to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

Electrospraying route to nanotechnology: An overview

Electrohydrodynamic spraying functioning modes - a critical-review

A review of liquid atomization by electrical means

Drops subjected to strong electric fields emit charged jets from their pointed tips. The disintegration of such jets into a spray consisting of charged droplets is common to electrospray ionization mass spectrometry, printing and coating processes, and raindrops in thunderclouds. Currently, there exist conflicting theories and measurements on the size and charge of these small electrospray droplets. We use theory and simulation to show that conductivity can be tuned to yield three scaling regimes for droplet radius and charge, a finding missed by previous studies. The amount of charge that electrospray droplets carry determines whether they are coulombically stable and charged below the Rayleigh limit of stability or are unstable and hence prone to further explosions once they are formed. Previous experiments reported droplet charge values ranging from 10% to in excess of . Simulations unequivocally show that electrospray droplets are coulombically stable at the instant they are created and that there exists a universal scaling law for droplet charge, .

https://www.pnas.org/content/pnas/110/13/4905.full.pdf

Universal scaling laws for the disintegration of electrified drops

The semiphenomenological classification of the modes of electrostatic dispersion of liquids

According to insight on the liquid drop instability as for the surface electric discharge it is shown that the initiation of lightning discharge can be related to instability of negatively charged water drops in an external electric field. In the total intracloud field (quasiconstant and chaotically varied in time) the large charged drops and the watered ice crystals (R > 30 μm) can be unstable and start to emit highly-dispersed (r ≤ 0.01R) heavily charged drops, unstable by the Rayleigh criterium of stability. These drops break down in a period of time of the order of tens of intrinsic oscillation periods ~10−6 s, emitting ~100 more smaller drops. The field strength of intrinsic charges near the drop surface is sufficiently large for effective autoelectron emission (108–109 V/cm). Due to the total electric field near the surface of the droplets the emitted electrons initiate electron avalanches whose characteristic linear sizes are determined by the dimensions of the space region where the coefficient of electron multiplication exceeds unity. The lifetime of one avalanche is ~10−7 s. The same is the time interval between the consistent emission of two electrons by a drop. A transit of the series of consistent avalanches following the same track leads to the development of a plasma frame in the avalanche track. The overlap of the frames of adjacent avalanches in the vicinity of a group of closely located drops gives rise to formation of a plasma region (PR). The PR polarization in an intracloud quasiconstant electric field E0 and its growth along E0 (which is controlled by electric charges arriving into the PR from the surrounding space and originates from the charged drops instability as well as development of the ionization-avalanche processes near the PR tops) enables one to explain the phenomenon of discharge channel branching and propose a physically clear mechanism of collection of negative electric charges from separate drops carried out by lightning. A stepped leader is treated as a result of the ion-sound waves drive in the discharge channel plasma.

The possible physical mechanism of initiation and growth of lightning

The theoretical consideration of physical regularities of electrostatic dispersion of liquids as aerosols

Influence of Disjoining Pressure upon Stability in the Electric Field of a Charged Liquid Film on the Surface of a Hard Core

The phenomenology of the instability of charged and uncharged drops and liquid menisci at the end of a capillary through which liquid is fed in constant and variable external electric fields, both uniform and nonuniform, is considered. Methods of investigating such instability are described. The steady-state deformation of drops in electric fields, the critical conditions under which their charged surface loses stability, and the laws of electrical dispersion of liquids in the final stage of instability are discussed.

A. I. Grigor’ev

Papers

The semiphenomenological classification of the modes of electrostatic dispersion of liquids

The possible physical mechanism of initiation and growth of lightning

The theoretical consideration of physical regularities of electrostatic dispersion of liquids as aerosols

Influence of Disjoining Pressure upon Stability in the Electric Field of a Charged Liquid Film on the Surface of a Hard Core

Capillary instabilities of charged drops and electrical dispersion of liquids