Equation electrohydrodynamic

Based on the numerical results obtained for the electrohydrodynamics equations the following conclusions can be drawn ... 

Wi transfer function associated with the electrohydrodynamic impedance for species i, see equation (15.29)... 

There are two trends in the existing hydrodynamic theories of free films, which differ in the way the (colloidal) interaction forces are taken into account. We mention first the method of Felderhof, who developed a systematic and consistent electrohydrodynamic theory for a nonviscous liquid. His theory was extended to include viscous behavior by Sche and Fijnaut. In these theories the interaction forces are included in the momentum equations. The other theoretical approach considers hydro-dynamic equations without the interaction forces. The influence of these interactions are considered only in the normal stress boundaiy conditions. 

The other model for the ionic friction concerns the dielectric response of solvent to the solute perturbation. When an ion is fixed in polar solvent, the solvent is polarized according to the electrostatic field from the ion. If the ion is displaced, the solvent polarization is not in equilibrium with a new position of the ion, and the relaxation of the polarization should take place in the solvent. The energy dissipation associated with this relaxation process may be identified as an extra friction. The extra friction, called the dielectric friction, decreases with increasing ionic radius, thereby, with decreasing electrostatic field from the ion. The dielectric friction model developed by Born [66], Fuoss [67], Boyd [68] and Zwanzig [69, 70] has taken a complete theoretical form due to the work by Hubbard and Onsager [71, 72] who proposed a set of continuum electrohydrodynamic equations in which the electrostatic as well as hydrodynamic strains are incorporated. 

Electrohydrodynamic Lithography of Functional Soft Materials for Advanced... introduces the further scaling equation ... 

A general electrohydrodynamic model of a weakly conductive viscous jet accelerated by an external electric field was also derived, by considering inertial, hydrostatic, viscous, electric, and surface tension forces. Nonlinear rheologic constitutive equation for the jet radius was derived,... 

This section deals with the influence of flexoelectricity on electroconvection with planar geometry and the most studied material parameter combination a < 0 and Oa > 0. The analysis makes use of the common nemato-electrohydrodynamic equations, 4s jjj addition the flexopolarization... 

Besides cell thickness d of the nematic layer, which has almost no effect in the case ei = es = 0, reveals a strong influence on EC for finite flexocoefficients. This is demonstrated in the lower panel of Fig. 4.5, where for d = 10 /Ltm the conductive branch is totally absent. Then, as with the conductive regime, one can find a transition from oblique to normal dielectric rolls above a Lifshitz frequency In a recent experiment the oblique dielectric rolls at small w have indeed been observed. The threshold characteristics Uc and qc and the obliqueness angle a. could be well reproduced by a theoretical analysis of the nemato-electrohydrodynamic equations including flexopolarization. ... 

Finally we would like to point out that the magnitude ao of the electrical conductivity in nematics plays an important role. For Fig. 4.5 we chose electrical conductivities in nematics commonly used in experiments. A closer look at the linear nemato-electrohydrodynamic equations (see the Appendix in Krekhov et shows that the thickness d as well as ao appear only through the dimensionless parameter Q oc Td/rq oc aocP. Thus decreasing ao by a factor of 16 is equivalent to a reduction of d by a factor of 4 (compare Fig. 4.5a and Fig. 4.5b). 

Hz, there are normal dielectric rolls and for f < /l oblique ones. The transition to flexodomains is observed at ft 0.1 Hz. A recent theoretical analysis has demonstrated that such a scenario is possible within the nemato-electrohydrodynamic equations including flexopolarization. 

A theoretical investigation of the stability of nematic liquid crystals with homeotropic orientation requires a three-dimensional approach. Helfrich s one-dimensional theory predicts the dependence of the threshold of the instability on the magnitude of Ae, as shown by curve 2 in Fig. 5.8, according to which the electrohydrodynamic instability should be observed when either Ae < 0 (and consequently the bend Frederiks effect reorientation will not take place), or when small Ae > 0. In Helfirich s model the destabilizing torque as dvzjdx is responsible for this instability, which replaces the destabilizing torque a dvzjdx in the equation for the director rotations (5.27). Although the torque is small ( a3 -C o 2 ) it is not compensated for (e.g., when Ae = 0) by anything else apart from the elastic torque. 

FIGURE 6.19. Dependence of the threshold voltage of an electrohydrodynamic instabihty of a cholesteric liquid crystal on cell thickness (hehcal pitch Po = 115 /mi). The solid lines indicate the experimental results. The dashed line shows the calculated values according to an equation similar to (6.38) without allowance for oscillations in the helical pitch [17]. 

This latter equation shows that the longitudinal electric field, while acting on the charges adsorbed at the interface, will induce an inevitable convective flow both exactly at the boundary and in adjacent areas of each phase. If either the strength E of the external field or the surface density F of the adsorbed particles is a function of time, one can analyze the stabihty of the system of the hydrodynamic equations (l)-(2) with an x-component for each phase velocity in the form of Eq. (26). Various kinds of electrohydrodynamic instability result from such an analysis and are described in the literature [28-31]. More details on the problems of electrohydrodynamic instability will be given below using particular model systems as examples. 

The injection of charge carriers leads to bulk liquid motion. This effect is also known as electrohydrodynamic motion (EHD). From measurements of the liquid velocity and from calculations of the flow by means of the Navier-Stokes equations it was found that also in the presence of liquid flow. Equation 71 for the SCL currents remains valid (Takashima et al., 1988). The mobility obtained from the slope, however, is the hydrodynamic mobility. For the electrode arrangement — razor blade/plane electrode — the injection current depends on the applied voltage as. 

An advantage of magnetic fields over electric fields for controlling alignment is that complications due to electrical conduction or electrohydrodynamic effects are not present. Competition between aligning fields has been used to obtain direct measurements of susceptibility anisotropies. The basis of the method can be understood from Eq. (5). A similar equation can be written for the free energy density of a liquid crystal in an electric field, such that ... 

This equation is known as the Poisson equation for electrostatics. The knowledge of charge density distribution pg) is required for obtaining potential distribution. If the charge density is zero, we get Laplace equation. These equations will be used in the electrohydrodynamic analysis of microfluidics. 

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