavitt, and Wortman,1981), rigid body rotation (Natarajan and R. A. Brown, 1987), the presence of counterions (Deserno, 2001), permittivity of the drop and the surrounding medium (Shrimpton, 2005) and internal inclusion of a highly charged ion (Oh et al.,2017).
All these work mentioned before specialize on spherical coordinate system with spherical harmonics for drop deformations. For drop shapes that are not perfectly spherical, method of spheroidal analysis was popularized by G. I. Taylor (1964). In his analysis, the shape of liquid body was assumed to be a prolate spheroid parametrized by its major radius and minor radius, which were found by satisfying the normal stress condition only at the pole and the equator but not at every point on the surface. Thinking alone the same lines, K. J. Cheng and Chaddock (1984) and K. J. Cheng and Chaddock (1986)
first developed a prototype of variational method from a consideration of the variation of free energy. Although a spheroidal liquid shape was still assumed in their work, instead of pursuing force balance on the surface of the spheroid they determined the deformation and equilibrium/instability criterion of spheroidal drops and bubbles from the extremization of total free energy of the conductive or dielectric liquid body over an eccentricity parameter subject to a constant-volume constraint. They were able to recover Taylor’s result and predicted that a minimum energy configuration is always possible for bubbles.
The condition which an arbitrarily shaped conductive liquid body must satisfy in order to be in equilibrium requires a coordinate-free approach such as the variational method.
Sujatha et al. (1983) and Chung, Cutler, Feuchtwang, et al. (1984) attempted a varia- tional formalism of the equilibrium configurations of a perfectly conductive fluid kept at a fixed voltage by minimizing the total free energy associated with a volume-conserving liquid body. However, they treated surface charge density as a local variable, i.e. a func- tion of the local geometric quantities described by the first and second derivatives only, rather than a shape functional of the complete surface. Hence their calculations may only be valid for very simple shapes. Later Ljepojevic and Forbes (1995) re-examined the variational problem of an electrified liquid using an Eulerian approach. By carefully tracking the leading order effect of geometric variations on the surface charge density, they managed to recovered the equilibrium condition, which is the familiar pressure bal- ance between capillary and Maxwell stresses. However, due to lack of precise treatments on shape geometry, their first order calculation cannot to be carried out to the second order variation, which is crucial to the stability of charged liquid drop.
It is worth noting that Rayleigh’s method is Eulerian: coordinates and potentials do not vary with deformation of the liquid body. The rational for the Eulerian point of view is challenging because functions defined on a reference domain need to be extended onto the deformed domain where these functions were not meant to exist in the first place. Despite the prevailing Eulerian approach in literature of free surface flow, Joseph (1973), inspired by his seminar work on the parametric domain dependence of eigenvalues of elliptic PDE (Joseph, 1967), drew attention to the conceptual and computational difficulties behind the Eulerian formalism commonly used in the derivation of the higher order water wave theory: expansion of the true velocity potential, which is defined for the wavy water domain, into a series of potential functions, which are defined only under the flat water surface, assumes these potentials can be continued analytically into that part of the wavy domain outside the flat domain. He then presented an alternative derivation by mapping the domain of complicated unknown configuration and the potentials defined within onto a relatively simple domain that is readily described by some curvilinear coordinate system. The method of domain variations, the prototype of which is attributed to Hadamard (1908), was developed based on this concept. Inspired
by the work of Joseph (1973), Feng (1997) revisited Rayleigh’s result of an electrically charged conductive drop by means of the domain variation technique. However his calculation is limited to perfect spherical and cylindrical geometries only.
In the field of modern control theory, the problem of finding Rayleigh charge limit by extremizing the total potential energy falls into the category of shape optimization, i.e. finding the optimal shape which minimizes a certain cost functional (e.g., potential energy of the liquid drop) subject to constraints (in our case electrostatic equation in vacuum and volume conservation of the liquid) where the method of domain variations plays a central role. For example, H. Wang, L. Liu, and D. Liu (2017) reformulated the problem of determining the equilibrium shape of the bubble in an applied electric field as an energy minimization problem, based on which a fixed mesh level-set gradient method was implemented to simulate equilibrium shapes of wall-contacting bubbles in an electric field. For further references and recent development in the topic of shape optimization, we refer to the comprehensive textbooks (Henrot and Pierre, 2005; Sokolowski and Zolesio, 1992). In this chapter, we adhere to the development laid out in the work of Bandle and Wagner (2015) who computed the first and second domain variations for functionals related to second order elliptic boundary value and eigenvalue problems subject to Robin boundary conditions. Finally, regarding the existence and nonexistence of equilibrium shapes of charged liquid drops with or without constraints, we refer the interested readers to a series of mathematically rigorous works (Goldman, Novaga, and Ruffini, 2014; Goldman and Ruffini,2017; Muratov, Novaga, and Ruffini, 2018) where the problem is carefully examined in a setting of functional analysis using Riesz potential and Riesz capacity.
In this chapter we would like to address three issues: first of all, we formulate the total potential energy of an isolated, charged, perfectly conductive, arbitrarily shaped liquid body based on the convective Lagrangian coordinates from continuum mechanics which allow systematic and geometrically precise treatments for arbitrary domain deformations.
Secondly, in contrast to the usual small amplitude deformations normal to a spherical surface typically considered in the literature, we rigorously derive the constrained first and second order volume-conserving shape variations to the potential energy (electro- static and surface energies) when liquid boundary undergoes both normal and tangential deformations. The equilibrium condition and stability criterion are also presented. Most importantly we discover that, for an equilibrium shape (if exits) with nonuniform mean curvature or surface charge distribution, there exist additional contributions to the sec- ond shape variation of the potential arising from the three-way coupling between normal, tangential deformation and mean curvature, which are entirely overlooked in the exist- ing literature due to inadequate treatment on geometric deformation. Lastly, we recover the classical Rayleigh charge limit by applying the shape variations derived earlier to a perfect sphere. In particular we show that when a spherical liquid drop is charged below
the Rayleigh limit, the second shape variation to its total potential energy is a strictly positive quadratic functional and hence implies local stability in the energy landscape.