Electric jets and Taylor cones. The most versatile
methods of controlling breakup, though, are achieved by
applying external forcing, either using an external flow or
by applying an external electric field or even combinations
thereof [356,357]. Either type of forcing can be used to mould
the fluid into an extremely fine jet, thus beating constraints
imposed by the nozzle size. In addition, the rapidly developing
microfluidic technology [358–360] has developed many ways
of controlling the formation of drops and bubbles in confined
geometries.
The technique of using electric fields to make extremely
fine sprays has a long history [361] and many important
applications [362], for example in biotechnology [346]. The
tendency of electric fields to ‘focus’ a fluid into very pointy
objects is epitomized by the static ‘Taylor cone’ solution, for
which both surface tension and electric forces become infinite
as the inverse distance r from the tip [363]. This means
the electric field has to diverge as r−1/2. Almost all fluids
in question have some, if small, conductivity [246], so the
appropriate boundary condition for an equilibrium situation is
that of a conductor, i.e. the tangential component of the electric
field vanishes. Using the solution for the electric field of a cone
with the proper divergence r−1/2 [364], this leads directly to
the condition
From the first zero of the Legendre function of degree 1/2 one
finds the famous result θ = 49.29◦ for the Taylor cone angle
in the case of a conducting fluid.
Figure 64 shows such a Taylor cone on a drop at the end
of a capillary, held in a strong electric field. Note the very fine
jet emerging from the apex of the cone (the so-called ‘cone-
jet’ mode [246, 366]), which is not part of Taylor’s analysis,
but which is our main interest below. A similar phenomenon
P1/2(cos(π − θ )) = 0. (252)
Electric jets and Taylor cones. The most versatilemethods of controlling breakup, though, are achieved byapplying external forcing, either using an external flow orby applying an external electric field or even combinationsthereof [356,357]. Either type of forcing can be used to mouldthe fluid into an extremely fine jet, thus beating constraintsimposed by the nozzle size. In addition, the rapidly developingmicrofluidic technology [358–360] has developed many waysof controlling the formation of drops and bubbles in confinedgeometries.The technique of using electric fields to make extremelyfine sprays has a long history [361] and many importantapplications [362], for example in biotechnology [346]. Thetendency of electric fields to ‘focus’ a fluid into very pointyobjects is epitomized by the static ‘Taylor cone’ solution, forwhich both surface tension and electric forces become infiniteas the inverse distance r from the tip [363]. This meansthe electric field has to diverge as r−1/2. Almost all fluidsin question have some, if small, conductivity [246], so theappropriate boundary condition for an equilibrium situation isthat of a conductor, i.e. the tangential component of the electricfield vanishes. Using the solution for the electric field of a conewith the proper divergence r−1/2 [364], this leads directly tothe conditionFrom the first zero of the Legendre function of degree 1/2 onefinds the famous result θ = 49.29◦ for the Taylor cone anglein the case of a conducting fluid.Figure 64 shows such a Taylor cone on a drop at the endof a capillary, held in a strong electric field. Note the very finejet emerging from the apex of the cone (the so-called ‘cone-jet’ mode [246, 366]), which is not part of Taylor’s analysis,but which is our main interest below. A similar phenomenonP1/2(cos(π − θ )) = 0. (252)
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