There are two most striking results in this project:
1. Liquid film motor. We have given the first theory explaining the rotation of a fluid in a thin liquid film in the presence of constant electric fields. This result can have a variety of applications in micro-technologies (such as micro-mixing).
2. The discovery of the vibrational Freedericksz transition in liquid crystals. This result shows that the externally imposed vibrations lead to the `overturning' of the nematic field in liquid crystals, very similar to that due to magnetic field. The possible high-impact applications of that result are related, for example, to the properties of liquid crystal displays in the condition of externally imposed vibrations.
Our main results are:
1.The mathematical model of a rotating electrohydrodynamic flow in a thin suspended liquid film is proposed and studied. The motion is driven by the given difference of potentials in one direction and constant external electrical field E in another direction in the plane of a film. To derive the model we employ the spatial averaging over the normal coordinate to a film that leads to the average Reynolds stress that is proportional to E^3. This stress generates tangential velocity in the vicinity of the edges of a film that, in turn, causes the rotational motion of a liquid. The proposed model is aimed to explain the experimental observations of the liquid film motor. Its mathematical model may stimulate the creation of effective micro-devices for micro-mixing in the nearest future.
2.A mathematical model describing a steady pH-gradient in the solution of ampholytes in water has been studied with the use of analytical, asymptotic, and numerical methods. We show that at the large values of an electric current a concentration distribution takes the form of a piecewise constant function that is drastically different from a classical Gaussian form. The correspondent pH-gradient takes a stepwise form, instead of being a linear function. A discovered anomalous pH-gradient can crucially affect the understanding of an isoelectric focusing process. Our approach allows generalizing the results of this paper to the infinite number of components.
3.The zonal electrophoresis in the channels of complex forms is considered mathematically with the use of computations. We show that for plane S-type rectangular channels stagnation regions can appear that cause the strong variations of the spatial distribution of an admixture. Besides, the shape of an admixture zone is strongly influenced by the effects of electromigration and by a convective mixing. Taking into account the zone spreading caused by electromigration, the influence of vertex points of cannel walls, and convection would explain the results of electrophoretic experiments, which are difficult to understand otherwise. Our computations show the important role of nonlinear interaction of waves and their interactions with the flow boundaries.
4.For nematic liquid crystals we have shown that the oscillations of channel's walls are equivalent to an additional external magnetic field that leads to the discovery of the vibrational Freedericksz transition. The topic of this research is close to the electrophoresis in an anisotropic medium. It also may allow introducing a new type of `phoresis' that can be called `vibrophoresis'.
5.For viscous fluid we have adapted the Vishik-Lyusternik method for the description of vibrating viscous flows. The asymptotic method developed here can be applied to the theory of the electrical double electrical layer (EDL). This method will allow us to study the structure of EDL and even to consider such an interesting problem as vibrational EDL.
Shiryaeva EV, Vladimirov VA, Zhukov MY. 2009. Theory of rotating electrohydrodynamic flow in a liquid film. Phys.Rev.E. 80(041603)(4 ):12-39
Vladimirov VA, Zhukov MY. 2007. Vibrational Freedericksz transition in liquid crystals. Physical Review E. (031706)
Vladimirov VA. 2008. Viscous Flows in a Half Space Caused by Tangential Vibrations on Its Boundary. Studies in Applied Mathematics. 121:337-367
Vladimirov VA, Zhukov MY, Shiryaeva EV. 2009. Modelling of Electromigration and Electroosmisis in a Planar Microchannel. Izvestiya vuzov Severo-Kavkazskii region, Series of Natural Sciences. (Special issue: Topical Problems in Mathematical Hydrodynamics):46-50.
Vladimirov VA. 2012. Magnetohydrodynamic drift equations: from Langmuir circulations to magnetohydrodynamic dynamo? Journal of Fluid Mechanics. 698:51-61.