Electrodynamic droplet deformation
The characteristic frequency and magnitude of the deformation is determined by a balance of electrodynamic, hydrodynamic, and capillary stresses acting on the droplet interface.
This phenomenon has been studied extensively both mathematically and experimentally because of the complex fluid dynamics that occur.
Characterization and modulation of electrodynamic droplet deformation is of particular interest for engineering applications because of the growing need to improve the performance of complex industrial processes(e.g. two-phase cooling,[1] crude oil demulsification).
The primary advantage of using oscillatory droplet deformation to improve these engineering processes is that the phenomenon does not require sophisticated machinery or the introduction of heat sources.
[2][3] The injected bubbles/droplets are typically of a lower density than the coolant and thus experience an upward buoyancy force.
They enhance the thermal performance of cooling systems because as they float upwards in heated pipes the coolant is forced to flow around the bubbles/droplets.
This approach modifies only bulk flow settings and does not provide engineers the option of control of directly modulating the mechanisms that govern the heat transfer dynamics.
Inducing oscillations in the bubbles/droplets is a promising approach to improving convective cooling because creates secondary and tertiary flow patterns that could improve heat transfer without introducing significant heat to the system.
Electrodynamic droplet deformation also of particular interest in crude oil processing as a method to improve the separation rate of water and salts from the bulk.
In its unprocessed form, crude oil cannot be used directly in industrial processes because the presence of salts can corrode heat exchangers and distillation equipment.
To avoid fouling due to these impurities it is necessary to first remove the salt, which is concentrated in suspended water droplets.
This can be easily shown by considering gravitational force, buoyancy, and Stokes flow drag.
[4] Taylor’s 1966 solution[5] to internal and external flow of a sphere induced by an electric field was the first to provide an argument that accounted for pressure induced by fluid flow both inside the droplet and in the external fluid field.
Unlike some of his contemporaries, Taylor argued that surface tension and a uniform internal pressure could not balance the spatially varying normal stress on a droplet interface that was resulted from the presence of a steady, uniform electric field.
[6] Taylor confirmed the validity of his solution by comparing it to images from flow visualization studies that observed circulation both inside and outside the droplet interface.
The droplet deformation ratio D is a quantity that expresses the relative extension and shortening of the vertical and horizontal dimensions of a sphere.
Since Torza treats the fluid inside the droplet and outside the droplet as having no net charge, the governing equation for the electric stress sub-problem reduces to Gauss's law with a spatial charge density of zero.
Separation of variables can be used to derive a solution to this equation of the form of a power series multiplied by the cosine of the polar angle taken relative to the direction of the electric field.
It is worth noting that because the electric field is in the form of a phasor, the scalar product and tensor product of electric field with itself, as are present in the Maxwell stress tensor, result in a doubling of the oscillation frequency.
Using this relation between surface pressures in conjunction with geometrical arguments derived by Taylor for small deformations, Torza was able to derive an analytical expression for the deformation ratio as the sum of a steady component and an oscillating component with a frequency that is twice that of the imposed electric field as shown.
The phi term is what Taylor and Torza refer to as a “discriminating function” because its value determines whether the droplet will tend to spend more time in either a prolate or oblate shape.
It is a function of all the material properties and the frequency of oscillation, but is completely independent of time.
The time varying cosine term shows that the droplet does in fact oscillate at twice the frequency of the imposed electric field but is also generally out of phase due to the constant alpha term that arises due to the mathematics.
The other variables are constants that depend on the geometric, electric, and thermodynamic properties of the relevant liquids in addition to the oscillation frequency.
In general, it is apparent that the magnitude of the droplet deformation is constrained by the interfacial tension, represented by gamma.
Although periodic droplet deformation is widely studied for its practical industrial applications, its implementation poses significant safety issues and physical limitations due to the use of electric field.
Research studies using water droplets suspended in silicone oil required root-mean-square values as high as 10^6 V/m .
Even for a small electrode spacing, this type of field requires electric potentials greater than 500V, which is roughly three times wall voltage in the United States.
Practically speaking, this large of an electric field can only be achieved if the electrode spacing is very small (~ O(0.1 mm)) or if a high-voltage amplifier is available.
It is for this reason that the majority of studies of this phenomenon are currently being conducted in research laboratories using small diameter tubes; tubes of this size are in fact present in industrial cooling systems, such as nuclear reactors.
